Spacetime

In physics, spacetime is any mathematical model which fuses the three dimensions of space and the one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why different observers perceive differently where and when events occur.

Until the 20th century, it was assumed that the three-dimensional geometry of the universe (its spatial expression in terms of coordinates, distances, and directions) was independent of one-dimensional time. The physicist Albert Einstein helped develop the idea of spacetime as part of his theory of relativity. Prior to his pioneering work, scientists had two separate theories to explain physical phenomena: Isaac Newton's laws of physics described the motion of massive objects, while James Clerk Maxwell's electromagnetic models explained the properties of light. However, in 1905, Einstein based a work on special relativity on two postulates:

The laws of physics are invariant (i.e., identical) in all inertial systems (i.e., non-accelerating frames of reference)
The speed of light in a vacuum is the same for all observers, regardless of the motion of the light source.

The logical consequence of taking these postulates together is the inseparable joining of the four dimensions—hitherto assumed as independent—of space and time. Many counterintuitive consequences emerge: in addition to being independent of the motion of the light source, the speed of light is constant regardless of the frame of reference in which it is measured; the distances and even temporal ordering of pairs of events change when measured in different inertial frames of reference (this is the relativity of simultaneity); and the linear additivity of velocities no longer holds true.

Einstein framed his theory in terms of kinematics (the study of moving bodies). His theory was an advance over Lorentz's 1904 theory of electromagnetic phenomena and Poincaré's electrodynamic theory. Although these theories included equations identical to those that Einstein introduced (i.e., the Lorentz transformation), they were essentially ad hoc models proposed to explain the results of various experiments—including the famous Michelson–Morley interferometer experiment—that were extremely difficult to fit into existing paradigms.

In 1908, Hermann Minkowski—once one of the math professors of a young Einstein in Zürich—presented a geometric interpretation of special relativity that fused time and the three spatial dimensions of space into a single four-dimensional continuum now known as Minkowski space. A key feature of this interpretation is the formal definition of the spacetime interval. Although measurements of distance and time between events differ for measurements made in different reference frames, the spacetime interval is independent of the inertial frame of reference in which they are recorded.^[1]

Minkowski's geometric interpretation of relativity was to prove vital to Einstein's development of his 1915 general theory of relativity, wherein he showed how mass and energy curve flat spacetime into a pseudo-Riemannian manifold.

Introduction

Definitions

Non-relativistic classical mechanics treats time as a universal quantity of measurement which is uniform throughout space, and separate from space. Classical mechanics assumes that time has a constant rate of passage, independent of the observer's state of motion, or anything external.^[2] Furthermore, it assumes that space is Euclidean; it assumes that space follows the geometry of common sense.^[3]

In the context of special relativity, time cannot be separated from the three dimensions of space, because the observed rate at which time passes for an object depends on the object's velocity relative to the observer. General relativity also provides an explanation of how gravitational fields can slow the passage of time for an object as seen by an observer outside the field.

In ordinary space, a position is specified by three numbers, known as dimensions. In the Cartesian coordinate system, these are called x, y, and z. A position in spacetime is called an event, and requires four numbers to be specified: the three-dimensional location in space, plus the position in time (Fig. 1). An event is represented by a set of coordinates x, y, z and t. Space time is thus four dimensional. Mathematical events have zero duration and represent a single point in spacetime.

The path of a particle through spacetime can be considered to be a succession of events. The series of events can be linked together to form a line which represents a particle's progress through spacetime. That line is called the particle's world line.^[4]^: 105

Mathematically, spacetime is a manifold, which is to say, it appears locally "flat" near each point in the same way that, at small enough scales, a globe appears flat.^[5] An extremely large scale factor, $c$ (conventionally called the speed-of-light) relates distances measured in space with distances measured in time. The magnitude of this scale factor (nearly 300,000 kilometres or 190,000 miles in space being equivalent to one second in time), along with the fact that spacetime is a manifold, implies that at ordinary, non-relativistic speeds and at ordinary, human-scale distances, there is little that humans might observe which is noticeably different from what they might observe if the world were Euclidean. It was only with the advent of sensitive scientific measurements in the mid-1800s, such as the Fizeau experiment and the Michelson–Morley experiment, that puzzling discrepancies began to be noted between observation versus predictions based on the implicit assumption of Euclidean space.^[6]

In special relativity, an observer will, in most cases, mean a frame of reference from which a set of objects or events is being measured. This usage differs significantly from the ordinary English meaning of the term. Reference frames are inherently nonlocal constructs, and according to this usage of the term, it does not make sense to speak of an observer as having a location. In Fig. 1-1, imagine that the frame under consideration is equipped with a dense lattice of clocks, synchronized within this reference frame, that extends indefinitely throughout the three dimensions of space. Any specific location within the lattice is not important. The latticework of clocks is used to determine the time and position of events taking place within the whole frame. The term observer refers to the entire ensemble of clocks associated with one inertial frame of reference.^[7]^: 17–22 In this idealized case, every point in space has a clock associated with it, and thus the clocks register each event instantly, with no time delay between an event and its recording. A real observer, however, will see a delay between the emission of a signal and its detection due to the speed of light. To synchronize the clocks, in the data reduction following an experiment, the time when a signal is received will be corrected to reflect its actual time were it to have been recorded by an idealized lattice of clocks.

In many books on special relativity, especially older ones, the word "observer" is used in the more ordinary sense of the word. It is usually clear from context which meaning has been adopted.

Physicists distinguish between what one measures or observes (after one has factored out signal propagation delays), versus what one visually sees without such corrections. Failure to understand the difference between what one measures/observes versus what one sees is the source of much error among beginning students of relativity.^[8]

History

Figure 1-2. Michelson and Morley expected that motion through the aether would cause a differential phase shift between light traversing the two arms of their apparatus. The most logical explanation of their negative result, aether dragging, was in conflict with the observation of stellar aberration.

By the mid-1800s, various experiments such as the observation of the Arago spot and differential measurements of the speed of light in air versus water were considered to have proven the wave nature of light as opposed to a corpuscular theory.^[9] Propagation of waves was then assumed to require the existence of a waving medium; in the case of light waves, this was considered to be a hypothetical luminiferous aether.^{[note 1]} However, the various attempts to establish the properties of this hypothetical medium yielded contradictory results. For example, the Fizeau experiment of 1851 demonstrated that the speed of light in flowing water was less than the sum of the speed of light in air plus the speed of the water by an amount dependent on the water's index of refraction. Among other issues, the dependence of the partial aether-dragging implied by this experiment on the index of refraction (which is dependent on wavelength) led to the unpalatable conclusion that aether simultaneously flows at different speeds for different colors of light.^[10] The famous Michelson–Morley experiment of 1887 (Fig. 1-2) showed no differential influence of Earth's motions through the hypothetical aether on the speed of light, and the most likely explanation, complete aether dragging, was in conflict with the observation of stellar aberration.^[6]

George Francis FitzGerald in 1889, and Hendrik Lorentz in 1892, independently proposed that material bodies traveling through the fixed aether were physically affected by their passage, contracting in the direction of motion by an amount that was exactly what was necessary to explain the negative results of the Michelson–Morley experiment. (No length changes occur in directions transverse to the direction of motion.)

By 1904, Lorentz had expanded his theory such that he had arrived at equations formally identical with those that Einstein was to derive later (i.e. the Lorentz transform), but with a fundamentally different interpretation. As a theory of dynamics (the study of forces and torques and their effect on motion), his theory assumed actual physical deformations of the physical constituents of matter.^[11]^{: 163–174} Lorentz's equations predicted a quantity that he called local time, with which he could explain the aberration of light, the Fizeau experiment and other phenomena. However, Lorentz considered local time to be only an auxiliary mathematical tool, a trick as it were, to simplify the transformation from one system into another.

Other physicists and mathematicians at the turn of the century came close to arriving at what is currently known as spacetime. Einstein himself noted, that with so many people unraveling separate pieces of the puzzle, "the special theory of relativity, if we regard its development in retrospect, was ripe for discovery in 1905."^[12]

Hendrik Lorentz

Henri Poincaré

Albert Einstein

Hermann Minkowski

Figure 1-3.

An important example is Henri Poincaré,^[13]^[14]^{: 73–80, 93–95} who in 1898 argued that the simultaneity of two events is a matter of convention.^[15]^{[note 2]} In 1900, he recognized that Lorentz's "local time" is actually what is indicated by moving clocks by applying an explicitly operational definition of clock synchronization assuming constant light speed.^{[note 3]} In 1900 and 1904, he suggested the inherent undetectability of the aether by emphasizing the validity of what he called the principle of relativity, and in 1905/1906^[16] he mathematically perfected Lorentz's theory of electrons in order to bring it into accordance with the postulate of relativity. While discussing various hypotheses on Lorentz invariant gravitation, he introduced the innovative concept of a 4-dimensional spacetime by defining various four vectors, namely four-position, four-velocity, and four-force.^[17]^[18] He did not pursue the 4-dimensional formalism in subsequent papers, however, stating that this line of research seemed to "entail great pain for limited profit", ultimately concluding "that three-dimensional language seems the best suited to the description of our world".^[18] Furthermore, even as late as 1909, Poincaré continued to believe in the dynamical interpretation of the Lorentz transform.^[11]^{: 163–174} For these and other reasons, most historians of science argue that Poincaré did not invent what is now called special relativity.^[14]^[11]

In 1905, Einstein introduced special relativity (even though without using the techniques of the spacetime formalism) in its modern understanding as a theory of space and time.^[14]^[11] While his results are mathematically equivalent to those of Lorentz and Poincaré, Einstein showed that the Lorentz transformations are not the result of interactions between matter and aether, but rather concern the nature of space and time itself. He obtained all of his results by recognizing that the entire theory can be built upon two postulates: The principle of relativity and the principle of the constancy of light speed.

Einstein performed his analysis in terms of kinematics (the study of moving bodies without reference to forces) rather than dynamics. His work introducing the subject was filled with vivid imagery involving the exchange of light signals between clocks in motion, careful measurements of the lengths of moving rods, and other such examples.^[19]^{[note 4]}

In addition, Einstein in 1905 superseded previous attempts of an electromagnetic mass–energy relation by introducing the general equivalence of mass and energy, which was instrumental for his subsequent formulation of the equivalence principle in 1907, which declares the equivalence of inertial and gravitational mass. By using the mass–energy equivalence, Einstein showed, in addition, that the gravitational mass of a body is proportional to its energy content, which was one of the early results in developing general relativity. While it would appear that he did not at first think geometrically about spacetime,^[21]^: 219 in the further development of general relativity Einstein fully incorporated the spacetime formalism.

When Einstein published in 1905, another of his competitors, his former mathematics professor Hermann Minkowski, had also arrived at most of the basic elements of special relativity. Max Born recounted a meeting he had made with Minkowski, seeking to be Minkowski's student/collaborator:^[22]

I went to Cologne, met Minkowski and heard his celebrated lecture 'Space and Time' delivered on 2 September 1908. [...] He told me later that it came to him as a great shock when Einstein published his paper in which the equivalence of the different local times of observers moving relative to each other was pronounced; for he had reached the same conclusions independently but did not publish them because he wished first to work out the mathematical structure in all its splendor. He never made a priority claim and always gave Einstein his full share in the great discovery.

Minkowski had been concerned with the state of electrodynamics after Michelson's disruptive experiments at least since the summer of 1905, when Minkowski and David Hilbert led an advanced seminar attended by notable physicists of the time to study the papers of Lorentz, Poincaré et al. However, it is not at all clear when Minkowski began to formulate the geometric formulation of special relativity that was to bear his name, or to which extent he was influenced by Poincaré's four-dimensional interpretation of the Lorentz transformation. Nor is it clear if he ever fully appreciated Einstein's critical contribution to the understanding of the Lorentz transformations, thinking of Einstein's work as being an extension of Lorentz's work.^[23]

On 5 November 1907 (a little more than a year before his death), Minkowski introduced his geometric interpretation of spacetime in a lecture to the Göttingen Mathematical society with the title, The Relativity Principle (Das Relativitätsprinzip).^{[note 5]} On 21 September 1908, Minkowski presented his famous talk, Space and Time (Raum und Zeit),^[24] to the German Society of Scientists and Physicians. The opening words of Space and Time include Minkowski's famous statement that "Henceforth, space for itself, and time for itself shall completely reduce to a mere shadow, and only some sort of union of the two shall preserve independence." Space and Time included the first public presentation of spacetime diagrams (Fig. 1-4), and included a remarkable demonstration that the concept of the invariant interval (discussed below), along with the empirical observation that the speed of light is finite, allows derivation of the entirety of special relativity.^{[note 6]}

The spacetime concept and the Lorentz group are closely connected to certain types of sphere, hyperbolic, or conformal geometries and their transformation groups already developed in the 19th century, in which invariant intervals analogous to the spacetime interval are used.^{[note 7]}

Einstein, for his part, was initially dismissive of Minkowski's geometric interpretation of special relativity, regarding it as überflüssige Gelehrsamkeit (superfluous learnedness). However, in order to complete his search for general relativity that started in 1907, the geometric interpretation of relativity proved to be vital, and in 1916, Einstein fully acknowledged his indebtedness to Minkowski, whose interpretation greatly facilitated the transition to general relativity.^[11]^{: 151–152} Since there are other types of spacetime, such as the curved spacetime of general relativity, the spacetime of special relativity is today known as Minkowski spacetime.

Spacetime in special relativity

Spacetime interval

In three dimensions, the distance $\Delta {d}$ between two points can be defined using the Pythagorean theorem:

(\Delta {d})^{2}=(\Delta {x})^{2}+(\Delta {y})^{2}+(\Delta {z})^{2}

Although two viewers may measure the x, y, and z position of the two points using different coordinate systems, the distance between the points will be the same for both (assuming that they are measuring using the same units). The distance is "invariant".

In special relativity, however, the distance between two points is no longer the same if measured by two different observers when one of the observers is moving, because of Lorentz contraction. The situation is even more complicated if the two points are separated in time as well as in space. For example, if one observer sees two events occur at the same place, but at different times, a person moving with respect to the first observer will see the two events occurring at different places, because (from their point of view) they are stationary, and the position of the event is receding or approaching. Thus, a different measure must be used to measure the effective "distance" between two events.

In four-dimensional spacetime, the analog to distance is the interval. Although time comes in as a fourth dimension, it is treated differently than the spatial dimensions. Minkowski space hence differs in important respects from four-dimensional Euclidean space. The fundamental reason for merging space and time into spacetime is that space and time are separately not invariant, which is to say that, under the proper conditions, different observers will disagree on the length of time between two events (because of time dilation) or the distance between the two events (because of length contraction). But special relativity provides a new invariant, called the spacetime interval, which combines distances in space and in time. All observers who measure the time and distance between any two events will end up computing the same spacetime interval. Suppose an observer measures two events as being separated in time by $\Delta t$ and a spatial distance $\Delta x.$ Then the spacetime interval $(\Delta {s})^{2}$ between the two events that are separated by a distance $\Delta {x}$ in space and by $\Delta {ct}=c\Delta t$ in the $ct$ -coordinate is:

(\Delta s)^{2}=(\Delta ct)^{2}-(\Delta x)^{2},

or for three space dimensions,

(\Delta s)^{2}=(\Delta ct)^{2}-(\Delta x)^{2}-(\Delta y)^{2}-(\Delta z)^{2}.

^[28]

The constant $c,$ the speed of light, converts time units (like seconds) into space units (like meters). Seconds times meters/second = meters.

Although for brevity, one frequently sees interval expressions expressed without deltas, including in most of the following discussion, it should be understood that in general, $x$ means $\Delta {x}$ , etc. We are always concerned with differences of spatial or temporal coordinate values belonging to two events, and since there is no preferred origin, single coordinate values have no essential meaning.

The equation above is similar to the Pythagorean theorem, except with a minus sign between the $(ct)^{2}$ and the $x^{2}$ terms. The spacetime interval is the quantity $s^{2},$ not $s$ itself. The reason is that unlike distances in Euclidean geometry, intervals in Minkowski spacetime can be negative. Rather than deal with square roots of negative numbers, physicists customarily regard $s^{2}$ as a distinct symbol in itself, rather than the square of something.^[21]^: 217

Because of the minus sign, the spacetime interval between two distinct events can be zero. If $s^{2}$ is positive, the spacetime interval is timelike, meaning that two events are separated by more time than space. If $s^{2}$ is negative, the spacetime interval is spacelike, meaning that two events are separated by more space than time. Spacetime intervals are zero when $x=\pm ct.$ In other words, the spacetime interval between two events on the world line of something moving at the speed of light is zero. Such an interval is termed lightlike or null. A photon arriving in our eye from a distant star will not have aged, despite having (from our perspective) spent years in its passage.

A spacetime diagram is typically drawn with only a single space and a single time coordinate. Fig. 2-1 presents a spacetime diagram illustrating the world lines (i.e. paths in spacetime) of two photons, A and B, originating from the same event and going in opposite directions. In addition, C illustrates the world line of a slower-than-light-speed object. The vertical time coordinate is scaled by $c$ so that it has the same units (meters) as the horizontal space coordinate. Since photons travel at the speed of light, their world lines have a slope of ±1. In other words, every meter that a photon travels to the left or right requires approximately 3.3 nanoseconds of time.

There are two sign conventions in use in the relativity literature:

s^{2}=(ct)^{2}-x^{2}-y^{2}-z^{2}

and

s^{2}=-(ct)^{2}+x^{2}+y^{2}+z^{2}

These sign conventions are associated with the metric signatures (+−−−) and (−+++). A minor variation is to place the time coordinate last rather than first. Both conventions are widely used within the field of study.

Reference frames

To gain insight in how spacetime coordinates measured by observers in different reference frames compare with each other, it is useful to work with a simplified setup with frames in a standard configuration. With care, this allows simplification of the math with no loss of generality in the conclusions that are reached. In Fig. 2-2, two Galilean reference frames (i.e. conventional 3-space frames) are displayed in relative motion. Frame S belongs to a first observer O, and frame S′ (pronounced "S prime") belongs to a second observer O′.

The x, y, z axes of frame S are oriented parallel to the respective primed axes of frame S′.
Frame S′ moves in the x-direction of frame S with a constant velocity v as measured in frame S.
The origins of frames S and S′ are coincident when time t = 0 for frame S and t′ = 0 for frame S′.^[4]^: 107

Fig. 2-3a redraws Fig. 2-2 in a different orientation. Fig. 2-3b illustrates a spacetime diagram from the viewpoint of observer O. Since S and S′ are in standard configuration, their origins coincide at times t = 0 in frame S and t′ = 0 in frame S′. The ct′ axis passes through the events in frame S′ which have x′ = 0. But the points with x′ = 0 are moving in the x-direction of frame S with velocity v, so that they are not coincident with the ct axis at any time other than zero. Therefore, the ct′ axis is tilted with respect to the ct axis by an angle θ given by

\tan(\theta )=v/c.

The x′ axis is also tilted with respect to the x axis. To determine the angle of this tilt, we recall that the slope of the world line of a light pulse is always ±1. Fig. 2-3c presents a spacetime diagram from the viewpoint of observer O′. Event P represents the emission of a light pulse at x′ = 0, ct′ = −a. The pulse is reflected from a mirror situated a distance a from the light source (event Q), and returns to the light source at x′ = 0, ct′ = a (event R).

The same events P, Q, R are plotted in Fig. 2-3b in the frame of observer O. The light paths have slopes = 1 and −1, so that △PQR forms a right triangle with PQ and QR both at 45 degrees to the x and ct axes. Since OP = OQ = OR, the angle between x′ and x must also be θ.^[4]^{: 113–118}

While the rest frame has space and time axes that meet at right angles, the moving frame is drawn with axes that meet at an acute angle. The frames are actually equivalent. The asymmetry is due to unavoidable distortions in how spacetime coordinates can map onto a Cartesian plane, and should be considered no stranger than the manner in which, on a Mercator projection of the Earth, the relative sizes of land masses near the poles (Greenland and Antarctica) are highly exaggerated relative to land masses near the Equator.

Light cone

In Fig. 2–4, event O is at the origin of a spacetime diagram, and the two diagonal lines represent all events that have zero spacetime interval with respect to the origin event. These two lines form what is called the light cone of the event O, since adding a second spatial dimension (Fig. 2-5) makes the appearance that of two right circular cones meeting with their apices at O. One cone extends into the future (t>0), the other into the past (t<0).

A light (double) cone divides spacetime into separate regions with respect to its apex. The interior of the future light cone consists of all events that are separated from the apex by more time (temporal distance) than necessary to cross their spatial distance at lightspeed; these events comprise the timelike future of the event O. Likewise, the timelike past comprises the interior events of the past light cone. So in timelike intervals Δct is greater than Δx, making timelike intervals positive. The region exterior to the light cone consists of events that are separated from the event O by more space than can be crossed at lightspeed in the given time. These events comprise the so-called spacelike region of the event O, denoted "Elsewhere" in Fig. 2-4. Events on the light cone itself are said to be lightlike (or null separated) from O. Because of the invariance of the spacetime interval, all observers will assign the same light cone to any given event, and thus will agree on this division of spacetime.^[21]^: 220

The light cone has an essential role within the concept of causality. It is possible for a not-faster-than-light-speed signal to travel from the position and time of O to the position and time of D (Fig. 2-4). It is hence possible for event O to have a causal influence on event D. The future light cone contains all the events that could be causally influenced by O. Likewise, it is possible for a not-faster-than-light-speed signal to travel from the position and time of A, to the position and time of O. The past light cone contains all the events that could have a causal influence on O. In contrast, assuming that signals cannot travel faster than the speed of light, any event, like e.g. B or C, in the spacelike region (Elsewhere), cannot either affect event O, nor can they be affected by event O employing such signalling. Under this assumption any causal relationship between event O and any events in the spacelike region of a light cone is excluded.^[29]

Relativity of simultaneity

All observers will agree that for any given event, an event within the given event's future light cone occurs after the given event. Likewise, for any given event, an event within the given event's past light cone occurs before the given event. The before–after relationship observed for timelike-separated events remains unchanged no matter what the reference frame of the observer, i.e. no matter how the observer may be moving. The situation is quite different for spacelike-separated events. Fig. 2-4 was drawn from the reference frame of an observer moving at v = 0. From this reference frame, event C is observed to occur after event O, and event B is observed to occur before event O. From a different reference frame, the orderings of these non-causally-related events can be reversed. In particular, one notes that if two events are simultaneous in a particular reference frame, they are necessarily separated by a spacelike interval and thus are noncausally related. The observation that simultaneity is not absolute, but depends on the observer's reference frame, is termed the relativity of simultaneity.^[30]

Fig. 2-6 illustrates the use of spacetime diagrams in the analysis of the relativity of simultaneity. The events in spacetime are invariant, but the coordinate frames transform as discussed above for Fig. 2-3. The three events (A, B, C) are simultaneous from the reference frame of an observer moving at v = 0. From the reference frame of an observer moving at v = 0.3c, the events appear to occur in the order C, B, A. From the reference frame of an observer moving at v = −0.5c, the events appear to occur in the order A, B, C. The white line represents a plane of simultaneity being moved from the past of the observer to the future of the observer, highlighting events residing on it. The gray area is the light cone of the observer, which remains invariant.

A spacelike spacetime interval gives the same distance that an observer would measure if the events being measured were simultaneous to the observer. A spacelike spacetime interval hence provides a measure of proper distance, i.e. the true distance = ${\sqrt {-s^{2}}}.$ Likewise, a timelike spacetime interval gives the same measure of time as would be presented by the cumulative ticking of a clock that moves along a given world line. A timelike spacetime interval hence provides a measure of the proper time = ${\sqrt {s^{2}}}.$ ^[21]^{: 220–221}

Invariant hyperbola

In Euclidean space (having spatial dimensions only), the set of points equidistant (using the Euclidean metric) from some point form a circle (in two dimensions) or a sphere (in three dimensions). In (1+1)-dimensional Minkowski spacetime (having one temporal and one spatial dimension), the points at some constant spacetime interval away from the origin (using the Minkowski metric) form curves given by the two equations

(ct)^{2}-x^{2}=\pm s^{2},

with $s^{2}$ some positive real constant. These equations describe two families of hyperbolae in an x–ct spacetime diagram, which are termed invariant hyperbolae.

In Fig. 2-7a, each magenta hyperbola connects all events having some fixed spacelike separation from the origin, while the green hyperbolae connect events of equal timelike separation.

The magenta hyperbolae, which cross the x axis, are timelike curves, which is to say that these hyperbolae represent actual paths that can be traversed by (constantly accelerating) particles in spacetime: Between any two events on one hyperbola a causality relation is possible, because the inverse of the slope—representing the necessary speed—for all secants is less than $c$ . On the other hand, the green hyperbolae, which cross the ct axis, are spacelike curves because all intervals along these hyperbolae are spacelike intervals: No causality is possible between any two points on one of these hyperbolae, because all secants represent speeds larger than $c$ .

Fig. 2-7b reflects the situation in (1+2)-dimensional Minkowski spacetime (one temporal and two spatial dimensions) with the corresponding hyperboloids. The invariant hyperbolae displaced by spacelike intervals from the origin generate hyperboloids of one sheet, while the invariant hyperbolae displaced by timelike intervals from the origin generate hyperboloids of two sheets.

The (1+2)-dimensional boundary between space- and timelike hyperboloids, established by the events forming a zero spacetime interval to the origin, is made up by degenerating the hyperboloids to the light cone. In (1+1)-dimensions the hyperbolae degenerate to the two grey 45°-lines depicted in Fig. 2-7a.

Time dilation and length contraction

Fig. 2-8 illustrates the invariant hyperbola for all events that can be reached from the origin in a proper time of 5 meters (approximately 1.67×10⁻⁸ s). Different world lines represent clocks moving at different speeds. A clock that is stationary with respect to the observer has a world line that is vertical, and the elapsed time measured by the observer is the same as the proper time. For a clock traveling at 0.3 c, the elapsed time measured by the observer is 5.24 meters (1.75×10⁻⁸ s), while for a clock traveling at 0.7 c, the elapsed time measured by the observer is 7.00 meters (2.34×10⁻⁸ s). This illustrates the phenomenon known as time dilation. Clocks that travel faster take longer (in the observer frame) to tick out the same amount of proper time, and they travel further along the x–axis within that proper time than they would have without time dilation.^[21]^{: 220–221} The measurement of time dilation by two observers in different inertial reference frames is mutual. If observer O measures the clocks of observer O′ as running slower in his frame, observer O′ in turn will measure the clocks of observer O as running slower.

Length contraction, like time dilation, is a manifestation of the relativity of simultaneity. Measurement of length requires measurement of the spacetime interval between two events that are simultaneous in one's frame of reference. But events that are simultaneous in one frame of reference are, in general, not simultaneous in other frames of reference.

Fig. 2-9 illustrates the motions of a 1 m rod that is traveling at 0.5 c along the x axis. The edges of the blue band represent the world lines of the rod's two endpoints. The invariant hyperbola illustrates events separated from the origin by a spacelike interval of 1 m. The endpoints O and B measured when t′ = 0 are simultaneous events in the S′ frame. But to an observer in frame S, events O and B are not simultaneous. To measure length, the observer in frame S measures the endpoints of the rod as projected onto the x-axis along their world lines. The projection of the rod's world sheet onto the x axis yields the foreshortened length OC.^[4]^: 125

(not illustrated) Drawing a vertical line through A so that it intersects the x′ axis demonstrates that, even as OB is foreshortened from the point of view of observer O, OA is likewise foreshortened from the point of view of observer O′. In the same way that each observer measures the other's clocks as running slow, each observer measures the other's rulers as being contracted.

In regards to mutual length contraction, Fig. 2-9 illustrates that the primed and unprimed frames are mutually rotated by a hyperbolic angle (analogous to ordinary angles in Euclidean geometry).^{[note 8]} Because of this rotation, the projection of a primed meter-stick onto the unprimed x-axis is foreshortened, while the projection of an unprimed meter-stick onto the primed x′-axis is likewise foreshortened.

Mutual time dilation and the twin paradox

Mutual time dilation

Mutual time dilation and length contraction tend to strike beginners as inherently self-contradictory concepts. If an observer in frame S measures a clock, at rest in frame S', as running slower than his', while S' is moving at speed v in S, then the principle of relativity requires that an observer in frame S' likewise measures a clock in frame S, moving at speed −v in S', as running slower than hers. How two clocks can run both slower than the other, is an important question that "goes to the heart of understanding special relativity."^[21]^: 198

This apparent contradiction stems from not correctly taking into account the different settings of the necessary, related measurements. These settings allow for a consistent explanation of the only apparent contradiction. It is not about the abstract ticking of two identical clocks, but about how to measure in one frame the temporal distance of two ticks of a moving clock. It turns out that in mutually observing the duration between ticks of clocks, each moving in the respective frame, different sets of clocks must be involved. In order to measure in frame S the tick duration of a moving clock W′ (at rest in S′), one uses two additional, synchronized clocks W₁ and W₂ at rest in two arbitrarily fixed points in S with the spatial distance d.

Two events can be defined by the condition "two clocks are simultaneously at one place", i.e., when W′ passes each W₁ and W₂. For both events the two readings of the collocated clocks are recorded. The difference of the two readings of W₁ and W₂ is the temporal distance of the two events in S, and their spatial distance is d. The difference of the two readings of W′ is the temporal distance of the two events in S′. In S′ these events are only separated in time, they happen at the same place in S′. Because of the invariance of the spacetime interval spanned by these two events, and the nonzero spatial separation d in S, the temporal distance in S′ must be smaller than the one in S: the smaller temporal distance between the two events, resulting from the readings of the moving clock W′, belongs to the slower running clock W′.

Conversely, for judging in frame S′ the temporal distance of two events on a moving clock W (at rest in S), one needs two clocks at rest in S′.

In this comparison the clock W is moving by with velocity −v. Recording again the four readings for the events, defined by "two clocks simultaneously at one place", results in the analogous temporal distances of the two events, now temporally and spatially separated in S′, and only temporally separated but collocated in S. To keep the spacetime interval invariant, the temporal distance in S must be smaller than in S′, because of the spatial separation of the events in S′: now clock W is observed to run slower.

The necessary recordings for the two judgements, with "one moving clock" and "two clocks at rest" in respectively S or S′, involves two different sets, each with three clocks. Since there are different sets of clocks involved in the measurements, there is no inherent necessity that the measurements be reciprocally "consistent" such that, if one observer measures the moving clock to be slow, the other observer measures the one's clock to be fast.^[21]^{: 198–199}

Figure 2-10. Mutual time dilation

Fig. 2-10 illustrates the previous discussion of mutual time dilation with Minkowski diagrams. The upper picture reflects the measurements as seen from frame S "at rest" with unprimed, rectangular axes, and frame S′ "moving with v > 0", coordinatized by primed, oblique axes, slanted to the right; the lower picture shows frame S′ "at rest" with primed, rectangular coordinates, and frame S "moving with −v < 0", with unprimed, oblique axes, slanted to the left.

Each line drawn parallel to a spatial axis (x, x′) represents a line of simultaneity. All events on such a line have the same time value (ct, ct′). Likewise, each line drawn parallel to a temporal axis (ct, ct′) represents a line of equal spatial coordinate values (x, x′).

One may designate in both pictures the origin O (= O′) as the event, where the respective "moving clock" is collocated with the "first clock at rest" in both comparisons. Obviously, for this event the readings on both clocks in both comparisons are zero. As a consequence, the worldlines of the moving clocks are the slanted to the right ct′-axis (upper pictures, clock W′) and the slanted to the left ct-axes (lower pictures, clock W). The worldlines of W₁ and W′₁ are the corresponding vertical time axes (ct in the upper pictures, and ct′ in the lower pictures).

In the upper picture the place for W₂ is taken to be A_x > 0, and thus the worldline (not shown in the pictures) of this clock intersects the worldline of the moving clock (the ct′-axis) in the event labelled A, where "two clocks are simultaneously at one place". In the lower picture the place for W′₂ is taken to be C_x′ < 0, and so in this measurement the moving clock W passes W′₂ in the event C.

In the upper picture the ct-coordinate A_t of the event A (the reading of W₂) is labeled B, thus giving the elapsed time between the two events, measured with W₁ and W₂, as OB. For a comparison, the length of the time interval OA, measured with W′, must be transformed to the scale of the ct-axis. This is done by the invariant hyperbola (see also Fig. 2-8) through A, connecting all events with the same spacetime interval from the origin as A. This yields the event C on the ct-axis, and obviously: OC < OB, the "moving" clock W′ runs slower.

To show the mutual time dilation immediately in the upper picture, the event D may be constructed as the event at x′ = 0 (the location of clock W′ in S′), that is simultaneous to C (OC has equal spacetime interval as OA) in S′. This shows that the time interval OD is longer than OA, showing that the "moving" clock runs slower.^[4]^: 124

In the lower picture the frame S is moving with velocity −v in the frame S′ at rest. The worldline of clock W is the ct-axis (slanted to the left), the worldline of W′₁ is the vertical ct′-axis, and the worldline of W′₂ is the vertical through event C, with ct′-coordinate D. The invariant hyperbola through event C scales the time interval OC to OA, which is shorter than OD; also, B is constructed (similar to D in the upper pictures) as simultaneous to A in S, at x = 0. The result OB > OC corresponds again to above.

The word "measure" is important. In classical physics an observer cannot affect an observed object, but the object's state of motion can affect the observer's observations of the object.

Twin paradox

Many introductions to special relativity illustrate the differences between Galilean relativity and special relativity by posing a series of "paradoxes". These paradoxes are, in fact, ill-posed problems, resulting from our unfamiliarity with velocities comparable to the speed of light. The remedy is to solve many problems in special relativity and to become familiar with its so-called counter-intuitive predictions. The geometrical approach to studying spacetime is considered one of the best methods for developing a modern intuition.^[31]

The twin paradox is a thought experiment involving identical twins, one of whom makes a journey into space in a high-speed rocket, returning home to find that the twin who remained on Earth has aged more. This result appears puzzling because each twin observes the other twin as moving, and so at first glance, it would appear that each should find the other to have aged less. The twin paradox sidesteps the justification for mutual time dilation presented above by avoiding the requirement for a third clock.^[21]^: 207 Nevertheless, the twin paradox is not a true paradox because it is easily understood within the context of special relativity.

The impression that a paradox exists stems from a misunderstanding of what special relativity states. Special relativity does not declare all frames of reference to be equivalent, only inertial frames. The traveling twin's frame is not inertial during periods when she is accelerating. Furthermore, the difference between the twins is observationally detectable: the traveling twin needs to fire her rockets to be able to return home, while the stay-at-home twin does not.^[32]^{[note 9]}

These distinctions should result in a difference in the twins' ages. The spacetime diagram of Fig. 2-11 presents the simple case of a twin going straight out along the x axis and immediately turning back. From the standpoint of the stay-at-home twin, there is nothing puzzling about the twin paradox at all. The proper time measured along the traveling twin's world line from O to C, plus the proper time measured from C to B, is less than the stay-at-home twin's proper time measured from O to A to B. More complex trajectories require integrating the proper time between the respective events along the curve (i.e. the path integral) to calculate the total amount of proper time experienced by the traveling twin.^[32]

Complications arise if the twin paradox is analyzed from the traveling twin's point of view.

Weiss's nomenclature, designating the stay-at-home twin as Terence and the traveling twin as Stella, is hereafter used.^[32]

Stella is not in an inertial frame. Given this fact, it is sometimes incorrectly stated that full resolution of the twin paradox requires general relativity:^[32]

A pure SR analysis would be as follows: Analyzed in Stella's rest frame, she is motionless for the entire trip. When she fires her rockets for the turnaround, she experiences a pseudo force which resembles a gravitational force.^[32] Figs. 2-6 and 2-11 illustrate the concept of lines (planes) of simultaneity: Lines parallel to the observer's x-axis (xy-plane) represent sets of events that are simultaneous in the observer frame. In Fig. 2-11, the blue lines connect events on Terence's world line which, from Stella's point of view, are simultaneous with events on her world line. (Terence, in turn, would observe a set of horizontal lines of simultaneity.) Throughout both the outbound and the inbound legs of Stella's journey, she measures Terence's clocks as running slower than her own. But during the turnaround (i.e. between the bold blue lines in the figure), a shift takes place in the angle of her lines of simultaneity, corresponding to a rapid skip-over of the events in Terence's world line that Stella considers to be simultaneous with her own. Therefore, at the end of her trip, Stella finds that Terence has aged more than she has.^[32]

Although general relativity is not required to analyze the twin paradox, application of the Equivalence Principle of general relativity does provide some additional insight into the subject. Stella is not stationary in an inertial frame. Analyzed in Stella's rest frame, she is motionless for the entire trip. When she is coasting her rest frame is inertial, and Terence's clock will appear to run slow. But when she fires her rockets for the turnaround, her rest frame is an accelerated frame and she experiences a force which is pushing her as if she were in a gravitational field. Terence will appear to be high up in that field and because of gravitational time dilation, his clock will appear to run fast, so much so that the net result will be that Terence has aged more than Stella when they are back together.^[32] The theoretical arguments predicting gravitational time dilation are not exclusive to general relativity. Any theory of gravity will predict gravitational time dilation if it respects the principle of equivalence, including Newton's theory.^[21]^: 16

Gravitation

This introductory section has focused on the spacetime of special relativity, since it is the easiest to describe. Minkowski spacetime is flat, takes no account of gravity, is uniform throughout, and serves as nothing more than a static background for the events that take place in it. The presence of gravity greatly complicates the description of spacetime. In general relativity, spacetime is no longer a static background, but actively interacts with the physical systems that it contains. Spacetime curves in the presence of matter, can propagate waves, bends light, and exhibits a host of other phenomena.^[21]^: 221 A few of these phenomena are described in the later sections of this article.

Basic mathematics of spacetime

Galilean transformations

A basic goal is to be able to compare measurements made by observers in relative motion. If there is an observer O in frame S who has measured the time and space coordinates of an event, assigning this event three Cartesian coordinates and the time as measured on his lattice of synchronized clocks (x, y, z, t) (see Fig. 1-1). A second observer O′ in a different frame S′ measures the same event in her coordinate system and her lattice of synchronized clocks (x′, y′, z′, t′). With inertial frames, neither observer is under acceleration, and a simple set of equations allows us to relate coordinates (x, y, z, t) to (x′, y′, z′, t′). Given that the two coordinate systems are in standard configuration, meaning that they are aligned with parallel (x, y, z) coordinates and that t = 0 when t′ = 0, the coordinate transformation is as follows:^[33]^[34]

x'=x-vt

y'=y

z'=z

t'=t.

Fig. 3-1 illustrates that in Newton's theory, time is universal, not the velocity of light.^[35]^: 36–37 Consider the following thought experiment: The red arrow illustrates a train that is moving at 0.4 c with respect to the platform. Within the train, a passenger shoots a bullet with a speed of 0.4 c in the frame of the train. The blue arrow illustrates that a person standing on the train tracks measures the bullet as traveling at 0.8 c. This is in accordance with our naive expectations.

More generally, assuming that frame S′ is moving at velocity v with respect to frame S, then within frame S′, observer O′ measures an object moving with velocity u′. Velocity u with respect to frame S, since x = ut, x′ = x − vt, and t = t′, can be written as x′ = ut − vt = (u − v)t = (u − v)t′. This leads to u′ = x′/t′ and ultimately

u'=u-v

or

u=u'+v,

which is the common-sense Galilean law for the addition of velocities.

Relativistic composition of velocities

The composition of velocities is quite different in relativistic spacetime. To reduce the complexity of the equations slightly, we introduce a common shorthand for the ratio of the speed of an object relative to light,

\beta =v/c

Fig. 3-2a illustrates a red train that is moving forward at a speed given by v/c = β = s/a. From the primed frame of the train, a passenger shoots a bullet with a speed given by u′/c = β′ = n/m, where the distance is measured along a line parallel to the red x′ axis rather than parallel to the black x axis. What is the composite velocity u of the bullet relative to the platform, as represented by the blue arrow? Referring to Fig. 3-2b:

From the platform, the composite speed of the bullet is given by u = c(s + r)/(a + b).
The two yellow triangles are similar because they are right triangles that share a common angle α. In the large yellow triangle, the ratio s/a = v/c = β.
The ratios of corresponding sides of the two yellow triangles are constant, so that r/a = b/s = n/m = β′. So b = u′s/c and r = u′a/c.
Substitute the expressions for b and r into the expression for u in step 1 to yield Einstein's formula for the addition of velocities:^[35]^: 42–48

u={v+u' \over 1+(vu'/c^{2})}.

The relativistic formula for addition of velocities presented above exhibits several important features:

If u′ and v are both very small compared with the speed of light, then the product vu′/c² becomes vanishingly small, and the overall result becomes indistinguishable from the Galilean formula (Newton's formula) for the addition of velocities: u = u′ + v. The Galilean formula is a special case of the relativistic formula applicable to low velocities.
If u′ is set equal to c, then the formula yields u = c regardless of the starting value of v. The velocity of light is the same for all observers regardless their motions relative to the emitting source.^[35]^: 49

Time dilation and length contraction revisited

It is straightforward to obtain quantitative expressions for time dilation and length contraction. Fig. 3-3 is a composite image containing individual frames taken from two previous animations, simplified and relabeled for the purposes of this section.

To reduce the complexity of the equations slightly, there are a variety of different shorthand notations for ct:

\mathrm {T} =ct

and

w=ct

are common.

One also sees very frequently the use of the convention

c=1.

In Fig. 3-3a, segments OA and OK represent equal spacetime intervals. Time dilation is represented by the ratio OB/OK. The invariant hyperbola has the equation $w = \sqrt x 2 + k 2$ where k = OK, and the red line representing the world line of a particle in motion has the equation w = x/β = xc/v. A bit of algebraic manipulation yields ${\textstyle OB=OK/{\sqrt {1-v^{2}/c^{2}}}.}$

The expression involving the square root symbol appears very frequently in relativity, and one over the expression is called the Lorentz factor, denoted by the Greek letter gamma $\gamma$ :^[36]

\gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}={\frac {1}{\sqrt {1-\beta ^{2}}}}

If v is greater than or equal to c, the expression for $\gamma$ becomes physically meaningless, implying that c is the maximum possible speed in nature. For any v greater than zero, the Lorentz factor will be greater than one, although the shape of the curve is such that for low speeds, the Lorentz factor is extremely close to one.

In Fig. 3-3b, segments OA and OK represent equal spacetime intervals. Length contraction is represented by the ratio OB/OK. The invariant hyperbola has the equation $x = \sqrt w 2 + k 2$ , where k = OK, and the edges of the blue band representing the world lines of the endpoints of a rod in motion have slope 1/β = c/v. Event A has coordinates (x, w) = (γk, γβk). Since the tangent line through A and B has the equation w = (x − OB)/β, we have γβk = (γk − OB)/β and

OB/OK=\gamma (1-\beta ^{2})={\frac {1}{\gamma }}

Lorentz transformations

The Galilean transformations and their consequent commonsense law of addition of velocities work well in our ordinary low-speed world of planes, cars and balls. Beginning in the mid-1800s, however, sensitive scientific instrumentation began finding anomalies that did not fit well with the ordinary addition of velocities.

Lorentz transformations are used to transform the coordinates of an event from one frame to another in special relativity.

The Lorentz factor appears in the Lorentz transformations:

{\begin{aligned}t'&=\gamma \left(t-{\frac {vx}{c^{2}}}\right)\\x'&=\gamma \left(x-vt\right)\\y'&=y\\z'&=z\end{aligned}}

The inverse Lorentz transformations are:

{\begin{aligned}t&=\gamma \left(t'+{\frac {vx'}{c^{2}}}\right)\\x&=\gamma \left(x'+vt'\right)\\y&=y'\\z&=z'\end{aligned}}

When v ≪ c and x is small enough, the v²/c² and vx/c² terms approach zero, and the Lorentz transformations approximate to the Galilean transformations.

$t'=\gamma (t-vx/c^{2}),$ $x'=\gamma (x-vt)$ etc., most often really mean $\Delta t'=\gamma (\Delta t-v\Delta x/c^{2}),$ $\Delta x'=\gamma (\Delta x-v\Delta t)$ etc. Although for brevity the Lorentz transformation equations are written without deltas, x means Δx, etc. We are, in general, always concerned with the space and time differences between events.

Calling one set of transformations the normal Lorentz transformations and the other the inverse transformations is misleading, since there is no intrinsic difference between the frames. Different authors call one or the other set of transformations the "inverse" set. The forwards and inverse transformations are trivially related to each other, since the S frame can only be moving forwards or reverse with respect to S′. So inverting the equations simply entails switching the primed and unprimed variables and replacing v with −v.^[37]^: 71–79

Example: Terence and Stella are at an Earth-to-Mars space race. Terence is an official at the starting line, while Stella is a participant. At time $t = t' = 0$ , Stella's spaceship accelerates instantaneously to a speed of 0.5 c. The distance from Earth to Mars is 300 light-seconds (about 90.0×10⁶ km). Terence observes Stella crossing the finish-line clock at $t = 600.00 s$ . But Stella observes the time on her ship chronometer to be ⁠ $t^{\prime }=\gamma \left(t-vx/c^{2}\right)=519.62\ {\text{s}}$ ⁠ as she passes the finish line, and she calculates the distance between the starting and finish lines, as measured in her frame, to be 259.81 light-seconds (about 77.9×10⁶ km). 1).

Deriving the Lorentz transformations

There have been many dozens of derivations of the Lorentz transformations since Einstein's original work in 1905, each with its particular focus. Although Einstein's derivation was based on the invariance of the speed of light, there are other physical principles that may serve as starting points. Ultimately, these alternative starting points can be considered different expressions of the underlying principle of locality, which states that the influence that one particle exerts on another can not be transmitted instantaneously.^[38]

The derivation given here and illustrated in Fig. 3-5 is based on one presented by Bais^[35]^: 64–66 and makes use of previous results from the Relativistic Composition of Velocities, Time Dilation, and Length Contraction sections. Event P has coordinates (w, x) in the black "rest system" and coordinates $(w', x')$ in the red frame that is moving with velocity parameter $β = v / c$ . To determine w′ and x′ in terms of w and x (or the other way around) it is easier at first to derive the inverse Lorentz transformation.

There can be no such thing as length expansion/contraction in the transverse directions. y' must equal y and z′ must equal z, otherwise whether a fast moving 1 m ball could fit through a 1 m circular hole would depend on the observer. The first postulate of relativity states that all inertial frames are equivalent, and transverse expansion/contraction would violate this law.^[37]^: 27–28
From the drawing, w = a + b and $x = r + s$
From previous results using similar triangles, we know that $s / a = b / r = v / c = β$ .
Because of time dilation, $a = γw'$
Substituting equation (4) into $s / a = β$ yields $s = γw' β$ .
Length contraction and similar triangles give us $r = γx'$ and $b = βr = βγx'$
Substituting the expressions for s, a, r and b into the equations in Step 2 immediately yield
$w=\gamma w'+\beta \gamma x'$

$x=\gamma x'+\beta \gamma w'$

The above equations are alternate expressions for the t and x equations of the inverse Lorentz transformation, as can be seen by substituting ct for w, ct′ for w′, and v/c for β. From the inverse transformation, the equations of the forwards transformation can be derived by solving for t′ and x′.

Linearity of the Lorentz transformations

The Lorentz transformations have a mathematical property called linearity, since x′ and t′ are obtained as linear combinations of x and t, with no higher powers involved. The linearity of the transformation reflects a fundamental property of spacetime that was tacitly assumed in the derivation, namely, that the properties of inertial frames of reference are independent of location and time. In the absence of gravity, spacetime looks the same everywhere.^[35]^: 67 All inertial observers will agree on what constitutes accelerating and non-accelerating motion.^[37]^: 72–73 Any one observer can use her own measurements of space and time, but there is nothing absolute about them. Another observer's conventions will do just as well.^[21]^: 190

A result of linearity is that if two Lorentz transformations are applied sequentially, the result is also a Lorentz transformation.

Example: Terence observes Stella speeding away from him at 0.500 c, and he can use the Lorentz transformations with $β = 0.500$ to relate Stella's measurements to his own. Stella, in her frame, observes Ursula traveling away from her at 0.250 c, and she can use the Lorentz transformations with $β = 0.250$ to relate Ursula's measurements with her own. Because of the linearity of the transformations and the relativistic composition of velocities, Terence can use the Lorentz transformations with $β = 0.666$ to relate Ursula's measurements with his own.

Doppler effect

The Doppler effect is the change in frequency or wavelength of a wave for a receiver and source in relative motion. For simplicity, we consider here two basic scenarios: (1) The motions of the source and/or receiver are exactly along the line connecting them (longitudinal Doppler effect), and (2) the motions are at right angles to the said line (transverse Doppler effect). We are ignoring scenarios where they move along intermediate angles.

Longitudinal Doppler effect

The classical Doppler analysis deals with waves that are propagating in a medium, such as sound waves or water ripples, and which are transmitted between sources and receivers that are moving towards or away from each other. The analysis of such waves depends on whether the source, the receiver, or both are moving relative to the medium. Given the scenario where the receiver is stationary with respect to the medium, and the source is moving directly away from the receiver at a speed of v_s for a velocity parameter of β_s, the wavelength is increased, and the observed frequency f is given by

f={\frac {1}{1+\beta _{s}}}f_{0}

On the other hand, given the scenario where source is stationary, and the receiver is moving directly away from the source at a speed of v_r for a velocity parameter of β_r, the wavelength is not changed, but the transmission velocity of the waves relative to the receiver is decreased, and the observed frequency f is given by

f=(1-\beta _{r})f_{0}

Light, unlike sound or water ripples, does not propagate through a medium, and there is no distinction between a source moving away from the receiver or a receiver moving away from the source. Fig. 3-6 illustrates a relativistic spacetime diagram showing a source separating from the receiver with a velocity parameter β, so that the separation between source and receiver at time w is βw. Because of time dilation, ⁠ $W=YW^{\prime }$ ⁠. Since the slope of the green light ray is −1, ⁠ ${T}=W+\beta w=\gamma w^{\prime }(1+\beta )$ ⁠. Hence, the relativistic Doppler effect is given by^[35]^: 58–59

f={\sqrt {\frac {1-\beta }{1+\beta }}}\,f_{0}.

Transverse Doppler effect

Suppose that a source and a receiver, both approaching each other in uniform inertial motion along non-intersecting lines, are at their closest approach to each other. It would appear that the classical analysis predicts that the receiver detects no Doppler shift. Due to subtleties in the analysis, that expectation is not necessarily true. Nevertheless, when appropriately defined, transverse Doppler shift is a relativistic effect that has no classical analog. The subtleties are these:^[39]^{: 541–543}

Fig. 3-7a. What is the frequency measurement when the receiver is geometrically at its closest approach to the source? This scenario is most easily analyzed from the frame S' of the source.^{[note 10]}
Fig. 3-7b. What is the frequency measurement when the receiver sees the source as being closest to it? This scenario is most easily analyzed from the frame S of the receiver.

Two other scenarios are commonly examined in discussions of transverse Doppler shift:

Fig. 3-7c. If the receiver is moving in a circle around the source, what frequency does the receiver measure?
Fig. 3-7d. If the source is moving in a circle around the receiver, what frequency does the receiver measure?

In scenario (a), the point of closest approach is frame-independent and represents the moment where there is no change in distance versus time (i.e. dr/dt = 0 where r is the distance between receiver and source) and hence no longitudinal Doppler shift. The source observes the receiver as being illuminated by light of frequency f′, but also observes the receiver as having a time-dilated clock. In frame S, the receiver is therefore illuminated by blueshifted light of frequency

f=f'\gamma =f'/{\sqrt {1-\beta ^{2}}}

In scenario (b) the illustration shows the receiver being illuminated by light from when the source was closest to the receiver, even though the source has moved on. Because the source's clocks are time dilated as measured in frame S, and since dr/dt was equal to zero at this point, the light from the source, emitted from this closest point, is redshifted with frequency

f=f'/\gamma =f'{\sqrt {1-\beta ^{2}}}

Scenarios (c) and (d) can be analyzed by simple time dilation arguments. In (c), the receiver observes light from the source as being blueshifted by a factor of $\gamma$ , and in (d), the light is redshifted. The only seeming complication is that the orbiting objects are in accelerated motion. However, if an inertial observer looks at an accelerating clock, only the clock's instantaneous speed is important when computing time dilation. (The converse, however, is not true.)^[39]^{: 541–543} Most reports of transverse Doppler shift refer to the effect as a redshift and analyze the effect in terms of scenarios (b) or (d).^{[note 11]}

Energy and momentum

Extending momentum to four dimensions

In classical mechanics, the state of motion of a particle is characterized by its mass and its velocity. Linear momentum, the product of a particle's mass and velocity, is a vector quantity, possessing the same direction as the velocity: $p = m v$ . It is a conserved quantity, meaning that if a closed system is not affected by external forces, its total linear momentum cannot change.

In relativistic mechanics, the momentum vector is extended to four dimensions. Added to the momentum vector is a time component that allows the spacetime momentum vector to transform like the spacetime position vector ⁠ $(x,t)$ ⁠. In exploring the properties of the spacetime momentum, we start, in Fig. 3-8a, by examining what a particle looks like at rest. In the rest frame, the spatial component of the momentum is zero, i.e. $p = 0$ , but the time component equals mc.

We can obtain the transformed components of this vector in the moving frame by using the Lorentz transformations, or we can read it directly from the figure because we know that ⁠ $(mc)^{\prime }=\gamma mc$ ⁠ and ⁠ $p^{\prime }=-\beta \gamma mc$ ⁠, since the red axes are rescaled by gamma. Fig. 3-8b illustrates the situation as it appears in the moving frame. It is apparent that the space and time components of the four-momentum go to infinity as the velocity of the moving frame approaches c.^[35]^: 84–87

We will use this information shortly to obtain an expression for the four-momentum.

Momentum of light

Light particles, or photons, travel at the speed of c, the constant that is conventionally known as the speed of light. This statement is not a tautology, since many modern formulations of relativity do not start with constant speed of light as a postulate. Photons therefore propagate along a light-like world line and, in appropriate units, have equal space and time components for every observer.

A consequence of Maxwell's theory of electromagnetism is that light carries energy and momentum, and that their ratio is a constant: ⁠ $E/p=c$ ⁠. Rearranging, ⁠ $E/c=p$ ⁠, and since for photons, the space and time components are equal, E/c must therefore be equated with the time component of the spacetime momentum vector.

Photons travel at the speed of light, yet have finite momentum and energy. For this to be so, the mass term in γmc must be zero, meaning that photons are massless particles. Infinity times zero is an ill-defined quantity, but E/c is well-defined.

By this analysis, if the energy of a photon equals E in the rest frame, it equals ⁠ $E^{\prime }=(1-\beta )\gamma E$ ⁠ in a moving frame. This result can be derived by inspection of Fig. 3-9 or by application of the Lorentz transformations, and is consistent with the analysis of Doppler effect given previously.^[35]^: 88

Mass-energy relationship

Consideration of the interrelationships between the various components of the relativistic momentum vector led Einstein to several famous conclusions.

In the low speed limit as $β = v/c$ approaches zero, $γ$ approaches 1, so the spatial component of the relativistic momentum ⁠ $\beta \gamma mc=\gamma mv$ ⁠ approaches mv, the classical term for momentum. Following this perspective, γm can be interpreted as a relativistic generalization of m. Einstein proposed that the relativistic mass of an object increases with velocity according to the formula ⁠ $m_{rel}=\gamma m$ ⁠.
Likewise, comparing the time component of the relativistic momentum with that of the photon, ⁠ $\gamma mc=m_{rel}c=E/c$ ⁠, so that Einstein arrived at the relationship ⁠ $E=m_{rel}c^{2}$ ⁠. Simplified to the case of zero velocity, this is Einstein's famous equation relating energy and mass.

Another way of looking at the relationship between mass and energy is to consider a series expansion of $γmc 2$ at low velocity:

E=\gamma mc^{2}={\frac {mc^{2}}{\sqrt {1-\beta ^{2}}}}

\approx mc^{2}+{\frac {1}{2}}mv^{2}...

The second term is just an expression for the kinetic energy of the particle. Mass indeed appears to be another form of energy.^[35]^: 90–92^[37]^{: 129–130, 180}

The concept of relativistic mass that Einstein introduced in 1905, m_rel, although amply validated every day in particle accelerators around the globe (or indeed in any instrumentation whose use depends on high velocity particles, such as electron microscopes,^[40] old-fashioned color television sets, etc.), has nevertheless not proven to be a fruitful concept in physics in the sense that it is not a concept that has served as a basis for other theoretical development. Relativistic mass, for instance, plays no role in general relativity.

For this reason, as well as for pedagogical concerns, most physicists currently prefer a different terminology when referring to the relationship between mass and energy.^[41] "Relativistic mass" is a deprecated term. The term "mass" by itself refers to the rest mass or invariant mass, and is equal to the invariant length of the relativistic momentum vector. Expressed as a formula,

E^{2}-p^{2}c^{2}=m^{2}c^{4}

This formula applies to all particles, massless as well as massive. For massless photons, it yields the same relationship as established earlier, ⁠ $E=\pm pc$ ⁠.^[35]^: 90–92

Four-momentum

Because of the close relationship between mass and energy, the four-momentum (also called 4-momentum) is also called the energy–momentum 4-vector. Using an uppercase P to represent the four-momentum and a lowercase p to denote the spatial momentum, the four-momentum may be written as

P\equiv (E/c,{\vec {p}})=(E/c,p_{x},p_{y},p_{z})

or alternatively,

P\equiv (E,{\vec {p}})=(E,p_{x},p_{y},p_{z})

using the convention that

c=1.

^[37]^{: 129–130, 180}

Conservation laws

In physics, conservation laws state that certain particular measurable properties of an isolated physical system do not change as the system evolves over time. In 1915, Emmy Noether discovered that underlying each conservation law is a fundamental symmetry of nature.^[42] The fact that physical processes don't care where in space they take place (space translation symmetry) yields conservation of momentum, the fact that such processes don't care when they take place (time translation symmetry) yields conservation of energy, and so on. In this section, we examine the Newtonian views of conservation of mass, momentum and energy from a relativistic perspective.

Total momentum

To understand how the Newtonian view of conservation of momentum needs to be modified in a relativistic context, we examine the problem of two colliding bodies limited to a single dimension.

In Newtonian mechanics, two extreme cases of this problem may be distinguished yielding mathematics of minimum complexity:

(1) The two bodies rebound from each other in a completely elastic collision.

(2) The two bodies stick together and continue moving as a single particle. This second case is the case of completely inelastic collision.

For both cases (1) and (2), momentum, mass, and total energy are conserved. However, kinetic energy is not conserved in cases of inelastic collision. A certain fraction of the initial kinetic energy is converted to heat.

In case (2), two masses with momentums ⁠ ${\boldsymbol {p}}_{\boldsymbol {1}}=m_{1}{\boldsymbol {v}}_{\boldsymbol {1}}$ ⁠ and ⁠ ${\boldsymbol {p}}_{\boldsymbol {2}}=m_{2}{\boldsymbol {v}}_{\boldsymbol {2}}$ ⁠ collide to produce a single particle of conserved mass ⁠ $m=m_{1}+m_{2}$ ⁠ traveling at the center of mass velocity of the original system, ${\boldsymbol {v_{cm}}}=\left(m_{1}{\boldsymbol {v_{1}}}+m_{2}{\boldsymbol {v_{2}}}\right)/\left(m_{1}+m_{2}\right)$ . The total momentum ⁠ ${\boldsymbol {p=p_{1}+p_{2}}}$ ⁠ is conserved.

Fig. 3-10 illustrates the inelastic collision of two particles from a relativistic perspective. The time components ⁠ $E_{1}/c$ ⁠ and ⁠ $E_{2}/c$ ⁠ add up to total E/c of the resultant vector, meaning that energy is conserved. Likewise, the space components ⁠ ${\boldsymbol {p_{1}}}$ ⁠ and ⁠ ${\boldsymbol {p_{2}}}$ ⁠ add up to form p of the resultant vector. The four-momentum is, as expected, a conserved quantity. However, the invariant mass of the fused particle, given by the point where the invariant hyperbola of the total momentum intersects the energy axis, is not equal to the sum of the invariant masses of the individual particles that collided. Indeed, it is larger than the sum of the individual masses: ⁠ $m>m_{1}+m_{2}$ ⁠.^[35]^: 94–97

Looking at the events of this scenario in reverse sequence, we see that non-conservation of mass is a common occurrence: when an unstable elementary particle spontaneously decays into two lighter particles, total energy is conserved, but the mass is not. Part of the mass is converted into kinetic energy.^[37]^{: 134–138}

Choice of reference frames

Figure 3-11.
(above) Lab Frame.
(right) Center of Momentum Frame.

The freedom to choose any frame in which to perform an analysis allows us to pick one which may be particularly convenient. For analysis of momentum and energy problems, the most convenient frame is usually the "center-of-momentum frame" (also called the zero-momentum frame, or COM frame). This is the frame in which the space component of the system's total momentum is zero. Fig. 3-11 illustrates the breakup of a high speed particle into two daughter particles. In the lab frame, the daughter particles are preferentially emitted in a direction oriented along the original particle's trajectory. In the COM frame, however, the two daughter particles are emitted in opposite directions, although their masses and the magnitude of their velocities are generally not the same.

Energy and momentum conservation

In a Newtonian analysis of interacting particles, transformation between frames is simple because all that is necessary is to apply the Galilean transformation to all velocities. Since ⁠ $v'=v-u$ ⁠, the momentum ⁠ $p'=p-mu$ ⁠. If the total momentum of an interacting system of particles is observed to be conserved in one frame, it will likewise be observed to be conserved in any other frame.^[37]^{: 241–245}

Conservation of momentum in the COM frame amounts to the requirement that $p = 0$ both before and after collision. In the Newtonian analysis, conservation of mass dictates that ⁠ $m=m_{1}+m_{2}$ ⁠. In the simplified, one-dimensional scenarios that we have been considering, only one additional constraint is necessary before the outgoing momenta of the particles can be determined—an energy condition. In the one-dimensional case of a completely elastic collision with no loss of kinetic energy, the outgoing velocities of the rebounding particles in the COM frame will be precisely equal and opposite to their incoming velocities. In the case of a completely inelastic collision with total loss of kinetic energy, the outgoing velocities of the rebounding particles will be zero.^[37]^{: 241–245}

Newtonian momenta, calculated as ⁠ $p=mv$ ⁠, fail to behave properly under Lorentzian transformation. The linear transformation of velocities ⁠ $v'=v-u$ ⁠ is replaced by the highly nonlinear ⁠ $v^{\prime }=(v-u){\Big /}\left(1-{\frac {vu}{c^{2}}}\right)$ ⁠ so that a calculation demonstrating conservation of momentum in one frame will be invalid in other frames. Einstein was faced with either having to give up conservation of momentum, or to change the definition of momentum. This second option was what he chose.^[35]^: 104

Figure 3-12a. Energy–momentum diagram for decay of a charged pion.

Figure 3-12b. Graphing calculator analysis of charged pion decay.

The relativistic conservation law for energy and momentum replaces the three classical conservation laws for energy, momentum and mass. Mass is no longer conserved independently, because it has been subsumed into the total relativistic energy. This makes the relativistic conservation of energy a simpler concept than in nonrelativistic mechanics, because the total energy is conserved without any qualifications. Kinetic energy converted into heat or internal potential energy shows up as an increase in mass.^[37]^: 127

Example: Because of the equivalence of mass and energy, elementary particle masses are customarily stated in energy units, where 1 MeV = 10⁶ electron volts. A charged pion is a particle of mass 139.57 MeV (approx. 273 times the electron mass). It is unstable, and decays into a muon of mass 105.66 MeV (approx. 207 times the electron mass) and an antineutrino, which has an almost negligible mass. The difference between the pion mass and the muon mass is 33.91 MeV.

π⁻
→
μ⁻
+
ν
_μ

Fig. 3-12a illustrates the energy–momentum diagram for this decay reaction in the rest frame of the pion. Because of its negligible mass, a neutrino travels at very nearly the speed of light. The relativistic expression for its energy, like that of the photon, is ⁠ $E_{v}=pc,$ ⁠ which is also the value of the space component of its momentum. To conserve momentum, the muon has the same value of the space component of the neutrino's momentum, but in the opposite direction.

Algebraic analyses of the energetics of this decay reaction are available online,^[43] so Fig. 3-12b presents instead a graphing calculator solution. The energy of the neutrino is 29.79 MeV, and the energy of the muon is 33.91 MeV − 29.79 MeV = 4.12 MeV. Most of the energy is carried off by the near-zero-mass neutrino.

Beyond the basics

The topics in this section are of significantly greater technical difficulty than those in the preceding sections and are not essential for understanding Introduction to curved spacetime.

Rapidity

Figure 4-1a. A ray through the unit circle x² + y² = 1 in the point (cos a, sin a), where a is twice the area between the ray, the circle, and the x-axis.

Figure 4-1b. A ray through the unit hyperbola x² − y² = 1 in the point (cosh a, sinh a), where a is twice the area between the ray, the hyperbola, and the x-axis.

Figure 4–2. Plot of the three basic Hyperbolic functions: hyperbolic sine (sinh), hyperbolic cosine (cosh) and hyperbolic tangent (tanh). Sinh is red, cosh is blue and tanh is green.

Lorentz transformations relate coordinates of events in one reference frame to those of another frame. Relativistic composition of velocities is used to add two velocities together. The formulas to perform the latter computations are nonlinear, making them more complex than the corresponding Galilean formulas.

This nonlinearity is an artifact of our choice of parameters.^[7]^: 47–59 We have previously noted that in an x–ct spacetime diagram, the points at some constant spacetime interval from the origin form an invariant hyperbola. We have also noted that the coordinate systems of two spacetime reference frames in standard configuration are hyperbolically rotated with respect to each other.

The natural functions for expressing these relationships are the hyperbolic analogs of the trigonometric functions. Fig. 4-1a shows a unit circle with sin(a) and cos(a), the only difference between this diagram and the familiar unit circle of elementary trigonometry being that a is interpreted, not as the angle between the ray and the x-axis, but as twice the area of the sector swept out by the ray from the x-axis. (Numerically, the angle and 2 × area measures for the unit circle are identical.) Fig. 4-1b shows a unit hyperbola with sinh(a) and cosh(a), where a is likewise interpreted as twice the tinted area.^[44] Fig. 4-2 presents plots of the sinh, cosh, and tanh functions.

For the unit circle, the slope of the ray is given by

{\text{slope}}=\tan a={\frac {\sin a}{\cos a}}.

In the Cartesian plane, rotation of point (x, y) into point (x', y') by angle θ is given by

{\begin{pmatrix}x'\\y'\\\end{pmatrix}}={\begin{pmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \\\end{pmatrix}}{\begin{pmatrix}x\\y\\\end{pmatrix}}.

In a spacetime diagram, the velocity parameter $\beta$ is the analog of slope. The rapidity, φ, is defined by^[37]^: 96–99

\beta \equiv \tanh \phi \equiv {\frac {v}{c}},

where

\tanh \phi ={\frac {\sinh \phi }{\cosh \phi }}={\frac {e^{\phi }-e^{-\phi }}{e^{\phi }+e^{-\phi }}}.

The rapidity defined above is very useful in special relativity because many expressions take on a considerably simpler form when expressed in terms of it. For example, rapidity is simply additive in the collinear velocity-addition formula;^[7]^: 47–59

\beta ={\frac {\beta _{1}+\beta _{2}}{1+\beta _{1}\beta _{2}}}=

{\frac {\tanh \phi _{1}+\tanh \phi _{2}}{1+\tanh \phi _{1}\tanh \phi _{2}}}=

\tanh(\phi _{1}+\phi _{2}),

or in other words, $\phi =\phi _{1}+\phi _{2}.$

The Lorentz transformations take a simple form when expressed in terms of rapidity. The γ factor can be written as

\gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}={\frac {1}{\sqrt {1-\tanh ^{2}\phi }}}

=\cosh \phi ,

\gamma \beta ={\frac {\beta }{\sqrt {1-\beta ^{2}}}}={\frac {\tanh \phi }{\sqrt {1-\tanh ^{2}\phi }}}

=\sinh \phi .

Transformations describing relative motion with uniform velocity and without rotation of the space coordinate axes are called boosts.

Substituting γ and γβ into the transformations as previously presented and rewriting in matrix form, the Lorentz boost in the x-direction may be written as

{\begin{pmatrix}ct'\\x'\end{pmatrix}}={\begin{pmatrix}\cosh \phi &-\sinh \phi \\-\sinh \phi &\cosh \phi \end{pmatrix}}{\begin{pmatrix}ct\\x\end{pmatrix}},

and the inverse Lorentz boost in the x-direction may be written as

{\begin{pmatrix}ct\\x\end{pmatrix}}={\begin{pmatrix}\cosh \phi &\sinh \phi \\\sinh \phi &\cosh \phi \end{pmatrix}}{\begin{pmatrix}ct'\\x'\end{pmatrix}}.

In other words, Lorentz boosts represent hyperbolic rotations in Minkowski spacetime.^[37]^: 96–99

The advantages of using hyperbolic functions are such that some textbooks such as the classic ones by Taylor and Wheeler introduce their use at a very early stage.^[7]^[45]^{[note 12]}

4‑vectors

Four‑vectors have been mentioned above in context of the energy–momentum 4‑vector, but without any great emphasis. Indeed, none of the elementary derivations of special relativity require them. But once understood, 4‑vectors, and more generally tensors, greatly simplify the mathematics and conceptual understanding of special relativity. Working exclusively with such objects leads to formulas that are manifestly relativistically invariant, which is a considerable advantage in non-trivial contexts. For instance, demonstrating relativistic invariance of Maxwell's equations in their usual form is not trivial, while it is merely a routine calculation (really no more than an observation) using the field strength tensor formulation. On the other hand, general relativity, from the outset, relies heavily on 4‑vectors, and more generally tensors, representing physically relevant entities. Relating these via equations that do not rely on specific coordinates requires tensors, capable of connecting such 4‑vectors even within a curved spacetime, and not just within a flat one as in special relativity. The study of tensors is outside the scope of this article, which provides only a basic discussion of spacetime.

Definition of 4-vectors

A 4-tuple, ⁠ $A=\left(A_{0},A_{1},A_{2},A_{3}\right)$ ⁠ is a "4-vector" if its component A_i transform between frames according to the Lorentz transformation.

If using ⁠ $(ct,x,y,z)$ ⁠ coordinates, A is a 4–vector if it transforms (in the x-direction) according to

{\begin{aligned}A_{0}'&=\gamma \left(A_{0}-(v/c)A_{1}\right)\\A_{1}'&=\gamma \left(A_{1}-(v/c)A_{0}\right)\\A_{2}'&=A_{2}\\A_{3}'&=A_{3}\end{aligned}}

which comes from simply replacing ct with A₀ and x with A₁ in the earlier presentation of the Lorentz transformation.

As usual, when we write x, t, etc. we generally mean Δx, Δt etc.

The last three components of a 4–vector must be a standard vector in three-dimensional space. Therefore, a 4–vector must transform like ⁠ $(c\Delta t,\Delta x,\Delta y,\Delta z)$ ⁠ under Lorentz transformations as well as rotations.^[31]^: 36–59

Properties of 4-vectors

Closure under linear combination: If A and B are 4-vectors, then ⁠ $C=aA+aB$ ⁠ is also a 4-vector.
Inner-product invariance: If A and B are 4-vectors, then their inner product (scalar product) is invariant, i.e. their inner product is independent of the frame in which it is calculated. Note how the calculation of inner product differs from the calculation of the inner product of a 3-vector. In the following, ${\vec {A}}$ and ${\vec {B}}$ are 3-vectors:

A\cdot B\equiv

A_{0}B_{0}-A_{1}B_{1}-A_{2}B_{2}-A_{3}B_{3}\equiv

A_{0}B_{0}-{\vec {A}}\cdot {\vec {B}}

In addition to being invariant under Lorentz transformation, the above inner product is also invariant under rotation in 3-space.

Two vectors are said to be orthogonal if

A\cdot B=0.

Unlike the case with 3-vectors, orthogonal 4-vectors are not necessarily at right angles with each other. The rule is that two 4-vectors are orthogonal if they are offset by equal and opposite angles from the 45° line which is the world line of a light ray. This implies that a lightlike 4-vector is orthogonal with itself.

Invariance of the magnitude of a vector: The magnitude of a vector is the inner product of a 4-vector with itself, and is a frame-independent property. As with intervals, the magnitude may be positive, negative or zero, so that the vectors are referred to as timelike, spacelike or null (lightlike). Note that a null vector is not the same as a zero vector. A null vector is one for which $A\cdot A=0,$ while a zero vector is one whose components are all zero. Special cases illustrating the invariance of the norm include the invariant interval $c^{2}t^{2}-x^{2}$ and the invariant length of the relativistic momentum vector $E^{2}-p^{2}c^{2}.$ ^[37]^{: 178–181}^[31]^: 36–59

Examples of 4-vectors

Displacement 4-vector: Otherwise known as the spacetime separation, this is (Δt, Δx, Δy, Δz), or for infinitesimal separations, (dt, dx, dy, dz).

dS\equiv (dt,dx,dy,dz)

Velocity 4-vector: This results when the displacement 4-vector is divided by $d\tau$ , where $d\tau$ is the proper time between the two events that yield dt, dx, dy, and dz.

V\equiv {\frac {dS}{d\tau }}={\frac {(dt,dx,dy,dz)}{dt/\gamma }}=

\gamma \left(1,{\frac {dx}{dt}},{\frac {dy}{dt}},{\frac {dz}{dt}}\right)=

(\gamma ,\gamma {\vec {v}})

Figure 4-3a. The momentarily comoving reference frames of an accelerating particle as observed from a stationary frame.

Figure 4-3b. The momentarily comoving reference frames along the trajectory of an accelerating observer (center).

The 4-velocity is tangent to the world line of a particle, and has a length equal to one unit of time in the frame of the particle.

An accelerated particle does not have an inertial frame in which it is always at rest. However, an inertial frame can always be found which is momentarily comoving with the particle. This frame, the momentarily comoving reference frame (MCRF), enables application of special relativity to the analysis of accelerated particles.

Since photons move on null lines,

d\tau =0

for a photon, and a 4-velocity cannot be defined. There is no frame in which a photon is at rest, and no MCRF can be established along a photon's path.

Energy–momentum 4-vector:

P\equiv (E/c,{\vec {p}})=(E/c,p_{x},p_{y},p_{z})

As indicated before, there are varying treatments for the energy-momentum 4-vector so that one may also see it expressed as

(E,{\vec {p}})

or

(E,{\vec {p}}c).

The first component is the total energy (including mass) of the particle (or system of particles) in a given frame, while the remaining components are its spatial momentum. The energy-momentum 4-vector is a conserved quantity.

Acceleration 4-vector: This results from taking the derivative of the velocity 4-vector with respect to $\tau .$

A\equiv {\frac {dV}{d\tau }}=

{\frac {d}{d\tau }}(\gamma ,\gamma {\vec {v}})=

\gamma \left({\frac {d\gamma }{dt}},{\frac {d(\gamma {\vec {v}})}{dt}}\right)

Force 4-vector: This is the derivative of the momentum 4-vector with respect to $\tau .$

F\equiv {\frac {dP}{d\tau }}=

\gamma \left({\frac {dE}{dt}},{\frac {d{\vec {p}}}{dt}}\right)=

\gamma \left({\frac {dE}{dt}},{\vec {f}}\right)

As expected, the final components of the above 4-vectors are all standard 3-vectors corresponding to spatial 3-momentum, 3-force etc.^[37]^{: 178–181}^[31]^: 36–59

4-vectors and physical law

The first postulate of special relativity declares the equivalency of all inertial frames. A physical law holding in one frame must apply in all frames, since otherwise it would be possible to differentiate between frames. Newtonian momenta fail to behave properly under Lorentzian transformation, and Einstein preferred to change the definition of momentum to one involving 4-vectors rather than give up on conservation of momentum.

Physical laws must be based on constructs that are frame independent. This means that physical laws may take the form of equations connecting scalars, which are always frame independent. However, equations involving 4-vectors require the use of tensors with appropriate rank, which themselves can be thought of as being built up from 4-vectors.^[37]^: 186

Acceleration

It is a common misconception that special relativity is applicable only to inertial frames, and that it is unable to handle accelerating objects or accelerating reference frames. Actually, accelerating objects can generally be analyzed without needing to deal with accelerating frames at all. It is only when gravitation is significant that general relativity is required.^[46]

Properly handling accelerating frames does require some care, however. The difference between special and general relativity is that (1) In special relativity, all velocities are relative, but acceleration is absolute. (2) In general relativity, all motion is relative, whether inertial, accelerating, or rotating. To accommodate this difference, general relativity uses curved spacetime.^[46]

In this section, we analyze several scenarios involving accelerated reference frames.

Dewan–Beran–Bell spaceship paradox

The Dewan–Beran–Bell spaceship paradox (Bell's spaceship paradox) is a good example of a problem where intuitive reasoning unassisted by the geometric insight of the spacetime approach can lead to issues.

In Fig. 4-4, two identical spaceships float in space and are at rest relative to each other. They are connected by a string which is capable of only a limited amount of stretching before breaking. At a given instant in our frame, the observer frame, both spaceships accelerate in the same direction along the line between them with the same constant proper acceleration.^{[note 13]} Will the string break?

When the paradox was new and relatively unknown, even professional physicists had difficulty working out the solution. Two lines of reasoning lead to opposite conclusions. Both arguments, which are presented below, are flawed even though one of them yields the correct answer.^[37]^{: 106, 120–122}

To observers in the rest frame, the spaceships start a distance L apart and remain the same distance apart during acceleration. During acceleration, L is a length contracted distance of the distance L' = γL in the frame of the accelerating spaceships. After a sufficiently long time, γ will increase to a sufficiently large factor that the string must break.
Let A and B be the rear and front spaceships. In the frame of the spaceships, each spaceship sees the other spaceship doing the same thing that it is doing. A says that B has the same acceleration that he has, and B sees that A matches her every move. So the spaceships stay the same distance apart, and the string does not break.^[37]^{: 106, 120–122}

The problem with the first argument is that there is no "frame of the spaceships." There cannot be, because the two spaceships measure a growing distance between the two. Because there is no common frame of the spaceships, the length of the string is ill-defined. Nevertheless, the conclusion is correct, and the argument is mostly right. The second argument, however, completely ignores the relativity of simultaneity.^[37]^{: 106, 120–122}

A spacetime diagram (Fig. 4-5) makes the correct solution to this paradox almost immediately evident. Two observers in Minkowski spacetime accelerate with constant magnitude $k$ acceleration for proper time $\sigma$ (acceleration and elapsed time measured by the observers themselves, not some inertial observer). They are comoving and inertial before and after this phase. In Minkowski geometry, the length along the line of simultaneity $A'B''$ turns out to be greater than the length along the line of simultaneity $AB$ .

The length increase can be calculated with the help of the Lorentz transformation. If, as illustrated in Fig. 4-5, the acceleration is finished, the ships will remain at a constant offset in some frame $S'.$ If $x_{A}$ and $x_{B}=x_{A}+L$ are the ships' positions in $S,$ the positions in frame $S'$ are:^[47]

{\begin{aligned}x'_{A}&=\gamma \left(x_{A}-vt\right)\\x'_{B}&=\gamma \left(x_{A}+L-vt\right)\\L'&=x'_{B}-x'_{A}=\gamma L\end{aligned}}

The "paradox", as it were, comes from the way that Bell constructed his example. In the usual discussion of Lorentz contraction, the rest length is fixed and the moving length shortens as measured in frame $S$ . As shown in Fig. 4-5, Bell's example asserts the moving lengths $AB$ and $A'B'$ measured in frame $S$ to be fixed, thereby forcing the rest frame length $A'B''$ in frame $S'$ to increase.

Accelerated observer with horizon

Certain special relativity problem setups can lead to insight about phenomena normally associated with general relativity, such as event horizons. In the text accompanying Fig. 2-7, the magenta hyperbolae represented actual paths that are tracked by a constantly accelerating traveler in spacetime. During periods of positive acceleration, the traveler's velocity just approaches the speed of light, while, measured in our frame, the traveler's acceleration constantly decreases.

Fig. 4-6 details various features of the traveler's motions with more specificity. At any given moment, her space axis is formed by a line passing through the origin and her current position on the hyperbola, while her time axis is the tangent to the hyperbola at her position. The velocity parameter $\beta$ approaches a limit of one as $ct$ increases. Likewise, $\gamma$ approaches infinity.

The shape of the invariant hyperbola corresponds to a path of constant proper acceleration. This is demonstrable as follows:

We remember that $\beta =ct/x.$
Since $c^{2}t^{2}-x^{2}=s^{2},$ we conclude that $\beta (ct)=ct/{\sqrt {c^{2}t^{2}-s^{2}}}.$
$\gamma =1/{\sqrt {1-\beta ^{2}}}=$ ${\sqrt {c^{2}t^{2}-s^{2}}}/s$
From the relativistic force law, $F=dp/dt=$ $dpc/d(ct)=d(\beta \gamma mc^{2})/d(ct).$
Substituting $\beta (ct)$ from step 2 and the expression for $\gamma$ from step 3 yields $F=mc^{2}/s,$ which is a constant expression.^[35]^{: 110–113}

Fig. 4-6 illustrates a specific calculated scenario. Terence (A) and Stella (B) initially stand together 100 light hours from the origin. Stella lifts off at time 0, her spacecraft accelerating at 0.01 c per hour. Every twenty hours, Terence radios updates to Stella about the situation at home (solid green lines). Stella receives these regular transmissions, but the increasing distance (offset in part by time dilation) causes her to receive Terence's communications later and later as measured on her clock, and she never receives any communications from Terence after 100 hours on his clock (dashed green lines).^[35]^{: 110–113}

After 100 hours according to Terence's clock, Stella enters a dark region. She has traveled outside Terence's timelike future. On the other hand, Terence can continue to receive Stella's messages to him indefinitely. He just has to wait long enough. Spacetime has been divided into distinct regions separated by an apparent event horizon. So long as Stella continues to accelerate, she can never know what takes place behind this horizon.^[35]^{: 110–113}

Introduction to curved spacetime

Basic propositions

Newton's theories assumed that motion takes place against the backdrop of a rigid Euclidean reference frame that extends throughout all space and all time. Gravity is mediated by a mysterious force, acting instantaneously across a distance, whose actions are independent of the intervening space.^{[note 14]} In contrast, Einstein denied that there is any background Euclidean reference frame that extends throughout space. Nor is there any such thing as a force of gravitation, only the structure of spacetime itself.^[7]^{: 175–190}

In spacetime terms, the path of a satellite orbiting the Earth is not dictated by the distant influences of the Earth, Moon and Sun. Instead, the satellite moves through space only in response to local conditions. Since spacetime is everywhere locally flat when considered on a sufficiently small scale, the satellite is always following a straight line in its local inertial frame. We say that the satellite always follows along the path of a geodesic. No evidence of gravitation can be discovered following alongside the motions of a single particle.^[7]^{: 175–190}

In any analysis of spacetime, evidence of gravitation requires that one observe the relative accelerations of two bodies or two separated particles. In Fig. 5-1, two separated particles, free-falling in the gravitational field of the Earth, exhibit tidal accelerations due to local inhomogeneities in the gravitational field such that each particle follows a different path through spacetime. The tidal accelerations that these particles exhibit with respect to each other do not require forces for their explanation. Rather, Einstein described them in terms of the geometry of spacetime, i.e. the curvature of spacetime. These tidal accelerations are strictly local. It is the cumulative total effect of many local manifestations of curvature that result in the appearance of a gravitational force acting at a long range from Earth.^[7]^{: 175–190}

Two central propositions underlie general relativity.

The first crucial concept is coordinate independence: The laws of physics cannot depend on what coordinate system one uses. This is a major extension of the principle of relativity from the version used in special relativity, which states that the laws of physics must be the same for every observer moving in non-accelerated (inertial) reference frames. In general relativity, to use Einstein's own (translated) words, "the laws of physics must be of such a nature that they apply to systems of reference in any kind of motion."^[48]^: 113 This leads to an immediate issue: In accelerated frames, one feels forces that seemingly would enable one to assess one's state of acceleration in an absolute sense. Einstein resolved this problem through the principle of equivalence.^[49]^{: 137–149}

The equivalence principle states that in any sufficiently small region of space, the effects of gravitation are the same as those from acceleration.

In Fig. 5-2, person A is in a spaceship, far from any massive objects, that undergoes a uniform acceleration of g. Person B is in a box resting on Earth. Provided that the spaceship is sufficiently small so that tidal effects are non-measurable (given the sensitivity of current gravity measurement instrumentation, A and B presumably should be Lilliputians), there are no experiments that A and B can perform which will enable them to tell which setting they are in.^[49]^{: 141–149}

An alternative expression of the equivalence principle is to note that in Newton's universal law of gravitation, F = GMm_g/r² = m_gg and in Newton's second law, F = m_ia, there is no a priori reason why the gravitational mass m_g should be equal to the inertial mass m_i. The equivalence principle states that these two masses are identical.^[49]^{: 141–149}

To go from the elementary description above of curved spacetime to a complete description of gravitation requires tensor calculus and differential geometry, topics both requiring considerable study. Without these mathematical tools, it is possible to write about general relativity, but it is not possible to demonstrate any non-trivial derivations.

Curvature of time

In the discussion of special relativity, forces played no more than a background role. Special relativity assumes the ability to define inertial frames that fill all of spacetime, all of whose clocks run at the same rate as the clock at the origin. Is this really possible? In a nonuniform gravitational field, experiment dictates that the answer is no. Gravitational fields make it impossible to construct a global inertial frame. In small enough regions of spacetime, local inertial frames are still possible. General relativity involves the systematic stitching together of these local frames into a more general picture of spacetime.^[31]^{: 118–126}

Years before publication of the general theory in 1916, Einstein used the equivalence principle to predict the existence of gravitational redshift in the following thought experiment: (i) Assume that a tower of height h (Fig. 5-3) has been constructed. (ii) Drop a particle of rest mass m from the top of the tower. It falls freely with acceleration g, reaching the ground with velocity $v = (2 gh) 1/2$ , so that its total energy E, as measured by an observer on the ground, is ⁠ $m+{\frac {{\frac {1}{2}}mv^{2}}{c^{2}}}=m+{\frac {mgh}{c^{2}}}$ ⁠ (iii) A mass-energy converter transforms the total energy of the particle into a single high energy photon, which it directs upward. (iv) At the top of the tower, an energy-mass converter transforms the energy of the photon E' back into a particle of rest mass m'.^[31]^{: 118–126}

It must be that $m = m'$ , since otherwise one would be able to construct a perpetual motion device. We therefore predict that $E' = m$ , so that

{\frac {E'}{E}}={\frac {h\nu \,'}{h\nu }}={\frac {m}{m+{\frac {mgh}{c^{2}}}}}=1-{\frac {gh}{c^{2}}}

A photon climbing in Earth's gravitational field loses energy and is redshifted. Early attempts to measure this redshift through astronomical observations were somewhat inconclusive, but definitive laboratory observations were performed by Pound & Rebka (1959) and later by Pound & Snider (1964).^[50]

Light has an associated frequency, and this frequency may be used to drive the workings of a clock. The gravitational redshift leads to an important conclusion about time itself: Gravity makes time run slower. Suppose we build two identical clocks whose rates are controlled by some stable atomic transition. Place one clock on top of the tower, while the other clock remains on the ground. An experimenter on top of the tower observes that signals from the ground clock are lower in frequency than those of the clock next to her on the tower. Light going up the tower is just a wave, and it is impossible for wave crests to disappear on the way up. Exactly as many oscillations of light arrive at the top of the tower as were emitted at the bottom. The experimenter concludes that the ground clock is running slow, and can confirm this by bringing the tower clock down to compare side by side with the ground clock.^[21]^: 16–18 For a 1 km tower, the discrepancy would amount to about 9.4 nanoseconds per day, easily measurable with modern instrumentation.

Clocks in a gravitational field do not all run at the same rate. Experiments such as the Pound–Rebka experiment have firmly established curvature of the time component of spacetime. The Pound–Rebka experiment says nothing about curvature of the space component of spacetime. But the theoretical arguments predicting gravitational time dilation do not depend on the details of general relativity at all. Any theory of gravity will predict gravitational time dilation if it respects the principle of equivalence.^[21]^: 16 This includes Newtonian gravitation. A standard demonstration in general relativity is to show how, in the "Newtonian limit" (i.e. the particles are moving slowly, the gravitational field is weak, and the field is static), curvature of time alone is sufficient to derive Newton's law of gravity.^[51]^{: 101–106}

Newtonian gravitation is a theory of curved time. General relativity is a theory of curved time and curved space. Given G as the gravitational constant, M as the mass of a Newtonian star, and orbiting bodies of insignificant mass at distance r from the star, the spacetime interval for Newtonian gravitation is one for which only the time coefficient is variable:^[21]^{: 229–232}

\Delta s^{2}=\left(1-{\frac {2GM}{c^{2}r}}\right)(c\Delta t)^{2}-\,(\Delta x)^{2}-(\Delta y)^{2}-(\Delta z)^{2}

Curvature of space

The $(1-2GM/(c^{2}r))$ coefficient in front of $(c\Delta t)^{2}$ describes the curvature of time in Newtonian gravitation, and this curvature completely accounts for all Newtonian gravitational effects. As expected, this correction factor is directly proportional to $G$ and $M$ , and because of the $r$ in the denominator, the correction factor increases as one approaches the gravitating body, meaning that time is curved.

But general relativity is a theory of curved space and curved time, so if there are terms modifying the spatial components of the spacetime interval presented above, shouldn't their effects be seen on, say, planetary and satellite orbits due to curvature correction factors applied to the spatial terms?

The answer is that they are seen, but the effects are tiny. The reason is that planetary velocities are extremely small compared to the speed of light, so that for planets and satellites of the solar system, the $(c\Delta t)^{2}$ term dwarfs the spatial terms.^[21]^{: 234–238}

Despite the minuteness of the spatial terms, the first indications that something was wrong with Newtonian gravitation were discovered over a century-and-a-half ago. In 1859, Urbain Le Verrier, in an analysis of available timed observations of transits of Mercury over the Sun's disk from 1697 to 1848, reported that known physics could not explain the orbit of Mercury, unless there possibly existed a planet or asteroid belt within the orbit of Mercury. The perihelion of Mercury's orbit exhibited an excess rate of precession over that which could be explained by the tugs of the other planets.^[52] The ability to detect and accurately measure the minute value of this anomalous precession (only 43 arc seconds per tropical century) is testimony to the sophistication of 19th century astrometry.

As the famous astronomer who had earlier discovered the existence of Neptune "at the tip of his pen" by analyzing wobbles in the orbit of Uranus, Le Verrier's announcement triggered a two-decades long period of "Vulcan-mania", as professional and amateur astronomers alike hunted for the hypothetical new planet. This search included several false sightings of Vulcan. It was ultimately established that no such planet or asteroid belt existed.^[53]

In 1916, Einstein was to show that this anomalous precession of Mercury is explained by the spatial terms in the curvature of spacetime. Curvature in the temporal term, being simply an expression of Newtonian gravitation, has no part in explaining this anomalous precession. The success of his calculation was a powerful indication to Einstein's peers that the general theory of relativity could be correct.

The most spectacular of Einstein's predictions was his calculation that the curvature terms in the spatial components of the spacetime interval could be measured in the bending of light around a massive body. Light has a slope of ±1 on a spacetime diagram. Its movement in space is equal to its movement in time. For the weak field expression of the invariant interval, Einstein calculated an exactly equal but opposite sign curvature in its spatial components.^[21]^{: 234–238}

\Delta s^{2}=\left(1-{\frac {2GM}{c^{2}r}}\right)(c\Delta t)^{2}

-\,\left(1+{\frac {2GM}{c^{2}r}}\right)\left[(\Delta x)^{2}+(\Delta y)^{2}+(\Delta z)^{2}\right]

In Newton's gravitation, the $(1-2GM/(c^{2}r))$ coefficient in front of $(c\Delta t)^{2}$ predicts bending of light around a star. In general relativity, the $(1+2GM/(c^{2}r))$ coefficient in front of $\left[(\Delta x)^{2}+(\Delta y)^{2}+(\Delta z)^{2}\right]$ predicts a doubling of the total bending.^[21]^{: 234–238}

The story of the 1919 Eddington eclipse expedition and Einstein's rise to fame is well told elsewhere.^[54]

Sources of spacetime curvature

In Newton's theory of gravitation, the only source of gravitational force is mass.

In contrast, general relativity identifies several sources of spacetime curvature in addition to mass. In the Einstein field equations, the sources of gravity are presented on the right-hand side in $T_{\mu \nu },$ the stress–energy tensor.

Fig. 5-5 classifies the various sources of gravity in the stress–energy tensor:

$T^{00}$ (red): The total mass–energy density, including any contributions to the potential energy from forces between the particles, as well as kinetic energy from random thermal motions.
$T^{0i}$ and $T^{i0}$ (orange): These are momentum density terms. Even if there is no bulk motion, energy may be transmitted by heat conduction, and the conducted energy will carry momentum.
$T^{ij}$ are the rates of flow of the i-component of momentum per unit area in the j-direction. Even if there is no bulk motion, random thermal motions of the particles will give rise to momentum flow, so the $i = j$ terms (green) represent isotropic pressure, and the $i \neq j$ terms (blue) represent shear stresses.^[55]

One important conclusion to be derived from the equations is that, colloquially speaking, gravity itself creates gravity.^{[note 15]} Energy has mass. Even in Newtonian gravity, the gravitational field is associated with an energy, ⁠ $E=mgh,$ ⁠ called the gravitational potential energy. In general relativity, the energy of the gravitational field feeds back into creation of the gravitational field. This makes the equations nonlinear and hard to solve in anything other than weak field cases.^[21]^: 240 Numerical relativity is a branch of general relativity using numerical methods to solve and analyze problems, often employing supercomputers to study black holes, gravitational waves, neutron stars and other phenomena in the strong field regime.

Energy-momentum

Figure 5-6. (left) Mass-energy warps spacetime. (right) Rotating mass–energy distributions with angular momentum J generate gravitomagnetic fields H.

In special relativity, mass-energy is closely connected to momentum. Just as space and time are different aspects of a more comprehensive entity called spacetime, mass–energy and momentum are merely different aspects of a unified, four-dimensional quantity called four-momentum. In consequence, if mass–energy is a source of gravity, momentum must also be a source. The inclusion of momentum as a source of gravity leads to the prediction that moving or rotating masses can generate fields analogous to the magnetic fields generated by moving charges, a phenomenon known as gravitomagnetism.^[56]

It is well known that the force of magnetism can be deduced by applying the rules of special relativity to moving charges. (An eloquent demonstration of this was presented by Feynman in volume II, chapter 13–6 of his Lectures on Physics, available online.^[57]) Analogous logic can be used to demonstrate the origin of gravitomagnetism. In Fig. 5-7a, two parallel, infinitely long streams of massive particles have equal and opposite velocities −v and +v relative to a test particle at rest and centered between the two. Because of the symmetry of the setup, the net force on the central particle is zero. Assume ⁠ $v\ll c$ ⁠ so that velocities are simply additive. Fig. 5-7b shows exactly the same setup, but in the frame of the upper stream. The test particle has a velocity of +v, and the bottom stream has a velocity of +2v. Since the physical situation has not changed, only the frame in which things are observed, the test particle should not be attracted towards either stream. But it is not at all clear that the forces exerted on the test particle are equal. (1) Since the bottom stream is moving faster than the top, each particle in the bottom stream has a larger mass energy than a particle in the top. (2) Because of Lorentz contraction, there are more particles per unit length in the bottom stream than in the top stream. (3) Another contribution to the active gravitational mass of the bottom stream comes from an additional pressure term which, at this point, we do not have sufficient background to discuss. All of these effects together would seemingly demand that the test particle be drawn towards the bottom stream.

The test particle is not drawn to the bottom stream because of a velocity-dependent force that serves to repel a particle that is moving in the same direction as the bottom stream. This velocity-dependent gravitational effect is gravitomagnetism.^[21]^{: 245–253}

Matter in motion through a gravitomagnetic field is hence subject to so-called frame-dragging effects analogous to electromagnetic induction. It has been proposed that such gravitomagnetic forces underlie the generation of the relativistic jets (Fig. 5-8) ejected by some rotating supermassive black holes.^[58]^[59]

Pressure and stress

Quantities that are directly related to energy and momentum should be sources of gravity as well, namely internal pressure and stress. Taken together, mass-energy, momentum, pressure and stress all serve as sources of gravity: Collectively, they are what tells spacetime how to curve.

General relativity predicts that pressure acts as a gravitational source with exactly the same strength as mass–energy density. The inclusion of pressure as a source of gravity leads to dramatic differences between the predictions of general relativity versus those of Newtonian gravitation. For example, the pressure term sets a maximum limit to the mass of a neutron star. The more massive a neutron star, the more pressure is required to support its weight against gravity. The increased pressure, however, adds to the gravity acting on the star's mass. Above a certain mass determined by the Tolman–Oppenheimer–Volkoff limit, the process becomes runaway and the neutron star collapses to a black hole.^[21]^{: 243, 280}

The stress terms become highly significant when performing calculations such as hydrodynamic simulations of core-collapse supernovae.^[60]

These predictions for the roles of pressure, momentum and stress as sources of spacetime curvature are elegant and play an important role in theory. In regards to pressure, the early universe was radiation dominated,^[61] and it is highly unlikely that any of the relevant cosmological data (e.g. nucleosynthesis abundances, etc.) could be reproduced if pressure did not contribute to gravity, or if it did not have the same strength as a source of gravity as mass–energy. Likewise, the mathematical consistency of the Einstein field equations would be broken if the stress terms did not contribute as a source of gravity.

Experimental test of the sources of spacetime curvature

Definitions: Active, passive, and inertial mass

Bondi distinguishes between different possible types of mass: (1) active mass ( $m_{a}$ ) is the mass which acts as the source of a gravitational field; (2)passive mass ( $m_{p}$ ) is the mass which reacts to a gravitational field; (3) inertial mass ( $m_{i}$ ) is the mass which reacts to acceleration.^[62]

$m_{p}$ is the same as gravitational mass ( $m_{g}$ ) in the discussion of the equivalence principle.

In Newtonian theory,

The third law of action and reaction dictates that $m_{a}$ and $m_{p}$ must be the same.
On the other hand, whether $m_{p}$ and $m_{i}$ are equal is an empirical result.

In general relativity,

The equality of $m_{p}$ and $m_{i}$ is dictated by the equivalence principle.
There is no "action and reaction" principle dictating any necessary relationship between $m_{a}$ and $m_{p}$ .^[62]

Pressure as a gravitational source

The classic experiment to measure the strength of a gravitational source (i.e. its active mass) was first conducted in 1797 by Henry Cavendish (Fig. 5-9a). Two small but dense balls are suspended on a fine wire, making a torsion balance. Bringing two large test masses close to the balls introduces a detectable torque. Given the dimensions of the apparatus and the measurable spring constant of the torsion wire, the gravitational constant G can be determined.

To study pressure effects by compressing the test masses is hopeless, because attainable laboratory pressures are insignificant in comparison with the mass-energy of a metal ball.

However, the repulsive electromagnetic pressures resulting from protons being tightly squeezed inside atomic nuclei are typically on the order of 10²⁸ atm ≈ 10³³ Pa ≈ 10³³ kg·s⁻²m⁻¹. This amounts to about 1% of the nuclear mass density of approximately 10¹⁸kg/m³ (after factoring in c² ≈ 9×10¹⁶m²s⁻²).^[63]

Figure 5-10. Lunar laser ranging experiment. (left) This retroreflector was left on the Moon by astronauts on the Apollo 11 mission. (right) Astronomers all over the world have bounced laser light off the retroreflectors left by Apollo astronauts and Russian lunar rovers to measure precisely the Earth-Moon distance.

If pressure does not act as a gravitational source, then the ratio $m_{a}/m_{p}$ should be lower for nuclei with higher atomic number Z, in which the electrostatic pressures are higher. L. B. Kreuzer (1968) did a Cavendish experiment using a Teflon mass suspended in a mixture of the liquids trichloroethylene and dibromoethane having the same buoyant density as the Teflon (Fig. 5-9b). Fluorine has atomic number $Z = 9$ , while bromine has $Z = 35$ . Kreuzer found that repositioning the Teflon mass caused no differential deflection of the torsion bar, hence establishing active mass and passive mass to be equivalent to a precision of 5×10⁻⁵.^[64]

Although Kreuzer originally considered this experiment merely to be a test of the ratio of active mass to passive mass, Clifford Will (1976) reinterpreted the experiment as a fundamental test of the coupling of sources to gravitational fields.^[65]

In 1986, Bartlett and Van Buren noted that lunar laser ranging had detected a 2 km offset between the moon's center of figure and its center of mass. This indicates an asymmetry in the distribution of Fe (abundant in the Moon's core) and Al (abundant in its crust and mantle). If pressure did not contribute equally to spacetime curvature as does mass–energy, the moon would not be in the orbit predicted by classical mechanics. They used their measurements to tighten the limits on any discrepancies between active and passive mass to about 10⁻¹².^[66]

Gravitomagnetism

The existence of gravitomagnetism was proven by Gravity Probe B (GP-B), a satellite-based mission which launched on 20 April 2004.^[67] The spaceflight phase lasted until 2005. The mission aim was to measure spacetime curvature near Earth, with particular emphasis on gravitomagnetism.

Initial results confirmed the relatively large geodetic effect (which is due to simple spacetime curvature, and is also known as de Sitter precession) to an accuracy of about 1%. The much smaller frame-dragging effect (which is due to gravitomagnetism, and is also known as Lense–Thirring precession) was difficult to measure because of unexpected charge effects causing variable drift in the gyroscopes. Nevertheless, by August 2008, the frame-dragging effect had been confirmed to within 15% of the expected result,^[68] while the geodetic effect was confirmed to better than 0.5%.^[69]^[70]

Subsequent measurements of frame dragging by laser-ranging observations of the LARES, LAGEOS-1 and LAGEOS-2 satellites has improved on the GP-B measurement, with results (as of 2016) demonstrating the effect to within 5% of its theoretical value,^[71] although there has been some disagreement on the accuracy of this result.^[72]

Another effort, the Gyroscopes in General Relativity (GINGER) experiment, seeks to use three 6 m ring lasers mounted at right angles to each other 1400 m below the Earth's surface to measure this effect.^[73]^[74]

Introduction to the mathematics of curved spacetime

The approach to tensors adopted here follows closely an older presentation by Lillian Lieber (1945, 2008)^[75]^{[note 16]}^{[note 17]} which was written to be accessible by anybody with a basic understanding of calculus. Lieber used the coordinate transformation approach to tensor analysis. The modern approach to tensor analysis stresses the geometrical nature of tensors rather than the transformation properties of their components.^[77]^: 77 Because of the coordinate-free nature of the abstract view, it is often considered more physical.^[78]^: 31 However, books on general relativity written in a manner intended to be usable by autodidacts (textbooks as well as semi-popularizations) usually adopt the coordinate transformation approach as requiring less mathematical sophistication on the part of the reader.^[76]^[79] Several textbooks, including that by Adler,^[78] provide side-by-side explanations in terms of both the classic view and the modern abstract view.

This non-rigorous introduction to the mathematics of general relativity stops at the vacuum field equations which are valid only in regions of space where the energy-momentum tensor is zero, which is to say, in regions devoid of mass-energy. Nevertheless, a variety of interesting results are possible with this limited approach, including derivation of the Schwarzschild metric and an exploration of some of its consequences.

Describing the shape of space and spacetime

Cartesian coordinates

Polar coordinates

Oblique coordinates

Spherical coordinates

Figure 6–1. Computing ds in different coordinate systems

In the section of this article on the Spacetime interval, the reader has been introduced to the concept of the interval $s^{2}$ and has been told, without detailed explanation, that the properties of this interval serve to characterize the geometric properties of the space (or spacetime) on which the interval has been defined.

For example, in a Euclidean plane, the Pythagorean theorem holds for right triangles drawn in that plane.

s^{2}=x^{2}+y^{2}

(A1)

Conversely, if the distance between two points on a surface is given by

s^{2}=x^{2}+y^{2}

then that surface is necessarily a Euclidean plane.^[75]^{: 113–125}

Failure of the Pythagorean theorem to hold implies that a surface has an intrinsic curvature. The intrinsic curvature of the surface can be ascertained solely from measurements made from within that surface, without external comparisons, and without information that might be obtained by measurements obtained from any higher-dimensional space in which the surface may be embedded. Intrinsic curvature is to be distinguished from extrinsic curvature. If one takes a flat sheet and rolls it into a cylinder, the surface has extrinsic curvature, but the Pythagorean theorem continues to hold for measurements made within the surface, so the surface has no intrinsic curvature. General relativity is concerned only with the intrinsic curvature of spacetime.^[77]^{: 153–154}

In differential calculus, the student learns how to apply the Pythagorean theorem in computing lengths along a curve, as in Fig. 6–1a, where the differential form of the theorem is

ds^{2}=dx^{2}+dy^{2}

(A2)

In subsequent discussion we will prefer to use generalized coordinates, substituting $x_{1}$ for $x$ and $x_{2}$ for $y,$ i.e.

ds^{2}=dx_{1}^{2}+dx_{2}^{2}

(A3)

The properties of a space do not depend on the coordinate system used to make measurements within that space. What would be the equivalent of (A2) for measurements made in other coordinate systems?

For polar coordinates, as shown in Fig. 6–1b, the relevant expression would be

ds^{2}=dr^{2}+r^{2}d\theta ^{2}

(A4)

where the equivalent expression using generalized coordinates, substituting $x_{1}$ for $r$ and $x_{2}$ for $\theta ,$ is

ds^{2}=dx_{1}^{2}+x_{1}^{2}dx_{2}^{2}\;.

(A5)

For oblique coordinates, as shown in Fig. 6–1c, the law of cosines allows us to write

ds^{2}=dx^{2}+dy^{2}-2dx\,dy\,\cos \alpha

(A6)

and the equivalent expression using generalized coordinates would be

ds^{2}=dx_{1}^{2}+dx_{2}^{2}-2dx_{1}\,dx_{2}\,\cos \alpha \;.

(A7)

What of surfaces with a bona fide intrinsic curvature? In Fig. 6–1d, we illustrate a sphere on which has been drawn the elements of the spherical coordinate system. With the understanding that $r=R\cos \beta ,$ we note that

ds^{2}=r^{2}d\alpha ^{2}+R^{2}d\beta ^{2}

(A8)

and the equivalent expression, replacing $\alpha$ with $x_{1}$ and $\beta$ with $x_{2}$ would be

ds^{2}=r^{2}dx_{1}^{2}+R^{2}dx_{2}^{2}

(A9)

The expression for $ds^{2}$ depends on both the intrinsic properties of the surface and the coordinate system used to describe that surface. Therefore, a cursory examination of $ds^{2}$ will not suffice to determine the characteristics of the surface that we are dealing with. To determine the characteristics of the surface starting from $ds^{2},$ we must determine the curvature tensor.^[75]^{: 113–125}

What are tensors?

In precalculus, one learns about scalars and vectors. Scalars are quantities that have magnitude only, while vectors have both magnitude and direction. Measurements such as temperature and age are scalars, whereas measurements of velocity, momentum, acceleration and force are vectors.

Tensors are a form of mathematical object that have found great use in science and engineering. "Tensor" is an inclusive term that includes scalars and vectors as special cases: A scalar is a tensor of rank zero, while a vector is a tensor of rank one.

A familiar engineering use of tensors is in the representation of compressive, tensile, and sheer stresses on an object. A pure force (a vector) acting uniformly on an entire object will not cause the object to deform; instead, the object will accelerate uniformly, and the object will not "feel" any effects of the force. It is the differential application of forces on different parts of an object that exerts stress on the object, causing mechanical strain.

In Fig 6–2, consider a small surface element which is being acted upon by the force $AB$ . The area and orientation of this surface element is represented by the vector $AG$ , which is perpendicular to the surface and whose magnitude represents the area of the surface element. The stress at $A$ depends on both vectors and is a tensor of rank two.^[75]^{: 127–140}

Tensors exist independently of any coordinate system. However, for computational purposes, it is convenient to decompose a tensor into components.

In Fig 6–3a, a force $F$ acts on a small surface $dS$ where $G$ is the vector that represents the area and orientation of this surface element. In Fig 6–3b, the projections of this surface element $dS_{x},dS_{y},$ and $dS_{z}$ on the $yz,xz,$ and $xy$ planes, respectively, are illustrated. The x, y, and z components of $G$ (not illustrated) represent the areas and orientations of these three projections.

The total effect of the force $F$ on $dS$ can be computed by considering the effect of each of its three components, $f_{x},\,f_{y},$ and $f_{z}$ on each of the three projections $dS_{x},\,dS_{y},\,$ and $dS_{z}.$

The x-component of $F,$ which is $f_{x},$ acts on each of the aforementioned projections, and the "pressure" (force per unit area) from $f_{x}$ acting on each of these projections is designated as $p_{xx},\,p_{xy},\,p_{xz},$ respectively. Since force equals pressure times area, we can write:^[75]^{: 127–140}

f_{x}=p_{xx}dS_{x}\,+\,p_{xy}dS_{y}\,+\,p_{xz}dS_{z}

Likewise, for $f_{y}$ and $f_{z},$ we write

f_{y}=p_{yx}dS_{x}\,+\,p_{yy}dS_{y}\,+\,p_{yz}dS_{z}

f_{z}=p_{zx}dS_{x}\,+\,p_{zy}dS_{y}\,+\,p_{zz}dS_{z}

The total stress $F$ on the surface $dS$ is $F=f_{x}\,+\,f_{y}\,+f_{z},$ so that

{\begin{aligned}F&=p_{xx}dS_{x}\,+\,p_{xy}dS_{y}\,+\,p_{xz}dS_{z}\\&+\,p_{yx}dS_{x}\,+\,p_{yy}dS_{y}\,+\,p_{yz}dS_{z}\\&+\,p_{zx}dS_{x}\,+\,p_{zy}dS_{y}\,+\,p_{zz}dS_{z}\end{aligned}}

(B1)

In three-dimensional space, force (a vector) has three components, but stress (a tensor of rank two) has nine components. A tensor of rank three will have n³ components and so forth.

In n-dimensional space, the n components of a vector are written in a single row, but the n² components of a tensor of rank two are written in a square array.

Effect of changes in the coordinate system

Relativity is concerned with finding the physical laws which hold good for all observers, regardless of their viewpoint (coordinate system). In 1905, with special relativity, Einstein considered changes in viewpoint due to differences in uniform relative velocity. In 1916, with general relativity, Einstein generalized the idea to include observers in much more complex relationships with each other. The concept of invariance that Einstein introduced is one of the most fundamental in all of physics. Tensors are objects that are intrinsically invariant under transformation of coordinate systems.^{[note 18]} In the following, we explore the effects of such transformation, beginning with a simple rotation of coordinates.^[75]^{: 141–150}

In Fig. 6–4, consider a conventional Cartesian coordinate system in the $xy$ plane. Suppose we transform to a new ${\bar {x}},\,{\bar {y}}$ coordinate system that is obtained from the $x,\,y$ system by rotating the coordinate axes by angle $\theta$ about the origin. If point $A$ has coordinates $x,\,y$ in the first coordinate system, its coordinates in the primed system are given by

{\bar {x}}=x\cos \theta +y\sin \theta

{\bar {y}}=-x\sin \theta +y\cos \theta

The inverse transformation, calculating $x$ and $y$ given ${\bar {x}}$ and ${\bar {y}},$ is readily obtained from this first transformation.

Through a series of steps, we will generalize this notation to encompass other transformations in an arbitrary number of dimensions. The generalized notation will allow an elegantly condensed method of writing the equations that simplifies complex manipulations.^[75]^{: 141–150}

Our first generalization is to rewrite the transformation so that it is no longer tied to a specific form of rotation:

{\bar {x}}=a\cdot x+b\cdot y

{\bar {y}}=c\cdot x+d\cdot y

where $a,\,b,\,c,\,d\,$ are functions of $\theta .\,$ In differential form, we may write the following:

d{\bar {x}}=a\cdot dx+b\cdot dy

d{\bar {y}}=c\cdot dx+d\cdot dy

We further generalize by using $dx^{1}$ and $dx^{2}$ instead of $dx$ and $dy$ , and by using the single letter $a$ with different subscripts instead of four different letters $a,\,b,\,c,\,d.$

We will henceforth mostly be using coordinates distinguished by superscripts rather than subscripts for reasons that will be discussed later. These superscripts are not to be confused with exponentiation:

{\begin{aligned}d{\bar {x}}^{1}=a_{11}dx^{1}+a_{12}dx^{2}\\d{\bar {x}}^{2}=a_{21}dx^{1}+a_{22}dx^{2}\end{aligned}}

(C1)

The subscripted $a$ 's are now understood as representing partial derivatives, with $a_{11}$ being the change in ${\bar {x}}^{1}$ due to a change in $x^{1}$ and so forth.^[75]^: 147

{\begin{aligned}d{\bar {x}}^{1}={\frac {\partial {\bar {x}}^{1}}{\partial x^{1}}}dx^{1}+{\frac {\partial {\bar {x}}^{1}}{\partial x^{2}}}dx^{2}\\d{\bar {x}}^{2}={\frac {\partial {\bar {x}}^{2}}{\partial x^{1}}}dx^{1}+{\frac {\partial {\bar {x}}^{2}}{\partial x^{2}}}dx^{2}\end{aligned}}

(C2)

Notational simplifications

The two equations in (C2) may be rewritten in a single line:

d{\bar {x}}^{\mu }=\sum \limits _{\sigma }{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\sigma }}}dx^{\sigma }\quad \quad {\begin{pmatrix}\mu =1,2\\\sigma =1,2\end{pmatrix}}

(D1)

The Einstein summation convention enables further abbreviation. Whenever a symbol occurs twice in a single term (e.g. the $\sigma$ in the right-hand member of (D1), it is understood that a summation is to be made on that subscript (or superscript). Hence, we may rewrite (D1) as follows:

d{\bar {x}}^{\mu }={\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\sigma }}}dx^{\sigma }\quad \quad {\begin{pmatrix}\mu =1,2\\\sigma =1,2\end{pmatrix}}

(D2)

Let $x^{\mu }$ be the coordinates of a point $P$ in a space of dimensionality n. Let $P'$ be a neighboring point having coordinates $x^{\mu }+dx^{\mu }$ as measured in the first frame. The coordinates of $P'$ in the second frame will be ${\bar {x}}^{\mu }+d{\bar {x}}^{\mu }.$ The n quantities $dx^{\mu }$ are understood to the components of the displacement vector ${\vec {PP'}}$ as measured in the first frame, while $d{\bar {x}}^{\mu }$ are the components of this same displacement vector as measured in the second frame. These are related to the components measured in the first frame by the transformation equation (D2).^[80]^: 89–90

The appearance of equation (D2) may be simplified further as follows: Given that $dx^{1}$ and $dx^{2}$ are the components of $ds$ in the unbarred system, we represent them more briefly by $V^{1}$ and $V^{2}.$ Likewise, given that $d{\bar {x}}^{1}$ and $d{\bar {x}}^{2}$ are the components of $ds$ in the barred system, we represent them more briefly by ${\bar {V}}^{1}$ and ${\bar {V}}^{2}.$

On the right side of (D2), $\mu ,$ which is not repeated, is known as a free index, while the repeated summation indices are known as dummy indices, since they disappear when performing the summation. Unless stated otherwise, any free index shall have the same range as the dummy indices.^[81]^: 2 Hence, in (D2),

{\begin{pmatrix}\mu =1,2\\\sigma =1,2\end{pmatrix}}

may be written as

(n=2).

These superscripts should not be confused with exponents. $V^{2}$ is not the square of $V.$ Rather, these superscripts are used for indexing purposes, the same as subscripts. Superscripts and subscripts are used for distinct purposes which will be explained shortly.

Hence, (D2) may be rewritten as follows:

{\bar {V}}^{\mu }={\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\sigma }}}V^{\sigma }

(D3)

Given a vector $V^{\sigma }$ , whose components are $V^{1}$ and $V^{2}$ in a given coordinate system, (D3) allows computation of its components in a new coordinate system related to the first by the transformation represented in (C1).

Actually, (D2) and (D3) are valid not merely for the transformation represented in (C1), but are valid for any transformation of coordinates (provided that the values of $x_{\sigma }$ and ${\bar {x}}_{\mu }$ are in one-to-one correspondence). In other words, in the transformation represented by

{\bar {x}}^{\mu }=f_{\mu }(x^{\sigma }),

where $f_{\mu }$ are completely arbitrary functions, (D2) and (D3) allow computation of the vector components in the transformed coordinate system.

Any set of quantities that transforms according to (D3) is, by definition, a vector, or more precisely, a contravariant vector. One should also note that (D3) is extensible to vectors of any number of dimensions. In the curved spacetime of general relativity, one cannot think of vectors as being directed line segments stretching from one point to another. A set of coordinates $x^{n}$ do not form a vector. In the case discussed here, a contravariant vector is the set of coordinate differentials $dx^{n}$ along some given curve.^[78]^: 39

Using this notation, a contravariant tensor of rank two is defined as follows:

{\bar {V}}^{\alpha \beta }={\frac {\partial {\bar {x}}^{\alpha }}{\partial x^{\gamma }}}{\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\delta }}}V^{\gamma \delta }

(D4)

Since $\gamma$ and $\delta$ each occur twice in the term on the right, it is understood that the term represents a sum for $\gamma$ and $\delta$ over their entire ranges. On the other hand, neither $\alpha$ nor $\beta$ occur twice in any single term. In three-space, $\alpha ,\,\beta ,\,\gamma ,\,\delta$ each range over $1,\,2,\,3,$ so the interpretation of (D4) is that it represents nine equations, each equation having the sum of nine terms on the right.

For example, given $\alpha =2,\,\beta =3,$ (D4) expands to the following:

{\bar {V}}^{23}={\frac {\partial {\bar {x}}^{2}}{\partial x^{1}}}{\frac {\partial {\bar {x}}^{3}}{\partial x^{1}}}V^{11}+{\frac {\partial {\bar {x}}^{2}}{\partial x^{1}}}{\frac {\partial {\bar {x}}^{3}}{\partial x^{2}}}V^{12}

+\;{\frac {\partial {\bar {x}}^{2}}{\partial x^{1}}}{\frac {\partial {\bar {x}}^{3}}{\partial x^{3}}}V^{13}

\quad \quad +\,{\frac {\partial {\bar {x}}^{2}}{\partial x^{2}}}{\frac {\partial {\bar {x}}^{3}}{\partial x^{1}}}V^{21}+{\frac {\partial {\bar {x}}^{2}}{\partial x^{2}}}{\frac {\partial {\bar {x}}^{3}}{\partial x^{2}}}V^{22}

+\;{\frac {\partial {\bar {x}}^{2}}{\partial x^{2}}}{\frac {\partial {\bar {x}}^{3}}{\partial x^{3}}}V^{23}

\quad \quad +\,{\frac {\partial {\bar {x}}^{2}}{\partial x^{3}}}{\frac {\partial {\bar {x}}^{3}}{\partial x^{1}}}V^{31}+{\frac {\partial {\bar {x}}^{2}}{\partial x^{3}}}{\frac {\partial {\bar {x}}^{3}}{\partial x^{2}}}V^{32}

+\;{\frac {\partial {\bar {x}}^{2}}{\partial x^{3}}}{\frac {\partial {\bar {x}}^{3}}{\partial x^{3}}}V^{33}

In four-space, (D4) expands to sixteen equations, each having a sum of sixteen terms on the right.

The notation presented here hence offers a concise representation of complex mathematical objects.^[75]^{: 151–159}

Tensor addition and multiplication

Tensor algebra includes various operations for making new tensors from old tensors. Here we begin with tensor addition, starting with tensors of rank one (vectors) in a plane.^[75]^{: 163–167}

Suppose we have two contravariant vectors in a plane, $A^{\alpha }$ with components $A^{1}$ and $A^{2}$ , and a second such vector, $B^{\alpha }$ with components $B^{1}$ and $B^{2}$ . Let us form another quantity, $C^{\alpha },$ by adding the corresponding components of $A^{\alpha }$ and $B^{\alpha }$ . In other words, $C^{1}=A^{1}+B^{1}$ and $C^{2}=A^{2}+B^{2}$ .

We ask whether the resulting quantity $C^{\alpha }$ is a vector, i.e. does it transform according to (D3)? Since $A^{\alpha }$ and $B^{\alpha }$ are contravariant vectors, we may write:

{\bar {A}}^{\lambda }={\frac {\partial {\bar {x}}^{\lambda }}{\partial x^{\alpha }}}A^{\alpha }

(E1)

{\bar {B}}^{\lambda }={\frac {\partial {\bar {x}}^{\lambda }}{\partial x^{\alpha }}}B^{\alpha }

(E2)

Taking the components one at a time, we may write, for the first components:

{\bar {A}}^{1}={\frac {\partial {\bar {x}}^{1}}{\partial x^{1}}}A^{1}+{\frac {\partial {\bar {x}}^{1}}{\partial x^{2}}}A^{2}

{\bar {B}}^{1}={\frac {\partial {\bar {x}}^{1}}{\partial x_{1}}}B^{1}+{\frac {\partial {\bar {x}}^{1}}{\partial x^{2}}}B^{2}

and likewise for the second components. Summing these, we obtain for the first and second components:

{\bar {A}}^{1}+{\bar {B}}^{1}={\frac {\partial {\bar {x}}^{1}}{\partial x^{1}}}(A^{1}+B^{1})

+\;{\frac {\partial {\bar {x}}^{1}}{\partial x^{2}}}(A^{2}+B^{2})

{\bar {A}}^{2}+{\bar {B}}^{2}={\frac {\partial {\bar {x}}^{2}}{\partial x^{1}}}(A^{1}+B^{1})

+\;{\frac {\partial {\bar {x}}^{2}}{\partial x^{2}}}(A^{2}+B^{2})

The above two equations may be rewritten more compactly as

{\bar {A}}^{\lambda }+{\bar {B}}^{\lambda }={\frac {\partial {\bar {x}}^{\lambda }}{\partial x^{\alpha }}}(A^{\alpha }+B^{\alpha })\quad \quad (n=2)

(E3)

or, using $C$ s to represent each summed component

{\bar {C}}^{\lambda }={\frac {\partial {\bar {x}}^{\lambda }}{\partial x^{\alpha }}}C^{\alpha }\quad \quad (n=2)

(E4)

Since $C^{\alpha }$ transforms according to (D3), we have established that the sum of two vectors is another vector. The same holds for tensors of higher rank.

Note in particular how (E4) may be obtained by summing (E1) and (E2) as if they were each single equations with a single term on the right, when in reality, each represents multiple equations with multiple terms on the right.

The notational system used here, developed by Ricci and Levi-Cevita about 1900, with later enhancements by Einstein, permits complex operations to be performed following a relatively simple algebraic process often termed "index juggling".^[78]^: 44 The notation automatically keeps track of whole sets of equations having many terms in each. We illustrate here with a process of multiplying tensors called "outer multiplication".

If we wish to multiply

{\bar {A}}^{\lambda }={\frac {\partial {\bar {x}}^{\lambda }}{\partial x^{\alpha }}}A^{\alpha }\quad \quad (n=2)

(E5)

by

{\bar {B}}^{\mu }={\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\beta }}}B^{\beta }\quad \quad (n=2)

(E6)

we can immediately write

{\bar {C}}^{\lambda \mu }={\frac {\partial {\bar {x}}^{\lambda }}{\partial x^{\alpha }}}{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\beta }}}C^{\alpha \beta }\quad \quad (n=2)

(E7)

In outer multiplication, each equation of (E5) is to be multiplied by each equation of (E6), so there would be four multiplications. Written in expanded form, the first equation of (E5), with $\lambda =1,$ and the first equation of (E6), with $\mu =1,$ are, respectively,

{\bar {A}}^{1}={\frac {\partial {\bar {x}}^{1}}{\partial x^{1}}}A^{1}+{\frac {\partial {\bar {x}}^{1}}{\partial x^{2}}}A^{2}\quad

and

\quad {\bar {B}}^{1}={\frac {\partial {\bar {x}}^{1}}{\partial x^{1}}}B^{1}+{\frac {\partial {\bar {x}}^{1}}{\partial x^{2}}}B^{2}

Following ordinary rules of algebra, we obtain, as the product, the following:

{\begin{aligned}{\bar {A}}^{1}{\bar {B}}^{1}&={\frac {\partial {\bar {x}}^{1}}{\partial x^{1}}}{\frac {\partial {\bar {x}}^{1}}{\partial x^{1}}}A^{1}B^{1}+{\frac {\partial {\bar {x}}^{1}}{\partial x^{2}}}{\frac {\partial {\bar {x}}^{1}}{\partial x^{1}}}A^{2}B^{1}\\&+\,{\frac {\partial {\bar {x}}^{1}}{\partial x^{1}}}{\frac {\partial {\bar {x}}^{1}}{\partial x^{2}}}A^{1}B^{2}+{\frac {\partial {\bar {x}}^{1}}{\partial x^{2}}}{\frac {\partial {\bar {x}}^{1}}{\partial x^{2}}}A^{2}B^{2}\end{aligned}}

(E8)

In like fashion, we obtain equations for ${\bar {A}}^{1}{\bar {B}}^{2},\,{\bar {A}}^{2}{\bar {B}}^{1},\,$ and ${\bar {A}}^{2}{\bar {B}}^{2}.$

To reiterate, according to the Einstein summation convention, since $\alpha$ and $\beta$ each occur twice on the right side of (E7), they must each take on all possible values to form a sum. For $\lambda =1,\mu =1,$ the terms sum to yield (E8), except that in (E7) we simplify the appearance by replacing $A^{\alpha }B^{\beta }$ with $C^{\alpha \beta }.$ In a similar fashion, we handle the other possible values of $\lambda$ and $\mu ,$ thus showing that (E7) completely represents the outer product of (E5) and (E6).^[75]^{: 163–167}

From (E7), it is evident that the outer product of two vectors is a tensor of rank two. In general, the product of two tensors of rank m and n is a tensor of rank m + n.^{[note 19]}

Covariant tensors

In Fig. 6–5, consider an object having varying density in different parts of the object. The density at any particular point is a scalar, but the change in density as we go from point to point is a directed quantity, i.e. a vector. If we designate the density at any particular point by $\psi$ , then

{\frac {\partial \psi }{\partial x^{1}}}

and

{\frac {\partial \psi }{\partial x^{2}}}

represent the partial variation of $\psi$ in the $x^{1}$ and $x^{2}$ directions. We will see that the transformation properties of this form of vector are different from those described before.^[75]^{: 167–172}

On top of the original coordinate system in Fig. 6–5, we overlay a changed coordinate system labeled with transformed coordinates. Given the unbarred coordinate components of the vector at point A, we wish to express its barred coordinate components. In other words, we wish to express

{\frac {\partial \psi }{\partial {\bar {x}}^{1}}}\,{\text{and}}\,{\frac {\partial \psi }{\partial {\bar {x}}^{2}}}

in terms of

{\frac {\partial \psi }{\partial x^{1}}}\,{\text{and}}\,{\frac {\partial \psi }{\partial x^{2}}}

The ${\bar {x}}^{1}$ and ${\bar {x}}^{2}$ coordinates of any point in the transformed coordinate system depend on both $x^{1}$ and $x^{2}$ of the nontransformed system. The transformed vector coordinates may be written as

{\frac {\partial \psi }{\partial {\bar {x}}^{1}}}=a_{11}{\frac {\partial \psi }{\partial x^{1}}}+a_{12}{\frac {\partial \psi }{\partial x^{2}}}

{\frac {\partial \psi }{\partial {\bar {x}}^{2}}}=a_{21}{\frac {\partial \psi }{\partial x^{1}}}+a_{22}{\frac {\partial \psi }{\partial x^{2}}}

where $a_{11}$ is the partial change in $x^{1}$ per change in ${\bar {x}}^{1}$ and so forth. Writing the equations out fully,

{\begin{aligned}{\frac {\partial \psi }{\partial {\bar {x}}^{1}}}={\frac {\partial \psi }{\partial x^{1}}}{\frac {\partial x^{1}}{\partial {\bar {x}}^{1}}}+{\frac {\partial \psi }{\partial x^{2}}}{\frac {\partial x^{2}}{\partial {\bar {x}}^{1}}}\\{\frac {\partial \psi }{\partial {\bar {x}}^{2}}}={\frac {\partial \psi }{\partial x^{1}}}{\frac {\partial x^{1}}{\partial {\bar {x}}^{2}}}+{\frac {\partial \psi }{\partial x^{2}}}{\frac {\partial x^{2}}{\partial {\bar {x}}^{2}}}\end{aligned}}

(F1)

As before, the above two equations may be combined using the summation convention:

{\frac {\partial \psi }{\partial {\bar {x}}^{\mu }}}={\frac {\partial \psi }{\partial x^{\sigma }}}{\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\mu }}}\quad \quad (n=2)

(F2)

Finally, using ${\overline {W}}_{\mu }$ to represent ${\frac {\partial \psi }{\partial {\bar {x}}^{\mu }}}$ and $W_{\sigma }$ to represent ${\frac {\partial \psi }{\partial x^{\sigma }}},$ we write (F2) as follows:

{\overline {W}}_{\mu }={\frac {\partial x^{\sigma }}{\partial {\bar {x}}^{\mu }}}W_{\sigma }\quad \quad (n=2)

(F3)

The transformation rule for vectors described by (F3) is different from the transformation rule for vectors described by (D3), in that the coefficient on the right in (F3) is the reciprocal of the corresponding coefficient in (D3). Equation (F3) is the mathematical definition of a covariant vector, i.e. a covariant tensor of rank one. A covariant vector is the gradient of a scalar.^[78]^: 39

A covariant tensor of rank two is defined as follows:^[75]^{: 167–172}

{\overline {W}}_{\alpha \beta }={\frac {\partial x^{\gamma }}{\partial {\bar {x}}^{\alpha }}}{\frac {\partial x^{\delta }}{\partial {\bar {x}}^{\beta }}}W_{\gamma \delta }

(F4)

Carefully compare (F3) with (D3), and (F4) with (D4).

Note that the indices of covariant tensors are subscripts, and the bars in the coefficients are in the denominators. In contrast, the indices of contravariant tensors are superscripts, and the bars in the coefficients are in the numerators.

Mixed tensors

Addition of covariant tensors can be performed in the same manner as contravariant tensors. Likewise, the outer multiplication of two covariant tensors of ranks m and n yields a covariant tensor of rank m + n. For example, the outer product of

{\bar {A}}_{\lambda }={\frac {\partial x^{\alpha }}{\partial {\bar {x}}^{\lambda }}}A_{\alpha }

and

{\bar {B}}_{\mu \nu }={\frac {\partial x^{\beta }}{\partial {\bar {x}}^{\mu }}}{\frac {\partial x^{\gamma }}{\partial {\bar {x}}^{\nu }}}B_{\beta \gamma }

is given by

{\bar {C}}_{\lambda \mu \nu }={\frac {\partial x^{\alpha }}{\partial {\bar {x}}^{\lambda }}}{\frac {\partial x^{\beta }}{\partial {\bar {x}}^{\mu }}}{\frac {\partial x^{\gamma }}{\partial {\bar {x}}^{\nu }}}C_{\alpha \beta \gamma }

On the other hand, outer multiplication of a covariant tensor of rank m by a contravariant tensor of rank n yields a product of rank m + n which has m indices of covariance and n indices of contravariance. For example the outer product of the covariant tensor

{\bar {A}}_{\lambda }={\frac {\partial x^{\alpha }}{\partial {\bar {x}}^{\lambda }}}A_{\alpha }

and the contravariant tensor

{\bar {B}}^{\mu }={\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\beta }}}B^{\beta }

is the mixed tensor^[75]^{: 173–178}

{\bar {C}}_{\lambda }^{\mu }={\frac {\partial x^{\alpha }}{\partial {\bar {x}}^{\lambda }}}{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\beta }}}C_{\alpha }^{\beta }

(G1)

Contraction

Consider the mixed tensor:

{\bar {A}}_{\gamma }^{\alpha \beta }={\frac {\partial x^{\nu }}{\partial {\bar {x}}^{\gamma }}}{\frac {\partial {\bar {x}}^{\alpha }}{\partial x^{\lambda }}}{\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\mu }}}A_{\nu }^{\lambda \mu }\quad \quad (n=2)

(H1)

This expression represents eight equations, each having eight terms on the right.

In the above, let us replace $\gamma$ by $\alpha$ , yielding

{\bar {A}}_{\alpha }^{\alpha \beta }={\frac {\partial x^{\nu }}{\partial {\bar {x}}^{\alpha }}}{\frac {\partial {\bar {x}}^{\alpha }}{\partial x^{\lambda }}}{\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\mu }}}A_{\nu }^{\lambda \mu }\quad \quad (n=2)

(H2)

On the left side, the summation convention means that we have two equations rather than eight. Moreover, the left side now has two terms rather than one.

On the right side, since $\alpha$ appears twice, the summation convention states that a sum needs to be taken over each value of $\nu$ and $\lambda$ . Note, however, that the $x\,{\text{'s}}$ are independent variables. Although functional relationships exist between the ${\bar {x}}\,{\text{'s}}$ and the $x\,{\text{'s}}$ , no such functional relationships exist among the $x\,{\text{'s}}$ themselves. What this means is that when $\nu \neq \lambda ,$ the terms drop out, since

{\frac {\partial x^{\nu }}{\partial {\bar {x}}^{\alpha }}}{\frac {\partial {\bar {x}}^{\alpha }}{\partial x^{\lambda }}}={\frac {\partial x^{\nu }}{\partial x^{\lambda }}}=0\quad \quad (\lambda \neq \nu )

On the other hand, when $\lambda =\nu ,$ we observe that

{\frac {\partial x^{\nu }}{\partial {\bar {x}}^{\alpha }}}{\frac {\partial {\bar {x}}^{\alpha }}{\partial x^{\lambda }}}={\frac {\partial x^{\lambda }}{\partial {\bar {x}}^{\alpha }}}{\frac {\partial {\bar {x}}^{\alpha }}{\partial x^{\lambda }}}=1\quad \quad (\lambda =\nu )

Equation (H2) therefore becomes

{\bar {A}}_{\alpha }^{\alpha \beta }={\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\mu }}}A_{\lambda }^{\lambda \mu }\quad \quad (n=2)

(H3)

To clarify the meaning of (H3), we expand the individual terms, noting that $\lambda$ and $\mu$ each appear twice on the right side:

{\bar {A}}_{1}^{11}+{\bar {A}}_{2}^{21}={\frac {\partial {\bar {x}}^{1}}{\partial x^{1}}}(A_{1}^{11}+A_{2}^{21})

+\;{\frac {\partial {\bar {x}}^{1}}{\partial x^{2}}}(A_{1}^{12}+A_{2}^{22})

{\bar {A}}_{1}^{12}+{\bar {A}}_{2}^{22}={\frac {\partial {\bar {x}}^{2}}{\partial x^{1}}}(A_{1}^{11}+A_{2}^{21})

+\;{\frac {\partial {\bar {x}}^{2}}{\partial x^{2}}}(A_{1}^{12}+A_{2}^{22})

In the above expressions, perform the following substitutions and apply the summation convention:

{\bar {C}}^{1}={\bar {A}}_{1}^{11}+{\bar {A}}_{2}^{21}

{\bar {C}}^{2}={\bar {A}}_{1}^{12}+{\bar {A}}_{2}^{22}

C^{1}=A_{1}^{11}+A_{2}^{21}

C^{2}=A_{1}^{12}+A_{2}^{22}

Then (H3) becomes

{\bar {C}}^{\beta }={\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\mu }}}C^{\mu }

(H4)

This process of reduction is called contraction and has reduced the starting rank 3 tensor (H1) to a tensor of rank one. In general, contraction enables construction of a tensor of rank n − 2 from one of rank n.^[75]^{: 178–183}

The general rule to contract a tensor is to set an upper index equal to a lower index and sum, yielding a tensor of reduced rank. For example, one possible contraction of $T_{\lambda \gamma }^{\alpha \beta }$ is $T_{\beta \gamma }^{\alpha \beta }=S_{\gamma }^{\alpha }$ .^[78]^: 44 Given several possible contractions, the one chosen would be dictated by the requirements of the physical problem being addressed.

If we multiply two tensors to form an outer product, and this product is a mixed tensor, contracting this mixed tensor results in an inner product. Hence, if the outer product of $A_{\alpha \beta }$ and $B^{\gamma }$ is the mixed tensor $C_{\alpha \beta }^{\gamma }\,$ , replacing $\gamma$ by $\beta$ results in the contracted tensor $D_{\alpha }$ , which is an inner product of $A_{\alpha \beta }$ and $B^{\gamma }$ .^[75]^{: 178–183} The importance of this procedure will be apparent later on.

The problem with "ordinary" differentiation

To be physically meaningful, the result of applying mathematical operations on tensors should be other tensors, since otherwise the operations lack coordinate independence. We have so far shown that addition, outer multiplication, and contraction of tensor variables do, in fact, yield tensors as their result. Ordinary differentiation, however, has issues.^[75]^{: 183–187}

Suppose we wish to compute the partial derivative of

{\bar {A}}^{\mu }={\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\sigma }}}A^{\sigma }

(I1)

with respect to ${\bar {x}}^{\nu }.$ Applying the product rule,^{[note 20]} we obtain:

{\frac {\partial {\bar {A}}^{\mu }}{\partial {\bar {x}}^{\nu }}}={\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\sigma }}}{\frac {\partial A^{\sigma }}{\partial {\bar {x}}^{\nu }}}+A^{\sigma }{\frac {\partial ^{2}{\bar {x}}^{\mu }}{\partial x^{\sigma }\partial {\bar {x}}^{\nu }}}

(I2)

The result does not match up at all with any of the tensor prototypes that we have thus far identified. This situation, however, can be partially rectified by a change of variables. Note that ${\frac {\partial A^{\sigma }}{\partial {\bar {x}}^{\nu }}}={\frac {\partial A^{\sigma }}{\partial x^{\tau }}}{\frac {\partial x^{\tau }}{\partial {\bar {x}}^{\nu }}}$

If we apply this substitution to the left term of (I2) and rearrange slightly,^{[note 21]} we obtain

{\frac {\partial {\bar {A}}^{\mu }}{\partial {\bar {x}}^{\nu }}}={\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\sigma }}}{\frac {\partial x^{\tau }}{\partial {\bar {x}}^{\nu }}}{\frac {\partial A^{\sigma }}{\partial x^{\tau }}}+{\frac {\partial ^{2}{\bar {x}}^{\mu }}{\partial x^{\sigma }\partial {\bar {x}}^{\nu }}}A^{\sigma }

(I3)

Close comparison of the left term of (I3) with other tensor prototypes presented thus far shows that the left term represents a mixed tensor of rank two. But the right term presents an issue.

For certain simple transformations, such as the rotation illustrated in Fig. 6–4, the right term vanishes, since the coefficients $\partial {\bar {x}}^{\mu }/\partial x^{\sigma }$ are constants. In such cases, (I3) will represent a tensor. In the general case, however, $\partial {\bar {x}}^{\mu }/\partial x^{\sigma }$ will not be constants, the right term will not vanish, and (I3) will not be a tensor. In general, therefore, ordinary differentiation of tensors does not represent a physically relevant operation.^[75]^{: 183–187}

The ordinary derivative of a tensor is a tensor if and only if coordinate changes are restricted to linear transformations.^[81]^: 68

We will shortly describe an operation called covariant differentiation which does always yield a tensor, and which is used in deriving the curvature tensor which plays an important role in general relativity.

The space(time) metric

As mentioned before, the expression for $ds^{2}$ is dependent both on the properties of the space(time) in question and on the coordinate system used. It turns out that all of the different expressions for $ds^{2}$ have the the common form

ds^{2}=g_{\mu \nu }dx^{\mu }dx^{\nu }

(J1)

This common form holds for all spaces and spacetimes, regardless of dimensionality.^[75]^{: 187–190}^{[note 22]}

In two dimensions, J1 may be expanded to

{\begin{aligned}ds^{2}&=g_{11}dx^{1}dx^{1}+g_{12}dx^{1}dx^{2}\\&+\,g_{21}dx^{2}dx^{1}+g_{22}dx^{2}dx^{2}\end{aligned}}

(J2)

For a Euclidean plane in Cartesian coordinates (A2), $g_{11}=1,$ $g_{12}=0,$ $g_{21}=0,$ and $g_{22}=1.$ This leads to $ds^{2}=(dx^{1})^{2}+(dx^{2})^{2}$

For polar coordinates (A4), $g_{11}=1,$ $g_{12}=0,$ $g_{21}=0,$ and $g_{22}=r^{2}.$

For oblique coordinates (A6), $g_{11}=1,$ $g_{12}=-\cos \alpha ,$ $g_{21}=-\cos \alpha ,$ and $g_{22}=1.$

For spherical coordinates (A8), $g_{11}=r^{2},$ $g_{12}=0,$ $g_{21}=0,$ and $g_{22}=R^{2}.$

Note that for each of the above, $g_{12}$ and $g_{21}$ have the same value.

In general, regardless of the dimensionality, the shape of the space(time), or the coordinate system employed,

g_{\mu \nu }=g_{\nu \mu }.

Any such set of $g\,{\text{'s}}$ form a covariant tensor of rank two. Demonstrating that the set of $g\,{\text{'s}}$ in (J1) form a tensor involves an application of the Quotient Theorem:

If the product (inner or outer) of a given quantity with a tensor of any specified type and arbitrary components is itself a tensor, then the given quantity is a tensor.^{[note 23]}

Given the Quotient theorem, demonstrating that $g_{\mu \nu }$ is a tensor is straightforward: Since $ds^{2}$ is a scalar, it is a tensor of rank zero. The product of $g_{\mu \nu }dx^{\mu }$ and $dx^{\nu }$ on the right-hand side of J1 is therefore also a tensor of rank zero. But $dx^{\nu }$ is a contravariant tensor of rank one (i.e. a vector), allowing us to deduce that $g_{\mu \nu }dx^{\mu }$ is a covariant tensor of rank one. But $dx^{\mu }$ is also a contravariant vector, demonstrating that $g_{\mu \nu }$ must be a covariant tensor of rank two.

The metric tensor $g_{\mu \nu }$ is the fundamental object of study in general relativity, since it characterizes the geometric properties of spacetime.^[75]^{: 187–190, 312–314}

Covariant derivatives of tensors

The covariant derivative discussed in this section is the natural generalization of the ordinary derivative, since it is a tensor, and since, in flat Euclidean space with Cartesian coordinates, it reduces to the ordinary derivative.^[78] The expression of the covariant derivative introduces two new symbols, (1) the contravariant metric tensor $g^{\mu \nu }$ (with raised indices), and (2) Christoffel's symbol of the second kind $\Gamma _{\mu \nu }^{\lambda }.$ ^[75]^{: 191–200}

For simplicity, we limit ourselves to two dimensions. In this environment, $g_{\mu \nu }$ will have four components, which can be arranged in a matrix: ${\begin{bmatrix}g_{11}&g_{12}\\g_{21}&g_{22}\end{bmatrix}}$

Since $g_{12}=g_{21},$ this is called a symmetric matrix, since it is symmetric with respect to the principal diagonal.

The determinant of this matrix, $|g_{\mu \nu }|,$ is often denoted simply by the letter $g\,.$

The inverse of this matrix is also symmetric, and its components transform as a contravariant tensor of rank two. The tensor represented by this matrix is $g^{\mu \nu }.$ The product of the two matrices is the identity matrix with ones along the diagonal and zeroes elsewhere. In tensor notation (note the summation upon $\lambda$ )

g^{\mu \lambda }g_{\lambda \nu }=\delta _{\lambda }^{\mu },\quad

where

\delta _{\lambda }^{\mu }

is the Kroneker delta:

$\delta _{ij}\equiv \delta _{j}^{i}\equiv \delta ^{ij}={\begin{cases}0&{\text{if }}i\neq j\\1&{\text{if }}i=j\end{cases}}$

Christoffel's symbol of the second kind is given by^{[note 24]}

\Gamma _{\mu \nu }^{\lambda }={\frac {1}{2}}g^{\lambda \alpha }\left({\frac {\partial g_{\mu \alpha }}{\partial x^{\nu }}}+{\frac {\partial g_{\nu \alpha }}{\partial x^{\mu }}}-{\frac {\partial g_{\mu \nu }}{\partial x^{\alpha }}}\right)

(K1)

Derivation of the Christoffel symbols is outside the scope of this simple introduction but may be found in most textbooks, a relatively accessible presentation being that of Grøn and Øyvind (2011).^[79]^{: 129–158} In two dimensional space, (K1) would represent eight equations. Remembering to sum over $\alpha ,$ we would have:

\Gamma _{11}^{1}={\frac {1}{2}}g^{11}\left({\frac {\partial g_{11}}{\partial x^{1}}}+{\frac {\partial g_{11}}{\partial x^{1}}}-{\frac {\partial g_{11}}{\partial x^{1}}}\right)

+\,{\frac {1}{2}}g^{12}\left({\frac {\partial g_{12}}{\partial x^{1}}}+{\frac {\partial g_{12}}{\partial x^{1}}}-{\frac {\partial g_{11}}{\partial x^{2}}}\right)

and similarly for the remaining seven values of $\Gamma _{\mu \nu }^{\lambda }.$

If $A_{\sigma }$ is a covariant tensor of rank one,^{[note 25]} its covariant derivative with respect to $x^{\tau }$ is defined as

A_{\sigma \tau }={\frac {\partial A_{\sigma }}{\partial x^{\tau }}}-\Gamma _{\sigma \tau }^{\alpha }A_{\alpha }

(K2)

$A_{\sigma \tau }$ is a covariant tensor of rank two.

If $A^{\sigma }$ is a contravariant tensor of rank one, its covariant derivative with respect to $x^{\tau }$ is defined as

A_{\tau }^{\sigma }={\frac {\partial A^{\sigma }}{\partial x^{\tau }}}+\Gamma _{\tau \epsilon }^{\sigma }A^{\epsilon }

(K3)

$A_{\tau }^{\sigma }$ is a mixed tensor of rank two.

If $A^{\sigma \tau }$ is a contravariant tensor of rank two, its covariant derivative with respect to $x^{\rho }$ is defined as

A_{\rho }^{\sigma \tau }={\frac {\partial A^{\sigma \tau }}{\partial x^{\rho }}}+\Gamma _{\rho \epsilon }^{\sigma }A^{\epsilon \tau }+\Gamma _{\rho \epsilon }^{\tau }A^{\sigma \epsilon }

(K4)

If $A_{\sigma }^{\tau }$ is a mixed tensor of rank two, its covariant derivative with respect to $x^{\rho }$ is defined as

A_{\sigma \rho }^{\tau }={\frac {\partial A_{\sigma }^{\tau }}{\partial x^{\rho }}}-\Gamma _{\rho \sigma }^{\epsilon }A_{\epsilon }^{\tau }+\Gamma _{\rho \epsilon }^{\tau }A_{\sigma }^{\epsilon }

(K5)

If $A_{\sigma \tau }$ is a covariant tensor of rank two, its covariant derivative with respect to $x^{\rho }$ is defined as

A_{\sigma \tau \rho }={\frac {\partial A_{\sigma \tau }}{\partial x^{\rho }}}-\Gamma _{\sigma \rho }^{\epsilon }A_{\epsilon \tau }-\Gamma _{\tau \rho }^{\epsilon }A_{\sigma \epsilon }

(K6)

In like fashion, we may obtain the covariant derivatives for tensors of higher ranks. In all cases, covariant differentiation leads to a tensor with one more rank of covariant character than the starting tensor.

In the special case where the $g{\text{'s}}$ are constants, as for instance when using Cartesian coordinates in a flat Euclidean plane, it is evident when looking at the definition of the Christoffel symbol (K1) that the symbols will all have value zero. In this case, (K3) becomes simply

A_{\tau }^{\sigma }={\frac {\partial A^{\sigma }}{\partial x^{\tau }}}

(K3)

In this special case, the covariant derivative is the same as the ordinary derivative.^[75]^{: 191–200}

The Riemann–Christoffel curvature tensor

Suppose that z is a function of x and y, for example z = x² + 2xy. The partial derivative of z with respect to x and y does not depend on the order of differentiation. In other words,

{\frac {\partial ^{2}z}{\partial x\partial y}}={\frac {\partial ^{2}z}{\partial y\partial x}}=2

On the other hand, order does matter in calculation of the second covariant derivative of a tensor due to the presence of Christoffel symbols.^[75]^{: 200–206}

To illustrate, we start by taking the covariant derivative of $A_{\sigma }$ with respect to $x^{\tau }$ :

A_{\sigma \tau }={\frac {\partial A_{\sigma }}{\partial x^{\tau }}}-\Gamma _{\sigma \tau }^{\alpha }A_{\alpha }

(L1)

Follow by taking the second covariant derivative with respect to $x^{\rho }$ :

A_{\sigma \tau \rho }={\frac {\partial A_{\sigma \tau }}{\partial x^{\rho }}}-\Gamma _{\sigma \rho }^{\epsilon }A_{\epsilon \tau }-\Gamma _{\tau \rho }^{\epsilon }A_{\sigma \epsilon }

(L2)

Substituting (L1) into (L2) yields

{\begin{aligned}A_{\sigma \tau \rho }&={\frac {\partial ^{2}A_{\sigma }}{\partial x^{\tau }x^{\rho }}}-\Gamma _{\sigma \tau }^{\alpha }{\frac {\partial A_{\alpha }}{\partial x^{\rho }}}-A_{\alpha }{\frac {\partial \Gamma _{\sigma \tau }^{\alpha }}{\partial x^{\rho }}}\\&-\Gamma _{\sigma \rho }^{\epsilon }{\frac {\partial A_{\epsilon }}{\partial x^{\tau }}}+\Gamma _{\sigma \rho }^{\epsilon }\Gamma _{\epsilon \tau }^{\alpha }A_{\alpha }\\&-\Gamma _{\tau \rho }^{\epsilon }{\frac {\partial A_{\sigma }}{\partial x^{\epsilon }}}+\Gamma _{\tau \rho }^{\epsilon }\Gamma _{\sigma \epsilon }^{\alpha }A_{\alpha }\end{aligned}}

(L3)

Taking the derivatives in reverse order yields

{\begin{aligned}A_{\sigma \rho \tau }&={\frac {\partial ^{2}A_{\sigma }}{\partial x^{\rho }x^{\tau }}}-\Gamma _{\sigma \rho }^{\alpha }{\frac {\partial A_{\alpha }}{\partial x^{\tau }}}-A_{\alpha }{\frac {\partial \Gamma _{\sigma \rho }^{\alpha }}{\partial x^{\tau }}}\\&-\Gamma _{\sigma \tau }^{\epsilon }{\frac {\partial A_{\epsilon }}{\partial x^{\rho }}}+\Gamma _{\sigma \tau }^{\epsilon }\Gamma _{\epsilon \rho }^{\alpha }A_{\alpha }\\&-\Gamma _{\rho \tau }^{\epsilon }{\frac {\partial A_{\sigma }}{\partial x^{\epsilon }}}+\Gamma _{\rho \tau }^{\epsilon }\Gamma _{\sigma \epsilon }^{\alpha }A_{\alpha }\end{aligned}}

(L4)

The first terms of (L3) and (L4) are equal:

{\frac {\partial ^{2}A_{\sigma }}{\partial x^{\tau }x^{\rho }}}={\frac {\partial ^{2}A_{\sigma }}{\partial x^{\rho }x^{\tau }}}

The second term of (L3) and the fourth term of (L4) are equal, since the choice of dummy symbol used for the summation makes no difference:

\Gamma _{\sigma \tau }^{\alpha }{\frac {\partial A_{\alpha }}{\partial x^{\rho }}}=\Gamma _{\sigma \tau }^{\epsilon }{\frac {\partial A_{\epsilon }}{\partial x^{\rho }}}

Likewise, the fourth term of (L3) and the second term of (L4) are equal:

\Gamma _{\sigma \rho }^{\epsilon }{\frac {\partial A_{\epsilon }}{\partial x^{\tau }}}=\Gamma _{\sigma \rho }^{\alpha }{\frac {\partial A_{\alpha }}{\partial x^{\tau }}}

The sixth and seventh terms of (L3) are equal to the sixth and seventh terms of (L4), since swapping the $\tau$ and $\rho$ leaves the value of $\Gamma _{\tau \rho }^{\epsilon }$ unchanged. This is easily seen in the definition of the Christoffel symbol (K1), remembering that $g_{\mu \nu }$ is symmetric. Likewise, the final terms of (L3) and (L4) are equal.

The third and fifth terms of (L3), however, are not equal to any of he terms of (L4). Subtracting (L4) from (L3) followed by rearrangement, we obtain

A_{\sigma \tau \rho }-A_{\sigma \rho \tau }=\left[{\frac {\partial \Gamma _{\sigma \rho }^{\alpha }}{\partial x^{\tau }}}-{\frac {\partial \Gamma _{\sigma \tau }^{\alpha }}{\partial x^{\rho }}}+\Gamma _{\sigma \rho }^{\epsilon }\Gamma _{\epsilon \tau }^{\alpha }-\Gamma _{\sigma \tau }^{\epsilon }\Gamma _{\epsilon \rho }^{\alpha }\right]A_{\alpha }

(L5)

The difference on the left-hand side of (L5) is a covariant vector of rank three. On the right-hand side of (L5), we had specified $A_{\alpha }$ as being an arbitrary covariant tensor of rank one. Since the inner product of $A_{\alpha }$ and the quantity in brackets is a covariant tensor of rank three, the Quotient Theorem tells us that the quantity in brackets must be a mixed tensor of rank four. This quantity is the Riemann-Christoffel curvature tensor:^[75]^{: 200–206}

R_{\sigma \tau \rho }^{\alpha }\equiv {\frac {\partial \Gamma _{\sigma \rho }^{\alpha }}{\partial x^{\tau }}}-{\frac {\partial \Gamma _{\sigma \tau }^{\alpha }}{\partial x^{\rho }}}+\Gamma _{\sigma \rho }^{\epsilon }\Gamma _{\epsilon \tau }^{\alpha }-\Gamma _{\sigma \tau }^{\epsilon }\Gamma _{\epsilon \rho }^{\alpha }

(L6)

Properties of the curvature tensor

The Riemann-Christoffel curvature tensor measures the failure of the second covariant derivative to commute.

If the Christoffel symbols on the right side of (L6) are expanded according to their definition in (K1), it is observed that the Riemann-Christoffel curvature tensor is an expression containing first and second derivatives of the $g\,{\text{'s}},$ which are themselves coefficients of (J1), the expression for $ds^{2}.$ ^[75]^{: 206–213}

In two dimensions, each of the indices of the curvature tensor has two possible values, so that $R_{\sigma \tau \rho }^{\alpha }$ has sixteen components. In three-space, the curvature tensor has 3⁴ or 81 components, while in the four dimensions of spacetime, the curvature tensor has 4⁴ or 256 components.

Various symmetries reduce the complexity of this expression. The first to note is that interchanging the $\tau$ and the $\rho$ of this expression merely changes its sign, so that of the sixteen possible combinations of $\tau$ and the $\rho$ , only six are independent.^[75]^{: 206–213} This may be seen as follows:

Suppose that we have sixteen quantities $a_{\alpha \beta }\;(n=4)$ arranged in a matrix: ${\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}\\a_{21}&a_{22}&a_{23}&a_{24}\\a_{31}&a_{32}&a_{33}&a_{34}\\a_{41}&a_{42}&a_{43}&a_{44}\\\end{bmatrix}}$ If we stipulate that $a_{\alpha \beta }=-a_{\beta \alpha },$ then the terms in the principal diagonal are necessarily zero, and the array becomes ${\begin{bmatrix}0&a_{12}&a_{13}&a_{14}\\-a_{12}&0&a_{23}&a_{24}\\-a_{13}&-a_{23}&0&a_{34}\\-a_{14}&-a_{24}&-a_{34}&0\\\end{bmatrix}}$ The above antisymmetric matrix has only six independent components rather than sixteen.

If, on the other hand, we had stipulated that $a_{\alpha \beta }=a_{\beta \alpha },$ the resulting symmetric matrix would have ten independent components.

The six independent combinations of $\tau$ and $\rho ,$ combined with the sixteen combinations of $\sigma$ and $\alpha$ gives 96 independent components rather than 256. Further symmetries reduce the total number of independent components from $n^{4}=256$ to ${\tfrac {1}{12}}n^{2}(n^{2}-1)=20.$ ^[76]^: 86^[78]^{: 115–117}

We had earlier shown that superficial examination of $ds^{2}$ does not reveal whether a space is flat or not, since the expression is dependent both on the properties of the space(time) in question and on the coordinate system used. The curvature tensor, however, allows us to make such a determination. If we apply $R_{\sigma \tau \rho }^{\alpha }$ to (A3), (A5), and (A7), we find its components are all zero, while if we apply it to (A9), the components are non-zero.

In the case of (A3), which applies to a Euclidean plane using ordinary Cartesian coordinates, the $g{\text{'s}}$ are constants, with $g_{11}=1,\,g_{22}=1$ with the others all zero. Hence the derivatives are all zero, the Christoffel symbols are all zero, and the components of the curvature tensor are all zero.

It would be a useful exercise for the reader to compute $R_{\sigma \tau \rho }^{\alpha }$ for (A5), which applies to a Euclidean plane using polar coordinates. Here, $g_{11}=1,\,g_{12}=g_{21}=0,\,g_{22}=(x^{1})^{2}.$

In summary,

R_{\sigma \tau \rho }^{\alpha }=0

(M1)

is a necessary and sufficient condition for the local space(time) to be flat. This holds regardless of dimensionality and the coordinate system used.^[75]^{: 206–213}

The vacuum field solution

In the development of general relativity, Einstein sought a means to relate spacetime curvature to mass and energy. However, the Riemann curvature tensor is of rank four, while the energy-momentum tensor is of rank two. Two tensors that are proportional to each other must be the same rank as well as have the same symmetries. Einstein, therefore, needed to derive a rank two tensor from the Riemann curvature tensor. (The alternative possibility, finding a rank four tensor expression of energy-momentum, makes no physical sense.) Of the three possible contractions of $R_{\sigma \tau \rho }^{\alpha }\,,$ contraction with the first subscript gives zero, while contraction with the second and third subscripts gives the same result but of opposite sign. Therefore, there was only one independent contraction of the curvature tensor that presented itself to Einstein.^[79]^{: 211–224}

Contracting (M1) with the third subscript yields the Ricci tensor, where

G_{11}=R_{111}^{1}+R_{112}^{2}+R_{113}^{3}+R_{114}^{4}=0

G_{12}=R_{121}^{1}+R_{122}^{2}+R_{123}^{3}+R_{124}^{4}=0

and so forth for each of the sixteen possible combinations of $\sigma$ and $\tau ,\,$ ultimately yielding

G_{\sigma \tau }=0\,.

(N1)

In examining (M1) before contracting it to yield (N1), we see that

G_{\sigma \tau }={\frac {\partial }{\partial x^{\tau }}}\Gamma _{\sigma \alpha }^{\alpha }-{\frac {\partial }{\partial x^{\alpha }}}\Gamma _{\sigma \tau }^{\alpha }+\Gamma _{\sigma \alpha }^{\epsilon }\Gamma _{\epsilon \tau }^{\alpha }-\Gamma _{\sigma \tau }^{\epsilon }\Gamma _{\epsilon \alpha }^{\alpha }

(N2)

From the definition of the Christoffel symbol, (N2) is revealed to be an expression containing first and second partial derivatives of the $g\,{\text{'s}}.$ Since $\sigma$ and $\tau$ may each take on four different values, (N2) represents sixteen equations. However symmetry considerations reduce this to ten equations, of which only six are independent.^[76]^: 89

Einstein proposed that (N1) should represent the vacuum field equations of general relativity, i.e. the equations that should be valid where the mass-energy density is zero.

Einstein's views on the equivalence principle had evolved significantly over the years since he first conceived of the principle in 1907. His early results in applying the equivalence principle, for example his deduction of the existence of gravitational time dilation and his early arguments on the bending of light in a gravitational field, used kinematic and dynamic analysis rather than geometric arguments. Stachel has identified Einstein's analysis of the rigid relativistic rotating disk as being key to the realization that he needed to adopt a geometric interpretation of spacetime, which he had formerly eschewed. (See Einstein's thought experiments: Non-Euclidean geometry and the rotating disk for a discussion of this point.) In later years, Einstein repeatedly stated that consideration of the rapidly rotating disk was of "decisive importance" to him because it showed that a gravitational field causes non-Euclidean arrangements of measuring rods.^[82]
The equivalence principle states that if we freefall in a gravitational field, gravity is locally eliminated. Since locally, we cannot distinguish a gravitational field from an inertial field resulting from uniform acceleration, gravitation should be regarded as an inertial force.^[76]^: 142
By 1912, Einstein had fully embraced the view that the paths of freely moving objects are determined by the geometry of the spacetime through which they travel. Freely moving objects always follow a straight line in their local inertial frames, which is to say, they always follow along the path of timelike geodesics. As indicated earlier in section Basic propositions, evidence of gravitation is observed by variation in the field rather than the field itself, as manifest in the relative accelerations of two separated particles. In Fig. 5-1, two separated particles, free-falling in the gravitational field of the Earth, exhibit tidal accelerations due to local inhomogeneities in the gravitational field such that each particle follows a different path through spacetime. The convergence or divergence of the test particles is described with the aid of the Riemann tensor.^[76]^: 142
The $g\,{\text{'s}}$ of the spacetime metric serve to quantify the shape of spacetime. In analogy with the field formulation of Newtonian gravitational theory, which we will discuss in the next section, (N1) represents a set of second-order partial differential equations for the potentials as field equations of the theory. These equations, of course, must be tensoral.^[76]^: 142

The equations of (N1) represent the simplest expression which is analogous to the field formulation of Newtonian gravitational theory (in regions of zero mass density). Predictions of this theory match up with the predictions of Newtonian gravitational theory in the low-speed, low-gravitation regime. These equations also predict additional effects that have been fully verified by observation and experiment.^[75]^{: 213–219}

The field formulation of Newtonian gravitation

Newton's law of universal gravitation is inherently non-relativistic. The most familiar expression of the law is in its action-at-a-distance form,

F=-G{\frac {m_{1}m_{2}}{r^{2}}},

(O1)

where $G$ in this case is the gravitational constant (not to be confused with the Ricci tensor), and the force is along a line connecting the two masses. The law requires that the forces between the gravitating bodies be transmitted instantaneously. Newton's law is incompatible with a finite speed of gravity. In 1805, Laplace concluded that the speed of gravitational interactions must be at least 7×10⁶ times the speed of light, otherwise the resulting orbital instabilities should long ago have caused the Earth to plunge into the Sun.^[83]^{[note 26]}

Einstein wanted to construct a theory of gravitation that adhered to relativistic principles. From his own work in 1905, he knew that Maxwell's theory of electromagnetism was consistent with special relativity. He also knew that it was Faraday's development of the field concept that led the way for Maxwell's inherently relativistic theory. Therefore, Einstein was certain that the general theory that he wanted to create would be a field theory rather than an action-at-a-distance theory.^[79]^{: 230–235}

In a field theory, changes in the field are expressed by means of differential equations. The gravitational potential $\phi$ is a function expressing the potential energy of a particle with unit mass in the gravitational field. The potential energy of a particle at position $P$ is the energy required to move the particle from an arbitrary position of zero energy to $P.$ This position of zero energy may be chosen freely. When performing calculations near the surface of the Earth, it is frequently chosen to be sea level. For celestial mechanics calculations, it is usually chosen to be from a position infinitely distant in space. The potential's value increases in the upward direction in the gravitational field.^[79]^{: 230–235}

To derive a field theory version of Newton's law, we first rearrange (O1) as follows:^[75]^{: 219–227}

{\frac {F}{m_{2}}}=-G{\frac {m_{1}}{r^{2}}}=a

On the left side of the equation, $F/m_{2}$ represents the acceleration of $m_{2}$ due to the gravitational field surrounding $m_{1}.$ Since $-Gm_{1}$ is a constant, we may rewrite the above equation as

a={\frac {C}{r^{2}}}

(O2)

Fig. 6–6 shows two axes of a three-dimensional diagram, the third $Z$ axis pointing out of the page towards the reader. Mass $m_{1}$ is at the origin, $m_{2}$ is at $P$ with coordinates $x,\,y,\,z,\,$ and $OP=r.\,$ Acceleration ${\vec {a}}$ is a vector quantity and may be split up into three components, ${\vec {a}}_{x},\,{\vec {a}}_{y},\,{\vec {a}}_{z}.\,$ It is evident that

a_{x}=-a\cdot {\frac {x}{r}},\ a_{y}=-a\cdot {\frac {y}{r}},\ a_{z}=-a\cdot {\frac {z}{r}}

Substituting in the value of $a$ from (O2), we get

a_{x}=-{\frac {Cx}{r^{3}}},\ a_{y}=-{\frac {Cy}{r^{3}}},\ a_{z}=-{\frac {Cz}{r^{3}}}

Taking the partial derivative of $a_{x}$ with respect to $x$ , we obtain

{\frac {\partial a_{x}}{\partial x}}=-Cr^{-3}+3Cxr^{-4}=\,

{\frac {-Cr^{3}+3Cxr^{2}\cdot \partial r/\partial x}{r^{6}}}

and likewise for $a_{y}$ and $a_{z}.\,$ But since $r^{2}=x^{2}+y^{2}+z^{2},$

{\frac {\partial r}{\partial x}}={\frac {x}{r}}.

Substituting this into the above equation,

{\frac {\partial a_{x}}{\partial x}}={\frac {-C(r^{2}-3x^{2})}{r^{5}}}

and likewise

{\frac {\partial a_{y}}{\partial y}}={\frac {-C(r^{2}-3y^{2})}{r^{5}}}\quad

and

\quad {\frac {\partial a_{z}}{\partial z}}={\frac {-C(r^{2}-3z^{2})}{r^{5}}}

Adding together the above equations, we obtain

{\frac {\partial a_{x}}{\partial x}}+{\frac {\partial a_{y}}{\partial y}}+{\frac {\partial a_{z}}{\partial z}}=0

(O3)

From the definition of gravitational potential, we may write

a_{x}={\frac {\partial \phi }{\partial x}},\;a_{y}={\frac {\partial \phi }{\partial y}},\;a_{z}={\frac {\partial \phi }{\partial z}}

Substituting into (O3), we obtain

{\frac {\partial ^{2}\phi }{\partial x^{2}}}+{\frac {\partial ^{2}\phi }{\partial y^{2}}}+{\frac {\partial ^{2}\phi }{\partial z^{2}}}=0

(O4)

The above field formulation of Newton's law of gravitation is known as Laplace's equation, valid for regions of zero mass density. It may be written more succinctly using the $\nabla ^{2}$ operator (pronounced "del square"):^{[note 27]}

\nabla ^{2}\phi =0

We observe in (O4) that the field formulation of Newton's law of gravitation is an equation containing second partial derivatives of the gravitational potential. By way of comparison, the vacuum solution of Einstein's field equation (N1) is a set of equations containing nothing higher than the second partial derivatives of the components of the metric tensor. Einstein's field equation expresses the equivalence principle by replacing the concept of a varying gravitational potential originating from action-at-a-distance forces, with the concept of a spacetime varying in shape.^[75]^{: 219–227}

We had noted before that each component of the Ricci tensor $G_{\sigma \tau }$ represents the sum of four components of the Riemann curvature tensor $R_{\sigma \tau \rho }^{\alpha }.$ If the components of the Riemann tensor are all zero, then spacetime is flat and the components of $G_{\sigma \tau }$ will all be zero. However, the converse is not true. If the components of $G_{\sigma \tau }$ are all zero, that does not imply that the components of the Riemann tensor need all be zero.

Even as, in Newtonian theory, $\nabla ^{2}\phi =0$ is the field equation for regions of zero mass density around gravitating bodies, so $G_{\sigma \tau }=0$ is the relativistic field equation for regions of zero mass-energy density around gravitating bodies.^[75]^{: 219–227}

Solving the vacuum field equations

The vacuum field solution of general relativity,

G_{\sigma \tau }=0

comprises six independent equations containing partial derivatives of the components of the metric tensor $g.$ To test these equations, we must use a form of the expression for $ds^{2}$ applicable to the physical situation which we are modeling and which preferably should be in a form convenient for calculation.^[75]^{: 227–237}

The classical tests for general relativity include observations of

Since the gravitational field of the Sun is very nearly spherically symmetric, a form of the expression for $ds^{2}$ which reflects this symmetry would be convenient for computation of anomalous perihelion precession and the deflection of light by the Sun. We begin by adopting spherical coordinates.^[75]^{: 227–237}

In three-dimensional Euclidean space, the expression for $ds^{2}$ in terms of spherical coordinates is

ds^{2}=dr^{2}+r^{2}d\theta ^{2}+r^{2}\sin ^{2}\theta \cdot d\phi ^{2}

as may be readily derived from $ds^{2}=(dx^{1})^{2}+(dx^{2})^{2}+(dx^{3})^{2}$ with the aid of Fig. 6–7.

The expression for flat Minkowski spacetime in four dimensions using Cartesian coordinates is

ds^{2}=-dx^{2}-dy^{2}-dz^{2}+c^{2}dt^{2}

which in spherical coordinates would be

ds^{2}=-dr^{2}-r^{2}d\theta ^{2}-r^{2}\sin ^{2}\theta \cdot d\phi ^{2}+c^{2}dt^{2}

However, general relativity involves consideration of curved spacetime. It is reasonable to assume that the expression for curved spacetime using spherical coordinates will have the form

ds^{2}=-e^{\lambda }dr^{2}

-\;e^{\mu }r^{2}(d\theta ^{2}+\sin ^{2}\theta \cdot d\phi ^{2})+e^{\nu }dt^{2}

{\text{or}}

(P1)

ds^{2}=-e^{\lambda }(dx^{1})^{2}-e^{\mu }(x^{1})^{2}((dx^{2})^{2}

+\;\sin ^{2}x^{2}\cdot (dx^{3})^{2})+e^{\nu }(dx^{4})^{2}

where $x^{1},\,x^{2},\,x^{3},\,x^{4}$ represent, respectively, the spherical coordinates $r,\,\theta ,\,\phi ,\,t,\;$ while $\lambda ,\,\mu ,\,\nu$ will be functions only of $x^{1}\equiv r.\,$ In other words, there will be no directional dependence of these functions, nor will there be any time dependence of these functions.

The requirement for spherical symmetry implies that $ds^{2}$ should not vary when $\theta$ and $\phi$ are varied, so that $\theta$ and $\phi$ only occur in the form $(d\theta ^{2}+\sin ^{2}\theta \cdot d\phi ^{2}).$ ^[76]^{: 184–186}

Furthermore, there are no product terms of the form $dx^{\sigma }dx^{\tau }$ where $\sigma \neq \tau .\;$ If terms like $dr\cdot d\theta ,\,d\theta \cdot d\phi ,\,$ or $dr\cdot d\phi$ existed, then the expression for $ds^{2}$ would be different if we turned in different directions. In particular, the metric needs to be invariant under the reflections $\theta \rightarrow \theta '=\pi -\theta$ and $\phi \rightarrow \phi '=-\phi .\,$ Likewise, since we are considering a static solution, we do not consider use of product terms such as $dr\cdot dt$ and so forth.

This eliminates all of the cross terms of the general expression for $ds^{2}$ presented in (J1). Only the squared terms $dr^{2},\,d\theta ^{2},\,d\phi ^{2},\,dt^{2}$ are used.

Functions $e^{\lambda },\,e^{\mu },\,e^{\nu }$ are inserted into the coefficients of (P1) to allow for the fact that the spacetime is curved. The form of these functions allows them to be adjusted to fit the scenario which we are modeling, and the expression of these functions as exponentials in the generalized formula is a mathematical convention that

ensures that their values are always positive, thus guaranteeing that the signature of the metric (i.e. the excess of plus signs over minus signs) is -2.^[76]^{: 184–186}
conveniently reduce in forthcoming calculations involving differentiation and the natural log.

Equation (P1) can be simplified by transforming coordinates:

e^{\mu }r^{2}\rightarrow {\bar {r}}^{2}

or, using generalized coordinates,

e^{\mu }(x^{1})^{2}\rightarrow ({\bar {x}}^{1})^{2}

By taking ${\bar {x}}^{1}$ as a new coordinate, it is possible to eliminate $e^{\mu }$ entirely. We may even drop the bar notation, since any change in $(dx^{1})^{2}$ resulting from the above substitution can be compensated for by modifying function $\lambda .$ Equation (P1) hence becomes

ds^{2}=-e^{\lambda }dr^{2}

-\;r^{2}(d\theta ^{2}+\sin ^{2}\theta \cdot d\phi ^{2})+e^{\nu }dt^{2}

{\text{or}}

(P2)

ds^{2}=-e^{\lambda }(dx^{1})^{2}-(x^{1})^{2}((dx^{2})^{2}

+\;\sin ^{2}x^{2}\cdot (dx^{3})^{2})+e^{\nu }(dx^{4})^{2}

The task now is to express $e^{\lambda }$ and $e^{\nu }$ as functions of $x^{1}.$ ^[75]^{: 227–237}

The Schwarzchild metric

From (P2), we have the following:

{\begin{aligned}&g_{11}=-e^{\lambda },\;g_{22}=-r^{2},\;g_{33}=-r^{2}\sin ^{2}\theta ,\;g_{44}=e^{\nu }\\&{\text{or}}\\&g_{11}=-e^{\lambda },\;g_{22}=-(x^{1})^{2},\;g_{33}=-(x^{1})^{2}\sin ^{2}x^{2},\;g_{44}=e^{\nu }\end{aligned}}

(Q1)

and $g_{\sigma \tau }=0$ when $\sigma \neq \tau .$

Hence the components of $g_{\mu \nu }$ form a diagonal matrix (i.e. have nonzero elements only along the principal diagonal). The determinant of $g_{\mu \nu }$ will therefore be simply equal to the product of the elements along the principal diagonal. Representing this determinant by the symbol $g,$ we have:

g=-e^{\lambda +\nu }(x^{1})^{4}\sin ^{2}x^{2}

(Q2)

Also in this case,

g^{\sigma \sigma }=1/g_{\sigma \sigma }

(meaning that $g^{11}=1/g_{11},\;g^{22}=1/g_{22}$ and so forth), and

g^{\sigma \tau }=0

when

\sigma \neq \tau .

The above relationships enable determining the coefficients $e^{\lambda }$ and $e^{\nu }$ of the metric tensor as well as enable establishing the form of the Ricci tensor $G_{\sigma \tau }$ , which represents the sixteen equations expressed by Equation (N2). In the following, these sixteen equations will be reduced to ten, then to six in the general solution. The Christoffel symbols in the solution will be categorized, and then each term will be individually addressed, ultimately leading to the Schwarzchild metric.^[75]^{: 237–255}

From sixteen equations to ten

We first show that $G_{\sigma \tau }$ is symmetric, which reduces $G_{\sigma \tau }=0$ to ten equations. Note the expression $\Gamma _{\sigma \alpha }^{\alpha }$ which is the first term on the right-hand side of (N2). From the definition of the Christoffel symbol (see (K1)),

\Gamma _{\sigma \alpha }^{\alpha }={\tfrac {1}{2}}g^{\alpha \epsilon }\left({\frac {\partial g_{\sigma \epsilon }}{\partial x^{\alpha }}}+{\frac {\partial g_{\alpha \epsilon }}{\partial x^{\sigma }}}-{\frac {\partial g_{\sigma \alpha }}{\partial x^{\epsilon }}}\right)

When the above expression is expanded using the Einstein summation convention, it is readily seen that most of the terms cancel out to yield

\Gamma _{\sigma \alpha }^{\alpha }={\tfrac {1}{2}}g^{\alpha \epsilon }{\frac {\partial g_{\sigma \epsilon }}{\partial x^{\sigma }}}

From the definition of the contravariant metric tensor $g^{\mu \nu },$ we obtain

{\tfrac {1}{2}}g^{\alpha \epsilon }{\frac {\partial g_{\sigma \epsilon }}{\partial x^{\sigma }}}={\frac {1}{2g}}{\frac {\partial g}{\partial x^{\sigma }}}

where $g$ is the determinant as described above. From basic calculus, we obtain

{\frac {1}{2g}}{\frac {\partial g}{\partial x^{\sigma }}}={\frac {\partial }{\partial x^{\sigma }}}\ln {\sqrt {-g}},

the negative of

g

being chosen so that the square root is real.

Hence,

\Gamma _{\sigma \alpha }^{\alpha }={\frac {\partial }{\partial x^{\sigma }}}\ln {\sqrt {-g}}

and by similar reasoning

\Gamma _{\epsilon \alpha }^{\alpha }={\frac {\partial }{\partial x^{\epsilon }}}\ln {\sqrt {-g}}

Substituting these into (K1), we obtain

{\begin{aligned}G_{\sigma \tau }&\equiv \Gamma _{\sigma \alpha }^{\epsilon }\Gamma _{\epsilon \tau }^{\alpha }+{\frac {\partial ^{2}}{\partial x^{\sigma }\partial x^{\tau }}}\ln {\sqrt {-g}}\\&\quad -{\frac {\partial }{\partial x^{\alpha }}}\Gamma _{\sigma \tau }^{\alpha }-\Gamma _{\sigma \tau }^{\epsilon }{\frac {\partial }{\partial x^{\epsilon }}}\ln {\sqrt {-g}}\\&=0\end{aligned}}

(Q3)

It is straightforward to demonstrate that interchange of $\sigma$ and $\tau$ in (Q3) leaves the equations unchanged. To start with, from the properties of the Christoffel symbol,

\Gamma _{\epsilon \tau }^{\alpha }=\Gamma _{\tau \epsilon }^{\alpha }

so that the two factors of the first term trade places but are otherwise unchanged ( $\epsilon$ and $\alpha$ are dummy variables that disappear upon expansion using the Einstein summation convention). The values of the second, third and fourth terms of (Q3) are likewise unaffected by swapping $\sigma$ and $\tau .$ Therefore,

G_{\sigma \tau }=G_{\tau \sigma }

so that the number of independent equations is reduced from sixteen to ten.^[75]^{: 237–255}

From ten equations to six

We refer the reader to treatments in standard textbooks such as Grøn & Næss (2011) for information on this step.^[79]^{: 217–224} The reduction of the ten equations of $G_{\mu \nu }=0$ to six is of considerable historical and physical importance, and took Einstein from 1913 to 1915 to resolve. He wished to be able to relate $G_{\mu \nu }$ to the energy-momentum tensor. Since energy and momentum are conserved, the four covariant derivatives of the energy-momentum tensor must be zero. Therefore the four covariant derivatives of the Einstein tensor must also be zero, but it was not obvious to Einstein how this should be the case. The mathematics demonstrating that this must be so had actually been developed many years earlier by Luigi Bianchi, but the Bianchi identities were unknown to Einstein in 1913. Furthermore, even if he could reduce the equations from ten to six, he still had the problem that the ten components of the metric tensor $g_{\mu \nu }$ would be underdetermined, since he would have only six equations to work with. It was not until the fall of 1915 that Einstein realized that he had a four-fold freedom in the choice of metric tensor, now called a gauge invariance, that reduced the ten $g\,{\text{'s}}$ to six, so that the number of unknowns would match the number of equations that he had available.^[75]^: 334

Categorizing the Christoffel symbols in the Ricci tensor

The Christoffel symbols in the expression for $G_{\sigma \tau }$ presented in (Q3) are highly degenerate, and over two hundred terms will drop out in the following analysis.^[75]^{: 237–255}

To accomplish this simplification, we first need to classify the Christoffel symbols in (Q3). We distinguish four classes of symbol:

Case A: Those where all the Greek letters are alike, i.e. $\Gamma _{\sigma \sigma }^{\sigma }$
Case B: Those of form $\Gamma _{\sigma \sigma }^{\tau }$
Case C: Those of form $\Gamma _{\sigma \tau }^{\tau }=\Gamma _{\tau \sigma }^{\tau }$
Case D: Those where the Greek letters are all different, i.e. $\Gamma _{\sigma \tau }^{\rho }$

According to the definition of the Christoffel symbol (K1),

\Gamma _{\sigma \sigma }^{\sigma }={\tfrac {1}{2}}g^{\sigma \alpha }\left({\frac {\partial g_{\sigma \alpha }}{\partial x^{\sigma }}}+{\frac {\partial g_{\sigma \alpha }}{\partial x^{\sigma }}}-{\frac {\partial g_{\sigma \sigma }}{\partial x^{\alpha }}}\right)

We had previously noted that $g_{\sigma \tau }=0$ when the indices are not alike. The $g\,{\text{'s}}$ non-zero only when the indices are the same. Furthermore, $g^{\sigma \sigma }=1/g_{\sigma \sigma }.$ We use these facts to simplify the above equation:

\Gamma _{\sigma \sigma }^{\sigma }={\frac {1}{2g_{\sigma \sigma }}}\left({\frac {\partial g_{\sigma \sigma }}{\partial x^{\sigma }}}+{\frac {\partial g_{\sigma \sigma }}{\partial x^{\sigma }}}-{\frac {\partial g_{\sigma \sigma }}{\partial x^{\sigma }}}\right)

Two terms cancel, so that

\Gamma _{\sigma \sigma }^{\sigma }={\frac {1}{2g_{\sigma \sigma }}}{\frac {\partial g_{\sigma \sigma }}{\partial x^{\sigma }}}

which yields, from basic calculus,

Case A:

\Gamma _{\sigma \sigma }^{\sigma }={\frac {1}{2}}{\frac {\partial }{\partial x^{\sigma }}}\ln g_{\sigma \sigma }

One handles the second case in similar fashion:

\Gamma _{\sigma \sigma }^{\tau }={\tfrac {1}{2}}g^{\tau \alpha }\left({\frac {\partial g_{\sigma \alpha }}{\partial x^{\sigma }}}+{\frac {\partial g_{\sigma \alpha }}{\partial x^{\sigma }}}-{\frac {\partial g_{\sigma \sigma }}{\partial x^{\alpha }}}\right)

Here, $g^{\tau \alpha }$ is non-zero only when $\alpha =\tau .$ This case is distinguished from the first case because $\tau \neq \sigma ,$ so that the first two terms within the parentheses are zero. Hence,

\Gamma _{\sigma \sigma }^{\tau }=-{\tfrac {1}{2}}g^{\tau \tau }{\frac {\partial g_{\sigma \sigma }}{\partial x^{\tau }}}

which yields

Case B:

\Gamma _{\sigma \sigma }^{\tau }=-{\frac {1}{2g_{\tau \tau }}}{\frac {\partial g_{\sigma \sigma }}{\partial x^{\tau }}}

Likewise,

Case C:

\Gamma _{\sigma \tau }^{\tau }=\Gamma _{\tau \sigma }^{\tau }={\frac {1}{2}}{\frac {\partial }{\partial x^{\sigma }}}\ln g_{\tau \tau }

Case D:

\Gamma _{\sigma \tau }^{\rho }=0

Term-by-term analysis of Case A

For $\sigma =1,$ and remembering the relationships in (Q1),

\Gamma _{11}^{1}={\frac {1}{2}}{\frac {\partial }{\partial x^{1}}}\ln g_{11}=

{\frac {1}{2}}{\frac {\partial }{\partial r}}\ln(-e^{\lambda })

Then

\Gamma _{11}^{1}={\frac {1}{2}}{\frac {-e^{\lambda }}{-e^{\lambda }}}{\frac {\partial \lambda }{\partial r}}=

{\frac {1}{2}}{\frac {\partial \lambda }{\partial r}}={\tfrac {1}{2}}\lambda '\,,

where $\lambda '$ represents $\partial \lambda /\partial x^{1}$ or $\partial \lambda /\partial r\,.$

For $\sigma =2$

\Gamma _{22}^{2}={\frac {1}{2}}{\frac {\partial }{\partial x^{2}}}\ln g_{22}=

{\frac {1}{2}}{\frac {\partial }{\partial x^{2}}}\ln(-x^{1})^{2}={\frac {1}{2}}{\frac {\partial }{\partial \theta }}\ln(-r^{2})=0\,,

since $r$ and $\theta$ are independent variables.

For $\sigma =3$ and $\sigma =4,$ we have:

\Gamma _{33}^{3}=\Gamma _{44}^{4}=0\,.

Term-by-term analysis of Case B

Let us first look at $\sigma =1,\,\tau =2\,:$

\Gamma _{11}^{2}=-{\frac {1}{2g_{22}}}{\frac {\partial }{\partial x^{2}}}g_{11}=-{\frac {1}{2g_{22}}}{\frac {\partial }{\partial x^{2}}}(-e^{\lambda })

Since $\lambda$ was defined as being a function of $x^{1}\equiv r$ only, the partial with respect to $x^{2}\equiv \theta$ is equal to zero,

\Gamma _{11}^{2}=0.

In like manner, we can work through all of the others through this case.

Complete list of non-zero Christoffel symbols in $G_{\sigma \tau }$

In all, there are 4 specific examples of Case A,
$4\cdot 3=12$ combinations of $\sigma$ and $\tau$ for Case B,
$4\cdot 3=12$ combinations of $\sigma$ and $\tau$ for Case C,
and $(4\cdot 3\cdot 2)/2=12$ combinations of $\sigma ,\,\tau ,\,\rho$ for Case D (since the value of the Christoffel symbol is unchanged when the two lower indices are swapped).

Hence, there are 40 distinct combinations, 31 of which reduce to zero. The complete list of non-zero Christoffel symbols in $G_{\sigma \tau }$ is:^[75]^{: 237–255}

\left.{\begin{aligned}&\Gamma _{11}^{1}={\tfrac {1}{2}}\lambda '\\&\Gamma _{12}^{2}=\Gamma _{21}^{2}={\frac {1}{r}}\\&\Gamma _{13}^{3}=\Gamma _{31}^{3}={\frac {1}{r}}\\&\Gamma _{14}^{4}=\Gamma _{41}^{4}={\tfrac {1}{2}}\nu '\\&\Gamma _{22}^{1}=-re^{-\lambda }\\&\Gamma _{23}^{3}=\cot \theta \\&\Gamma _{33}^{1}=-r\sin ^{2}\theta \cdot e^{-\lambda }\\&\Gamma _{33}^{2}=-\sin \theta \cdot \cos \theta \\&\Gamma _{44}^{1}={\tfrac {1}{2}}e^{\nu -\lambda }\cdot \nu '\end{aligned}}\right\}

(Q4)

where $\nu '\equiv {\frac {\partial \nu }{\partial x^{1}}}\equiv {\frac {\partial \nu }{\partial r}}$ After dropping all of the (over 200) zero terms from (Q3), there remain only five equations with a much reduced number of terms. Here are the remaining equations of $G_{\sigma \tau }=0$ after the zero terms have been eliminated:^[75]^{: 237–255} ${\begin{aligned}G_{11}=&\;0\\=&\;\Gamma _{11}^{1}\Gamma _{11}^{1}+\Gamma _{12}^{2}\Gamma _{21}^{2}+\Gamma _{13}^{3}\Gamma _{31}^{3}+\Gamma _{14}^{4}\Gamma _{41}^{4}\\&-{\frac {\partial }{\partial x^{1}}}\Gamma _{11}^{1}+{\frac {\partial ^{2}}{\partial (x^{1})^{2}}}\ln {\sqrt {-g}}\\&-\Gamma _{11}^{1}{\frac {\partial }{\partial x^{1}}}\ln {\sqrt {-g}}\end{aligned}}$

${\begin{aligned}G_{22}=&\;0\\=&\;2\,\Gamma _{22}^{1}\Gamma _{12}^{2}+\Gamma _{23}^{3}\Gamma _{23}^{3}\\&-{\frac {\partial }{\partial x^{1}}}\Gamma _{22}^{1}+{\frac {\partial ^{2}}{\partial (x^{2})^{2}}}\ln {\sqrt {-g}}\\&-\Gamma _{22}^{1}{\frac {\partial }{\partial x^{1}}}\ln {\sqrt {-g}}\end{aligned}}$

${\begin{aligned}G_{33}=&\;0\\=&\;2\,\Gamma _{33}^{1}\Gamma _{13}^{3}+2\,\Gamma _{33}^{2}\Gamma _{23}^{3}\\&-\Gamma _{33}^{1}{\frac {\partial }{\partial x^{1}}}\ln {\sqrt {-g}}\\&-\Gamma _{33}^{2}{\frac {\partial }{\partial x^{2}}}\ln {\sqrt {-g}}\end{aligned}}$

${\begin{aligned}G_{44}=&\;0\\=&\;2\,\Gamma _{44}^{1}\Gamma _{14}^{4}-{\frac {\partial }{\partial x^{1}}}\Gamma _{44}^{1}\\&-\Gamma _{44}^{1}{\frac {\partial }{\partial x^{1}}}\ln {\sqrt {-g}}\end{aligned}}$

${\begin{aligned}G_{12}=&\;0\\=&\,\Gamma _{13}^{3}\Gamma _{23}^{3}-\Gamma _{12}^{2}{\frac {\partial }{\partial x^{2}}}\ln {\sqrt {-g}}\end{aligned}}$

We now substitute into the above five equations the values from (Q4) and the value of $g$ from (Q2):^[75]^{: 237–255}

${\begin{aligned}G_{11}=&\;0\\=&\;{\tfrac {1}{4}}\lambda '^{2}+{\frac {1}{r^{2}}}+{\frac {1}{r^{2}}}+{\tfrac {1}{4}}\nu '^{2}-{\tfrac {1}{2}}\lambda ''\\&\;+\left({\tfrac {1}{2}}\lambda ''+{\tfrac {1}{2}}\nu ''-{\frac {2}{r^{2}}}\right)-{\tfrac {1}{2}}\lambda '\left({\tfrac {1}{2}}\lambda '+{\tfrac {1}{2}}\nu '+{\frac {2}{r}}\right)\\=&\;{\tfrac {1}{4}}\nu '^{2}+{\tfrac {1}{2}}\nu ''-{\tfrac {1}{4}}\lambda '\nu '-{\frac {\lambda '}{r}}\end{aligned}}$

${\begin{aligned}G_{22}=&\;0\\=&\;e^{-\lambda }\left[1+{\tfrac {1}{2}}r\left(\nu '-\lambda '\right)\right]-1\end{aligned}}$

${\begin{aligned}G_{33}=&\;0\\=&\;\sin ^{2}\theta \cdot e^{-\lambda }\left[1+{\tfrac {1}{2}}r\left(\nu '-\lambda '\right)\right]-\sin ^{2}\theta \end{aligned}}$

${\begin{aligned}G_{44}=&\;0\\=&\;e^{\nu -\lambda }\left(-{\tfrac {1}{2}}\nu ''+{\tfrac {1}{4}}\lambda '\nu '-{\tfrac {1}{4}}\nu '^{2}-{\frac {\nu '}{r}}\right)\end{aligned}}$

where $\lambda ''={\frac {\partial ^{2}\lambda }{\partial r^{2}}}\,$ and $\,\nu ''={\frac {\partial ^{2}\nu }{\partial r^{2}}}$ ^{[note 28]}

On the other hand,

G_{12}={\frac {1}{r}}\cot \theta -{\frac {1}{r}}\cot \theta

which is identically zero and is therefore eliminated, leaving four equations.

Also note that the expression for $G_{33}$ contains the expression for $G_{22}.$ The two equations are not independent, so we are left with only three independent equations.

Solving for $e^{\lambda }$ and $e^{\nu }$ : The Schwarzschild metric

If we divide $G_{44}$ by $e^{\nu -\lambda }$ and add to $G_{11},$ we get

\lambda '=-\nu '

(Q5)

Integrating (Q5) yields $\,\lambda =-\nu +C\,$ where $C$ is a constant of integration. The value of the constant can be found by noting the following boundary condition on (P2): At points infinitely distant from gravitating masses, spacetime is flat so that the coefficients $e^{\lambda }$ and $e^{\nu }$ of $dr^{2}$ and $dt^{2}$ are both equal to one, i.e.

{\begin{aligned}&ds^{2}=-dr^{2}-r^{2}(d\theta ^{2}+\sin ^{2}\theta \cdot d\phi ^{2})+dt^{2}\\&{\text{or}}\\&ds^{2}=-(dx^{1})^{2}-(x^{1})^{2}((dx^{2})^{2}+\sin ^{2}x^{2}\cdot (dx^{3})^{2})+(dx^{4})^{2}\end{aligned}}

(Q6)

Infinitely distant from gravitating masses, therefore, $\lambda =-\nu =0$ and so $C$ must be zero.^[75]^{: 237–255} Hence,

\lambda =-\nu

(Q7)

Substituting (Q5) and (Q7) into the expression for $G_{22}$ above yields ${\begin{aligned}G_{22}&=0\\&=e^{\nu }(1+r\nu ')-1\end{aligned}}$

which informs us that

e^{\nu }(1+r\nu ')=1

(Q8)

Let $\;\gamma =e^{\nu }\,$ which implies $\,\gamma '=e^{\nu }\nu '.\,$ Substituting into (Q8) and rearranging, we get the separable differential equation $\,\gamma +r\gamma '=1\,$ which yields

\gamma =1-{\frac {2m}{r}}

(Q9)

where $2m$ is a constant of integration expressed as such for reasons that will be discussed later on.^{[note 29]}

We have thus determined $e^{\lambda }$ and $e^{\nu }$

e^{\nu }\;=\;1/e^{\lambda }\;=\;\gamma \;=\;

1-{\frac {2m}{r}}\;=\;1-{\frac {2m}{x^{1}}}

Equation (P2) therefore becomes

ds^{2}=-\gamma ^{-1}dr^{2}

-\;r^{2}(d\theta ^{2}+\sin ^{2}\theta \cdot d\phi ^{2})+\gamma dt^{2}

{\text{or}}

(Q10)

ds^{2}=-(1-{\frac {2m}{r}})^{-1}dr^{2}

-\;r^{2}(d\theta ^{2}+\sin ^{2}\theta \cdot d\phi ^{2})+(1-{\frac {2m}{r}})dt^{2}

This is the famous Schwarzschild metric.^[75]^{: 237–255}

Anomalous perihelion precession of Mercury

Movement along geodesics

According to Newton's laws of motion, a planet orbiting the Sun would move in a straight line except for being pulled off course by the Sun's gravity. According to general relativity, there is no such thing as gravitational force. Rather, as discussed in section Basic propositions, a planet orbiting the Sun continuously follows the local "nearest thing to a straight line", which is to say, it follows a geodesic path.^[75]^{: 255–265}

Finding the equation of a geodesic requires knowing something about the calculus of variations, which is outside the scope of the typical undergraduate math curriculum, so we will not go into details of the analysis.^{[note 30]}

Determining the straightest path between two points resembles the task of finding the maximum or minimum of a function. In ordinary calculus, given the function $y=f(x),\,$ an "extremum" or "stationary point" may be found wherever the derivative of the function is zero.

In the calculus of variations, we seek to minimize the value of the functional between the start and end points. In the example shown in Fig. 6–8, this is by finding the function for which

\delta \int _{A}^{B}ds=0

where $\delta$ is the variation and the integral of $ds$ is the world-line.

Skipping the details of the derivation, the general formula for the equation of a geodesic is

{\frac {d^{2}x^{\sigma }}{ds^{2}}}+\Gamma _{\alpha \beta }^{\sigma }{\frac {dx^{\alpha }}{ds}}{\frac {dx^{\beta }}{ds}}=0

(R1)

valid for all dimensionalities and shapes of space(time).

Let us consider a flat, three dimensional Euclidean space using Cartesian coordinates. For such a space,

g_{11}=g_{22}=g_{33}=1\,

and

g_{\mu \nu }=0\,

for

\mu \neq \nu

The derivatives of the $g\,{\text{'s}}$ in the Christoffel symbol (K1) are all zero, so (R1) becomes

{\frac {d^{2}x^{\sigma }}{ds^{2}}}=0\quad \quad (n=3)

(R2)

After replacing $ds$ by the proper time $dt$ , the time along the timelike world line (the time experienced by the moving object), and expanding R2, we get

{\frac {d^{2}x^{1}}{dt^{2}}}=0,\quad {\frac {d^{2}x^{2}}{dt^{2}}}=0,\quad {\frac {d^{2}x^{3}}{dt^{2}}}=0

(R3)

which is to say, an object freely moving in Euclidean three-space travels with unaccelerated motion along a straight line.^[75]^{: 255–265}

Orbital motion: Stability of the orbital plane

Equation (R1) is a general expression for the geodesic. To apply it to the gravitational field around the Sun, the $g\,{\text{'s}}$ in the Christoffel symbols must be replaced with those specific to the Schwarzschild metric.^[75]^{: 266–268}

Equations (Q4) present the values of $\Gamma _{\alpha \beta }^{\sigma }$ in terms of $\lambda ,\,\nu ,\,r,\,\theta$ while (Q7) allows simplification of the expression to terms of $\nu ,\,r,\,\theta .\,$ Since $e^{\nu }=\gamma$ and (Q9) allows us to express $\gamma$ in terms of $r$ , we can thus express $\Gamma _{\alpha \beta }^{\sigma }$ in terms of $r$ and $\theta .$

Remember that (R1) is actually four equations. In particular, $x^{\sigma }$ for $\sigma =2$ corresponds to $\theta$ in Fig. 6-7. Suppose we launched an object into orbit around the Sun with $\theta =\pi /2$ and an initial velocity in the $xy$ plane? How would the object subsequently behave? Equation (R1) for $x^{2}\equiv \theta$ becomes

{\frac {d^{2}\theta }{ds^{2}}}+\Gamma _{\alpha \beta }^{2}{\frac {dx^{\alpha }}{ds}}{\frac {dx^{\beta }}{ds}}=0

(R4)

From (Q7), we know that the non-zero Christoffel symbols for $\sigma =2$ are

\Gamma _{12}^{2}=\Gamma _{21}^{2}={\frac {1}{r}}

and

\Gamma _{33}^{2}=-\sin \theta \cdot \cos \theta

so that in summing (R4) over all values of $\alpha$ and $\beta ,$ we get

{\frac {d^{2}\theta }{ds^{2}}}+{\frac {2}{r}}{\frac {dr}{ds}}{\frac {d\theta }{ds}}-\sin \theta \cdot \cos \theta \left({\frac {d\phi }{ds}}\right)^{2}=0

(R5)

Since we stipulated an initial $\theta =\pi /2$ and an initial velocity in the $xy$ plane, $\cos \theta =0$ and $d\theta /ds=0$ so that (R5) becomes

{\frac {d^{2}\theta }{ds^{2}}}=0

(R6)

In other words, a planet launched into orbit around the Sun remains in orbit around the same plane in which it was launched, the same as in Newtonian physics.^[75]^{: 266–268}

Orbital motion: Modified Keplerian ellipses

Starting with (R1), we explore the behavior of the other variables of the geodesic equation applied to the Schwarzschild metric:^[75]^{: 268–272}

For $\sigma =1,$ (R1) becomes

{\frac {d^{2}x^{1}}{ds^{2}}}+\Gamma _{11}^{1}\left({\frac {dx^{1}}{ds}}\right)^{2}+\Gamma _{22}^{1}\left({\frac {dx^{2}}{ds}}\right)^{2}

+\;\Gamma _{33}^{1}\left({\frac {dx^{3}}{ds}}\right)^{2}+\Gamma _{44}^{1}\left({\frac {dx^{4}}{ds}}\right)^{2}=0

or

{\frac {d^{2}r}{ds^{2}}}+{\tfrac {1}{2}}\lambda '\left({\frac {dr}{ds}}\right)^{2}-re^{-\lambda }\left({\frac {d\theta }{ds}}\right)^{2}

-\;r\cdot \sin ^{2}\theta \cdot e^{-\lambda }\left({\frac {d\phi }{ds}}\right)^{2}+{\tfrac {1}{2}}e^{\nu -\lambda }\nu '\left({\frac {dt}{ds}}\right)^{2}=0

Since we have stipulated that $\theta =\pi /2,\;$ $d\theta /ds=0\,$ and $\,\sin \theta =1.$ The above equation therefore becomes

{\frac {d^{2}r}{ds^{2}}}+{\tfrac {1}{2}}\lambda '\left({\frac {dr}{ds}}\right)^{2}-re^{-\lambda }\left({\frac {d\phi }{ds}}\right)^{2}+{\tfrac {1}{2}}e^{\nu -\lambda }\nu '\left({\frac {dt}{ds}}\right)^{2}=0

(R7)

Likewise, for $\sigma =3\,$ and $\,\sigma =4,\,$ we get

{\frac {d^{2}\phi }{ds^{2}}}+{\frac {2}{r}}{\frac {dr}{ds}}{\frac {d\phi }{ds}}=0

(R8)

{\frac {d^{2}t}{ds^{2}}}+\nu '{\frac {dr}{ds}}{\frac {dt}{ds}}=0

(R9)

(Q10), (R7), (R8), and (R9) may be combined to get:^[75]^{: 335–336}^[76]^{: 195–196}

\left.{\begin{aligned}&{\frac {d^{2}u}{d\phi ^{2}}}+u={\frac {m}{h^{2}}}+3mu^{2}\\&r^{2}{\frac {d\phi }{ds}}=h\end{aligned}}\right\}

(R10)

where $m$ and $h$ are constants of integration and $u=1/r.$

The equations above are those of an object in orbit around a central mass. The second of the two equations is essentially a statement of the conservation of angular momentum. The first of the two equations is expressed in this form so that it may be compared with the Binet equation, devised by Jacques Binet in the 1800s while exploring the shapes of orbits under alternative force laws.

For an inverse square law, the Binet equation predicts, in agreement with Newton, that orbits are conic sections.^[75]^{: 336–338} Given a Newtonian inverse square law, the equations of motion are:

\left.{\begin{aligned}&{\frac {d^{2}u}{d\phi ^{2}}}+u={\frac {m}{h^{2}}}\\&r^{2}{\frac {d\phi }{dt}}=h\end{aligned}}\right\}

(R11)

where $m$ is the mass of the Sun, $r$ is the orbital radius, and $d\phi /dt$ is the angular velocity of the planet.

The relativistic equations for orbital motion (R10) are observed to be nearly identical to the Newtonian equations (R11) except for the presence of $3mu^{2}$ in the relativistic equations and the use of $ds$ rather than $dt.$

The Binet equation provides the physical meaning of $m,$ which we had introduced as an arbitrary constant of integration in the derivation of the Schwarzschild metric in (Q9).^[75]^{: 268–272}

Orbital motion: Anomalous precession

The presence of the term $3mu^{2}$ in (R10) means that the orbit does not form a closed loop, but rather shifts slightly with each revolution, as illustrated (in much exaggerated form) in Fig. 6–9.^[75]^{: 272–276}

Now in fact, there are a number of effects in the Solar System that cause the perihelia of planets to deviate from closed Keplerian ellipses even in the absence of relativity. Newtonian theory predicts closed ellipses only for an isolated two-body system. The presence of other planets perturb each others' orbits, so that Mercury's orbit, for instance, would precess by slightly over 532 arcsec/century due to these Newtonian effects.^[84]

In 1859, Urbain Le Verrier, after extensive extensive analysis of historical data on timed transits of Mercury over the Sun's disk from 1697 to 1848, concluded that there was a significant excess deviation of Mercury's orbit from the precession predicted by these Newtonian effects amounting to 38 arcseconds/century (This estimate was later refined to 43 arcseconds/century by Simon Newcomb in 1882). Over the next half-century, extensive observations definitively ruled out the hypothetical planet Vulcan proposed by Le Verrier as orbiting between Mercury and the Sun that might account for this discrepancy.

Starting from (R10), the excess angular advance of Mercury's perihelion per orbit may be calculated:^[75]^{: 338–341}^[76]^{: 195–198}

\Delta \phi _{orbit}={\frac {6\pi m}{a(1-e^{2})}}={\frac {6\pi GM/c^{2}}{a(1-e^{2})}}\;,

(R12)

The first equality is in relativistic units, while the second equality is in MKS units. In the second equality, we replace $m,$ the geometric mass (units of length) with M, the mass in kilograms.

G

is the gravitational constant (6.672 × 10^-11 m³/kg-s²)

M

is the mass of the Sun (1.99 × 10³⁰ kg)

c

is the speed of light (2.998 × 10⁸ m/s)

a

is Mercury's perihelion (5.791 × 10¹⁰ m)

e

is Mercury's orbital eccentricity (0.20563)

We find that

\Delta \phi _{orbit}=5.021\times 10^{-7}{\text{radian}}

which works out to 43 arcsec/century.^[75]^{: 338–341}^[76]^{: 195–198}

Deflection of light in a gravitational field

The most famous of the early tests of general relativity was the measurement of the gravitational deflection of starlight passing near the Sun. As noted before, anything moving freely in spacetime travels along the path of a geodesic. This includes light.

Consider Fig. 6–10. Line $AE$ represents the straight-line path of a ray of light in the absence of any large mass along its path. If the ray passes near the Sun, however, its path is deflected so that it follows the curved line $AF,\,$ which we illustrate as just grazing the Sun of radius $R.\,$ An observer situated at $F$ sees the star as apparently being at position $B$ rather than at its true position $A.$ The angle $\alpha$ is the angle between the true position of the star and its apparent position.^[75]^{: 276–289}

We have learned above, in the Spacetime interval section of this article, that the interval between two events on the world line of a particle moving at the speed of light is zero. Equations (R10) present the geodesic equation (R4) applied to the Schwarzschild metric (Q10). Substituting $\,ds=0\,$ in the second equation of (R10) gives $\,h=\infty ,\,$ which results in the first equation of (R10) becoming

{\frac {d^{2}u}{d\phi ^{2}}}+u=3mu^{2}

which is hence a differential equation describing the path of light passing by a massive spherical object. Solving this differential equation yields, in Cartesian coordinates:^[75]^{: 341–342}^[76]^{: 199–201}

x=R-{\frac {m}{R}}{\frac {x^{2}+2y^{2}}{\sqrt {x^{2}+y^{2}}}}

Given $\alpha$ a very small angle, the asymptotes of this curve are:

x=R\pm 2y{\frac {m}{R}},

where $m,\,$ in relativistic units, is a length.

The angle $\alpha$ may be calculated from the slopes of the asymptotes:

\tan \alpha ={\frac {4Rm}{R^{2}-4m^{2}}}

(S1)

which for very small $\,\alpha \,$ and $\,m\ll R\,$ becomes

\alpha ={\frac {4m}{R}}

(S2)

Plugging in $R=6.955\times 10^{5}{\text{km}}$ and $m=1.477\,{\text{km}},\,$ we get

\alpha =8.494\times 10^{-6}{\text{rad}}=1.75\,{\text{arcsec}}

The earliest measurement of the gravitational deflection of light, the 1919 Eddington experiment, established the validity of this figure to within broad limits. Modern measurements have validated the accuracy of this prediction to the 0.03% level.^[85]

Gravitational redshift

The third of the classical tests of relativity is the prediction of gravitational red shift. This was initially thought to represent an important test of general relativity because the Schwarzschild solution was employed in its derivation. However, as demonstrated above in the section Curvature of time, red shift is predicted by any theory of gravitation that is consistent with the equivalence principle. This includes Newtonian gravitation.^[76]^{: 201–204}

The derivation presented in Curvature of time uses kinematic arguments and does not make use of the field equations. Nevertheless, it is instructive to compare the kinematic arguments presented earlier with the more geometric approach accorded by use of the Schwarzschild solution.

Let $ds$ represent the invariant proper time of the period (i.e. inverse frequency) of some well-defined spectral line of an element. We know from special relativity that although observers in different frames may measure different $dx,\,dy,\,dz,\,dt$ for an interval, that the interval does not change with change of frame. Likewise the proper time of the period should not change with position in a gravitational potential field. Assume that a distant observer is at rest relative to an atom at the surface of the Sun as it emits light. In the Schwarzschild solution (Q10), we may write $dr=d\theta =d\phi =0,$ leaving $dt$ as the only non-zero term. The Schwarzschild solution reduces to

ds={\sqrt {1-{\frac {2m}{r}}}}dt

If $m\ll r$ ,

dt=(1+m/r)ds

(T1)

Plugging in the values for the Sun's geometric mass and radius, we conclude that the distant observer should observe the light emitted by the atom as being redshifted by a factor $1+1.477/695500=1.00000212\,.$ ^[75]^{: 289–299}

This is an extremely small factor of redshift, and confirmation took many years. See Gravitational redshift and time dilation for details.

Technical topics

Is spacetime really curved?

In Poincaré's conventionalist views, the essential criteria according to which one should select a Euclidean versus non-Euclidean geometry would be economy and simplicity. A realist would say that Einstein discovered spacetime to be non-Euclidean. A conventionalist would say that Einstein merely found it more convenient to use non-Euclidean geometry. The conventionalist would maintain that Einstein's analysis said nothing about what the geometry of spacetime really is.^[86]

Such being said,

1. Is it possible to represent general relativity in terms of flat spacetime?

2. Are there any situations where a flat spacetime interpretation of general relativity may be more convenient than the usual curved spacetime interpretation?

In response to the first question, a number of authors including Deser, Grishchuk, Rosen, Weinberg, etc. have provided various formulations of gravitation as a field in a flat manifold. Those theories are variously called "bimetric gravity", the "field-theoretical approach to general relativity", and so forth.^[87]^[88]^[89]^[90] Kip Thorne has provided a popular review of these theories.^[91]^{: 397–403}

The flat spacetime paradigm posits that matter creates a gravitational field that causes rulers to shrink when they are turned from circumferential orientation to radial, and that causes the ticking rates of clocks to dilate. The flat spacetime paradigm is fully equivalent to the curved spacetime paradigm in that they both represent the same physical phenomena. However, their mathematical formulations are entirely different. Working physicists routinely switch between using curved and flat spacetime techniques depending on the requirements of the problem. The flat spacetime paradigm turns out to be especially convenient when performing approximate calculations in weak fields. Hence, flat spacetime techniques will be used when solving gravitational wave problems, while curved spacetime techniques will be used in the analysis of black holes.^[91]^{: 397–403}

Asymptotic symmetries

The spacetime symmetry group for Special Relativity is the Poincaré group, which is a ten-dimensional group of three Lorentz boosts, three rotations, and four spacetime translations. It is logical to ask what symmetries if any might apply in General Relativity. A tractable case might be to consider the symmetries of spacetime as seen by observers located far away from all sources of the gravitational field. The naive expectation for asymptotically flat spacetime symmetries might be simply to extend and reproduce the symmetries of flat spacetime of special relativity, viz., the Poincaré group.

In 1962 Hermann Bondi, M. G. van der Burg, A. W. Metzner^[92] and Rainer K. Sachs^[93] addressed this asymptotic symmetry problem in order to investigate the flow of energy at infinity due to propagating gravitational waves. Their first step was to decide on some physically sensible boundary conditions to place on the gravitational field at light-like infinity to characterize what it means to say a metric is asymptotically flat, making no a priori assumptions about the nature of the asymptotic symmetry group — not even the assumption that such a group exists. Then after designing what they considered to be the most sensible boundary conditions, they investigated the nature of the resulting asymptotic symmetry transformations that leave invariant the form of the boundary conditions appropriate for asymptotically flat gravitational fields. What they found was that the asymptotic symmetry transformations actually do form a group and the structure of this group does not depend on the particular gravitational field that happens to be present. This means that, as expected, one can separate the kinematics of spacetime from the dynamics of the gravitational field at least at spatial infinity. The puzzling surprise in 1962 was their discovery of a rich infinite-dimensional group (the so-called BMS group) as the asymptotic symmetry group, instead of the finite-dimensional Poincaré group, which is a subgroup of the BMS group. Not only are the Lorentz transformations asymptotic symmetry transformations, there are also additional transformations that are not Lorentz transformations but are asymptotic symmetry transformations. In fact, they found an additional infinity of transformation generators known as supertranslations. This implies the conclusion that General Relativity (GR) does not reduce to special relativity in the case of weak fields at long distances.^[94]^: 35

Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, defined as smooth manifolds with a Riemannian metric (an inner product on the tangent space at each point that varies smoothly from point to point). This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions.

Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "Ueber die Hypothesen, welche der Geometrie zu Grunde liegen" ("On the Hypotheses on which Geometry is Based").^[95] It is a very broad and abstract generalization of the differential geometry of surfaces in R³. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions. It enabled the formulation of Einstein's general theory of relativity, made profound impact on group theory and representation theory, as well as analysis, and spurred the development of algebraic and differential topology.

Curved manifolds

For physical reasons, a spacetime continuum is mathematically defined as a four-dimensional, smooth, connected Lorentzian manifold $(M,g)$ . This means the smooth Lorentz metric $g$ has signature $(3,1)$ . The metric determines the geometry of spacetime, as well as determining the geodesics of particles and light beams. About each point (event) on this manifold, coordinate charts are used to represent observers in reference frames. Usually, Cartesian coordinates $(x,y,z,t)$ are used. Moreover, for simplicity's sake, units of measurement are usually chosen such that the speed of light $c$ is equal to 1.^[96]

A reference frame (observer) can be identified with one of these coordinate charts; any such observer can describe any event $p$ . Another reference frame may be identified by a second coordinate chart about $p$ . Two observers (one in each reference frame) may describe the same event $p$ but obtain different descriptions.^[96]

Usually, many overlapping coordinate charts are needed to cover a manifold. Given two coordinate charts, one containing $p$ (representing an observer) and another containing $q$ (representing another observer), the intersection of the charts represents the region of spacetime in which both observers can measure physical quantities and hence compare results. The relation between the two sets of measurements is given by a non-singular coordinate transformation on this intersection. The idea of coordinate charts as local observers who can perform measurements in their vicinity also makes good physical sense, as this is how one actually collects physical data—locally.^[96]

For example, two observers, one of whom is on Earth, but the other one who is on a fast rocket to Jupiter, may observe a comet crashing into Jupiter (this is the event $p$ ). In general, they will disagree about the exact location and timing of this impact, i.e., they will have different 4-tuples $(x,y,z,t)$ (as they are using different coordinate systems). Although their kinematic descriptions will differ, dynamical (physical) laws, such as momentum conservation and the first law of thermodynamics, will still hold. In fact, relativity theory requires more than this in the sense that it stipulates these (and all other physical) laws must take the same form in all coordinate systems. This introduces tensors into relativity, by which all physical quantities are represented.

Geodesics are said to be time-like, null, or space-like if the tangent vector to one point of the geodesic is of this nature. Paths of particles and light beams in spacetime are represented by time-like and null (light-like) geodesics, respectively.^[96]

Privileged character of 3+1 spacetime

There are two kinds of dimensions: spatial (bidirectional) and temporal (unidirectional).^[98] Let the number of spatial dimensions be N and the number of temporal dimensions be T. That N = 3 and T = 1, setting aside the compactified dimensions invoked by string theory and undetectable to date, can be explained by appealing to the physical consequences of letting N differ from 3 and T differ from 1. The argument is often of an anthropic character and possibly the first of its kind, albeit before the complete concept came into vogue.

The implicit notion that the dimensionality of the universe is special is first attributed to Gottfried Wilhelm Leibniz, who in the Discourse on Metaphysics suggested that the world is "the one which is at the same time the simplest in hypothesis and the richest in phenomena".^[99] Immanuel Kant argued that 3-dimensional space was a consequence of the inverse square law of universal gravitation. While Kant's argument is historically important, John D. Barrow said that it "gets the punch-line back to front: it is the three-dimensionality of space that explains why we see inverse-square force laws in Nature, not vice-versa" (Barrow 2002:204).^{[note 31]}

In 1920, Paul Ehrenfest showed that if there is only a single time dimension and more than three spatial dimensions, the orbit of a planet about its Sun cannot remain stable. The same is true of a star's orbit around the center of its galaxy.^[100] Ehrenfest also showed that if there are an even number of spatial dimensions, then the different parts of a wave impulse will travel at different speeds. If there are $5+2k$ spatial dimensions, where k is a positive whole number, then wave impulses become distorted. In 1922, Hermann Weyl claimed that Maxwell's theory of electromagnetism can be expressed in terms of an action only for a four-dimensional manifold.^[101] Finally, Tangherlini showed in 1963 that when there are more than three spatial dimensions, electron orbitals around nuclei cannot be stable; electrons would either fall into the nucleus or disperse.^[102]

Max Tegmark expands on the preceding argument in the following anthropic manner.^[103] If T differs from 1, the behavior of physical systems could not be predicted reliably from knowledge of the relevant partial differential equations. In such a universe, intelligent life capable of manipulating technology could not emerge. Moreover, if T > 1, Tegmark maintains that protons and electrons would be unstable and could decay into particles having greater mass than themselves. (This is not a problem if the particles have a sufficiently low temperature.)^[103] Lastly, if N < 3, gravitation of any kind becomes problematic, and the universe would probably be too simple to contain observers. For example, when N < 3, nerves cannot cross without intersecting.^[103] Hence anthropic and other arguments rule out all cases except N = 3 and T = 1, which describes the world around us.

On the other hand, in view of creating black holes from an ideal monatomic gas under its self-gravity, Wei-Xiang Feng showed that (3 + 1)-dimensional spacetime is the marginal dimensionality. Moreover, it is the unique dimensionality that can afford a "stable" gas sphere with a "positive" cosmological constant. However, a self-gravitating gas cannot be stably bound if the mass sphere is larger than ~10²¹ solar masses, due to the small positivity of the cosmological constant observed.^[104]

In 2019, James Scargill argued that complex life may be possible with two spatial dimensions. According to Scargill, a purely scalar theory of gravity may enable a local gravitational force, and 2D networks may be sufficient for complex neural networks.^[105]^[106]

Notes

^ luminiferous from the Latin lumen, light, + ferens, carrying; aether from the Greek αἰθήρ (aithēr), pure air, clear sky
^ By stating that simultaneity is a matter of convention, Poincaré meant that to talk about time at all, one must have synchronized clocks, and the synchronization of clocks must be established by a specified, operational procedure (convention). This stance represented a fundamental philosophical break from Newton, who conceived of an absolute, true time that was independent of the workings of the inaccurate clocks of his day. This stance also represented a direct attack against the influential philosopher Henri Bergson, who argued that time, simultaneity, and duration were matters of intuitive understanding.^[15]
^ The operational procedure adopted by Poincaré was essentially identical to what is known as Einstein synchronization, even though a variant of it was already a widely used procedure by telegraphers in the middle 19th century. Basically, to synchronize two clocks, one flashes a light signal from one to the other, and adjusts for the time that the flash takes to arrive.^[15]
^ A hallmark of Einstein's career, in fact, was his use of visualized thought experiments (Gedanken–Experimente) as a fundamental tool for understanding physical issues. For special relativity, he employed moving trains and flashes of lightning for his most penetrating insights. For curved spacetime, he considered a painter falling off a roof, accelerating elevators, blind beetles crawling on curved surfaces and the like. In his great Solvay Debates with Bohr on the nature of reality (1927 and 1930), he devised multiple imaginary contraptions intended to show, at least in concept, means whereby the Heisenberg uncertainty principle might be evaded. Finally, in a profound contribution to the literature on quantum mechanics, Einstein considered two particles briefly interacting and then flying apart so that their states are correlated, anticipating the phenomenon known as quantum entanglement.^[20]
^ In the original version of this lecture, Minkowski continued to use such obsolescent terms as the ether, but the posthumous publication in 1915 of this lecture in the Annals of Physics (Annalen der Physik) was edited by Sommerfeld to remove this term. Sommerfeld also edited the published form of this lecture to revise Minkowski's judgement of Einstein from being a mere clarifier of the principle of relativity, to being its chief expositor.^[22]
^ (In the following, the group G_∞ is the Galilean group and the group G_c the Lorentz group.) "With respect to this it is clear that the group G_c in the limit for c = ∞, i.e. as group G_∞, exactly becomes the full group belonging to Newtonian Mechanics. In this state of affairs, and since G_c is mathematically more intelligible than G_∞, a mathematician may, by a free play of imagination, hit upon the thought that natural phenomena actually possess an invariance, not for the group G_∞, but rather for a group G_c, where c is definitely finite, and only exceedingly large using the ordinary measuring units."^[24]
^ For instance, the Lorentz group is a subgroup of the conformal group in four dimensions.^[25]^: 41–42 The Lorentz group is isomorphic to the Laguerre group transforming planes into planes,^[25]^: 39–42 it is isomorphic to the Möbius group of the plane,^[26]^: 22 and is isomorphic to the group of isometries in hyperbolic space which is often expressed in terms of the hyperboloid model.^[27]^: 3.2.3
^ In a Cartesian plane, ordinary rotation leaves a circle unchanged. In spacetime, hyperbolic rotation preserves the hyperbolic metric.
^ Even with no (de)acceleration i.e. using one inertial frame O for constant, high-velocity outward journey and another inertial frame I for constant, high-velocity inward journey – the sum of the elapsed time in those frames (O and I) is shorter than the elapsed time in the stationary inertial frame S. Thus acceleration and deceleration is not the cause of shorter elapsed time during the outward and inward journey. Instead the use of two different constant, high-velocity inertial frames for outward and inward journey is really the cause of shorter elapsed time total. Granted, if the same twin has to travel outward and inward leg of the journey and safely switch from outward to inward leg of the journey, the acceleration and deceleration is required. If the travelling twin could ride the high-velocity outward inertial frame and instantaneously switch to high-velocity inward inertial frame the example would still work. The point is that real reason should be stated clearly. The asymmetry is because of the comparison of sum of elapsed times in two different inertial frames (O and I) to the elapsed time in a single inertial frame S.
^ The ease of analyzing a relativistic scenario often depends on the frame in which one chooses to perform the analysis. In this linked image, we present alternative views of the transverse Doppler shift scenario where source and receiver are at their closest approach to each other. (a) If we analyze the scenario in the frame of the receiver, we find that the analysis is more complicated than it should be. The apparent position of a celestial object is displaced from its true position (or geometric position) because of the object's motion during the time it takes its light to reach an observer. The source would be time-dilated relative to the receiver, but the redshift implied by this time dilation would be offset by a blueshift due to the longitudinal component of the relative motion between the receiver and the apparent position of the source. (b) It is much easier if, instead, we analyze the scenario from the frame of the source. An observer situated at the source knows, from the problem statement, that the receiver is at its closest point to him. That means that the receiver has no longitudinal component of motion to complicate the analysis. Since the receiver's clocks are time-dilated relative to the source, the light that the receiver receives is therefore blue-shifted by a factor of gamma.
^ Not all experiments characterize the effect in terms of a redshift. For example, the Kündig experiment was set up to measure transverse blueshift using a Mössbauer source setup at the center of a centrifuge rotor and an absorber at the rim.
^ Rapidity arises naturally as a coordinates on the pure boost generators inside the Lie algebra algebra of the Lorentz group. Likewise, rotation angles arise naturally as coordinates (modulo 2 $π$ ) on the pure rotation generators in the Lie algebra. (Together they coordinatize the whole Lie algebra.) A notable difference is that the resulting rotations are periodic in the rotation angle, while the resulting boosts are not periodic in rapidity (but rather one-to-one). The similarity between boosts and rotations is formal resemblance.
^ In relativity theory, proper acceleration is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily at rest relative to the object being measured.
^ Newton himself was acutely aware of the inherent difficulties with these assumptions, but as a practical matter, making these assumptions was the only way that he could make progress. In 1692, he wrote to his friend Richard Bentley: "That Gravity should be innate, inherent and essential to Matter, so that one body may act upon another at a distance thro' a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it."
^ More precisely, the gravitational field couples to itself. In Newtonian gravity, the potential due to two point masses is simply the sum of the potentials of the two masses, but this does not apply to GR. This can be thought of as the result of the equivalence principle: If gravitation did not couple to itself, two particles bound by their mutual gravitational attraction would not have the same inertial mass (due to negative binding energy) as their gravitational mass.^[51]^{: 112–113}
^ D'Inverno writes that Lieber's book inspired him, as a young teenager, to take up relativity as his life's work. He warns, however: "This is a very bizarre book in appearance. The book is not set out in the usual way but rather as though it were concrete poetry. Moreover, it is interspersed by surrealist drawings by Hugh Lieber involving the symbols from the text. I must confess that at first sight the book looks rather cranky, but it is not." ^[76]^: 11
^ Lieber, as did Einstein, preferred to use subscripted $dx_{i}\,{\text{'s}}$ rather than superscripted $dx^{i}\,{\text{'s}}$ in the tensor formulas. Current practice is to use superscripts to emphasize that the $dx^{i}\,{\text{'s}}$ are displacement vectors that transform as contravariant vectors. Also, we have preferred to use $\Gamma _{\mu \nu }^{\lambda }$ rather than the outmoded $\{\mu \nu ,\lambda \}$ notation for the Christoffel symbol. Furthermore, Lieber used $B_{\sigma \tau \rho }^{\alpha }$ rather than the current more commonly used $R_{\sigma \rho \tau }^{\alpha }=-R_{\sigma \tau \rho }^{\alpha }$ for the Riemann-Christoffel curvature tensor. Note that some textbook authors have adopted a definition of the curvature tensor that is of reverse sign to the definition adopted here.
^ An important theorem states that if a tensor equation is true in one system of coordinates, then it is true in all systems, whether they be Cartesian, cylindrical, spherical, rotated or in relative motion, etc. This theorem provides a powerful method of proof for a tensor equation: It needs only be proven to be true in one coordinate system (chosen for its ease of calculation) to be true for all.^[78]^: 45–46
^ Note: Certain superficially plausible manipulations in tensor calculus, performed by mistaken analogy with common algebraic manipulations, are in fact incorrect, as can be shown by expanding the terms following the notational rules that have been given. Contrast the following identities with the similar-looking but incorrect non-identities:^[81]^: 3
$a_{ij}(x_{j}+y_{j})\equiv a_{ij}x_{j}+a_{ij}y_{j}$

$a_{ij}(x_{i}+y_{j})\not \equiv a_{ij}x_{i}+a_{ij}y_{j}\quad$ NO!

$a_{ij}x_{i}y_{j}\equiv a_{ij}y_{j}x_{i}$

$a_{ij}x_{i}x_{j}\equiv a_{ji}x_{i}x_{j}$

$a_{ij}x_{i}y_{j}\not \equiv a_{ij}y_{i}x_{j}\quad$ NO!

$(a_{ij}+a_{ji})x_{i}x_{j}\equiv 2a_{ij}x_{i}x_{j}$

$(a_{ij}+a_{ji})x_{i}y_{j}\not \equiv 2a_{ij}x_{i}y_{j}\quad$ NO!

$(a_{ij}-a_{ji})x_{i}x_{j}\equiv 0$
^ Although one should be careful about accidentally misapplying concepts of single-variable calculus to multivariable calculus, the product rule in multivariable calculus looks almost identical to the rule in single-variable calculus: ${\frac {\partial }{\partial x}}(uv)=u{\frac {\partial v}{\partial x}}+v{\frac {\partial u}{\partial x}}$
^ Although this rearrangement of terms in the product is legitimate, various other manipulations that are common when working with full derivatives are not. In particular, one may not treat partial derivatives like fractions. Partial derivatives must be treated as complete entities whose numerators and denominators cannot be separated. So we should never pull them apart like ${\frac {\partial f}{\partial t}}=kxt^{2}\;\implies \;\partial f=kxt^{2}\partial t.$ Never do this. With full derivatives, this is permissible because full derivatives represent the ratio of two differentials. But there are no such things as partial differentials. $\partial f$ and $\partial t$ do not separately exist.
^ Except at "singular" points in space, which are points where matter is located.
^ It is sufficient to prove the Quotient Theorem true for a particular case, since it will be evident that the argument is of general application. For example, suppose $X_{\gamma \delta ...}^{\alpha \beta ...}A_{\alpha }$ is known to be a contravariant vector for all choices of the covariant vector $A_{\alpha }.$ Since $X_{\gamma \delta ...}^{\alpha \beta ...}A_{\alpha }$ is a contravariant vector, it follows the pattern of (D3):
${\bar {X}}_{\gamma \delta ...}^{\alpha \beta ...}{\bar {A}}_{\alpha }={\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\nu }}}X_{\gamma \delta ...}^{\alpha \nu ...}A_{\alpha }$
Since we are given that $A_{\alpha }$ is a covariant vector,
${\bar {A}}_{\alpha }={\frac {\partial x^{\mu }}{\partial {\bar {x}}^{\alpha }}}A_{\mu }\quad$ or $\quad A_{\alpha }={\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}{\bar {A}}_{\mu }$
Substituting,
${\bar {X}}_{\gamma \delta ...}^{\alpha \beta ...}{\bar {A}}_{\alpha }={\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\nu }}}X_{\gamma \delta ...}^{\alpha \nu ...}{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}{\bar {A}}_{\mu }$
Swapping the dummy indices $\alpha$ and $\mu$ on the right-hand side, then rearranging, we get
$\left[{\bar {X}}_{\gamma \delta ...}^{\alpha \beta ...}-{\frac {\partial {\bar {x}}^{\alpha }}{\partial x^{\mu }}}{\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\nu }}}X_{\gamma \delta ...}^{\mu \nu ...}\right]{\bar {A}}_{\alpha }=0$
${\bar {A}}_{\alpha }$ would not generally be zero, therefore
${\bar {X}}_{\gamma \delta ...}^{\alpha \beta ...}={\frac {\partial {\bar {x}}^{\alpha }}{\partial x^{\mu }}}{\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\nu }}}X_{\gamma \delta ...}^{\mu \nu ...}$
Comparison with (D4) shows that $X_{\gamma \delta ...}^{\mu \nu ...}$ transforms as a contravariant tensor of rank two.^[75]^{: 312–314}^[80]^: 94–95
^ Einstein introduced a powerful comma notation for the partial derivative of a function. He would simplify the appearance of (K1) as follows:^[79]^{: 149, 157} $\Gamma _{\mu \nu }^{\lambda }={\frac {1}{2}}g^{\lambda \alpha }\left(g_{\mu \alpha ,\nu }+g_{\nu \alpha ,\mu }-g_{\mu \nu ,\alpha }\right)$ We won't use this notation, but it is frequently found in the literature.
^ Especially in the older literature, one often sees covariant tensors of rank one referred to as "covectors", while contravariant tensors of rank one are referred to simply as "vectors".
^ The precise consequences of a finite speed of light depend on the mechanism assumed to underlie Newtonian gravitation. Laplace was considering a mechanism whereby gravity is caused by "the impulse of a fluid directed towards the centre of the attracting body". In an alternative mechanistic theory, the Earth would always be pulled toward the optical position of the Sun, which is displaced forward from its geometric position due to aberration. This would cause a pull ahead of the Earth, which would cause the orbit of the Earth to rapidly spiral outward. In reality, however, any finite speed of gravity would result in the violation of conservation of energy and conservation of angular momentum. Gravitational wave astronomers have confirmed that the speed of gravity equals c to a high degree of accuracy. The seeming paradox between the measured finite speed of gravity and the stability of the Earth's orbit is resolved by general relativity.
^ In the older literature, the recommended pronunciation is often given as "nabla square"
^ ${\frac {\partial }{\partial r}}\ln {\sqrt {-g}}={\frac {\partial }{\partial r}}(\ln {\sqrt {e^{\lambda +\nu }r^{4}\sin ^{2}\theta }})=\,$ ${\tfrac {1}{2}}\lambda '+{\tfrac {1}{2}}\nu '+{\frac {2}{r}}$
$\;{\frac {\partial ^{2}}{\partial r^{2}}}\ln {\sqrt {-g}}=\,$ ${\tfrac {1}{2}}\lambda ''+{\tfrac {1}{2}}\nu ''-{\frac {2}{r^{2}}}$
^ The constant $m$ is the mass of the central particle in relativistic units. It has dimensions of length and is often called the geometric mass. The identification of $m$ with geometric mass is often expressed as a boundary condition argument, for instance in Adler (2021),^[78]^{: 125–129} but in actuality, as explained in D'Inverno (1992),^[76]^{: 186–190} the field equations force this interpretation.
^ Very basic treatments of the subject may be found in D'Inverno (1992)^[76]^{: 82–83, 99–101} and in Lawden (2002).^[80]^{: 114–117}
^ This is because the law of gravitation (or any other inverse-square law) follows from the concept of flux and the proportional relationship of flux density and field strength. If N = 3, then 3-dimensional solid objects have surface areas proportional to the square of their size in any selected spatial dimension. In particular, a sphere of radius r has a surface area of 4πr². More generally, in a space of N dimensions, the strength of the gravitational attraction between two bodies separated by a distance of r would be inversely proportional to r^N−1.

Additional details

^ Different reporters viewing the scenarios presented in this figure interpret the scenarios differently depending on their knowledge of the situation. (i) A first reporter, at the center of mass of particles 2 and 3 but unaware of the large mass 1, concludes that a force of repulsion exists between the particles in scenario A while a force of attraction exists between the particles in scenario B. (ii) A second reporter, aware of the large mass 1, smiles at the first reporter's naiveté. This second reporter knows that in reality, the apparent forces between particles 2 and 3 really represent tidal effects resulting from their differential attraction by mass 1. (iii) A third reporter, trained in general relativity, knows that there are, in fact, no forces at all acting between the three objects. Rather, all three objects move along geodesics in spacetime.

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External links

Media related to Spacetime at Wikimedia Commons
Albert Einstein on space–time 13th edition Encyclopædia Britannica Historical: Albert Einstein's 1926 article
Encyclopedia of Space–time and gravitation Scholarpedia Expert articles
Stanford Encyclopedia of Philosophy: "Space and Time: Inertial Frames" by Robert DiSalle.

[10] luminiferous from the Latin lumen, light, + ferens, carrying; aether from the Greek αἰθήρ (aithēr), pure air, clear sky

[17] By stating that simultaneity is a matter of convention, Poincaré meant that to talk about time at all, one must have synchronized clocks, and the synchronization of clocks must be established by a specified, operational procedure (convention). This stance represented a fundamental philosophical break from Newton, who conceived of an absolute, true time that was independent of the workings of the inaccurate clocks of his day. This stance also represented a direct attack against the influential philosopher Henri Bergson, who argued that time, simultaneity, and duration were matters of intuitive understanding.^[15]

[18] The operational procedure adopted by Poincaré was essentially identical to what is known as Einstein synchronization, even though a variant of it was already a widely used procedure by telegraphers in the middle 19th century. Basically, to synchronize two clocks, one flashes a light signal from one to the other, and adjusts for the time that the flash takes to arrive.^[15]

[24] A hallmark of Einstein's career, in fact, was his use of visualized thought experiments (Gedanken–Experimente) as a fundamental tool for understanding physical issues. For special relativity, he employed moving trains and flashes of lightning for his most penetrating insights. For curved spacetime, he considered a painter falling off a roof, accelerating elevators, blind beetles crawling on curved surfaces and the like. In his great Solvay Debates with Bohr on the nature of reality (1927 and 1930), he devised multiple imaginary contraptions intended to show, at least in concept, means whereby the Heisenberg uncertainty principle might be evaded. Finally, in a profound contribution to the literature on quantum mechanics, Einstein considered two particles briefly interacting and then flying apart so that their states are correlated, anticipating the phenomenon known as quantum entanglement.^[20]

[28] In the original version of this lecture, Minkowski continued to use such obsolescent terms as the ether, but the posthumous publication in 1915 of this lecture in the Annals of Physics (Annalen der Physik) was edited by Sommerfeld to remove this term. Sommerfeld also edited the published form of this lecture to revise Minkowski's judgement of Einstein from being a mere clarifier of the principle of relativity, to being its chief expositor.^[22]

[30] (In the following, the group G_∞ is the Galilean group and the group G_c the Lorentz group.) "With respect to this it is clear that the group G_c in the limit for c = ∞, i.e. as group G_∞, exactly becomes the full group belonging to Newtonian Mechanics. In this state of affairs, and since G_c is mathematically more intelligible than G_∞, a mathematician may, by a free play of imagination, hit upon the thought that natural phenomena actually possess an invariance, not for the group G_∞, but rather for a group G_c, where c is definitely finite, and only exceedingly large using the ordinary measuring units."^[24]

[34] For instance, the Lorentz group is a subgroup of the conformal group in four dimensions.^[25]^: 41–42 The Lorentz group is isomorphic to the Laguerre group transforming planes into planes,^[25]^: 39–42 it is isomorphic to the Möbius group of the plane,^[26]^: 22 and is isomorphic to the group of isometries in hyperbolic space which is often expressed in terms of the hyperboloid model.^[27]^: 3.2.3

[38] In a Cartesian plane, ordinary rotation leaves a circle unchanged. In spacetime, hyperbolic rotation preserves the hyperbolic metric.

[41] Even with no (de)acceleration i.e. using one inertial frame O for constant, high-velocity outward journey and another inertial frame I for constant, high-velocity inward journey – the sum of the elapsed time in those frames (O and I) is shorter than the elapsed time in the stationary inertial frame S. Thus acceleration and deceleration is not the cause of shorter elapsed time during the outward and inward journey. Instead the use of two different constant, high-velocity inertial frames for outward and inward journey is really the cause of shorter elapsed time total. Granted, if the same twin has to travel outward and inward leg of the journey and safely switch from outward to inward leg of the journey, the acceleration and deceleration is required. If the travelling twin could ride the high-velocity outward inertial frame and instantaneously switch to high-velocity inward inertial frame the example would still work. The point is that real reason should be stated clearly. The asymmetry is because of the comparison of sum of elapsed times in two different inertial frames (O and I) to the elapsed time in a single inertial frame S.

[49] The ease of analyzing a relativistic scenario often depends on the frame in which one chooses to perform the analysis. In this linked image, we present alternative views of the transverse Doppler shift scenario where source and receiver are at their closest approach to each other. (a) If we analyze the scenario in the frame of the receiver, we find that the analysis is more complicated than it should be. The apparent position of a celestial object is displaced from its true position (or geometric position) because of the object's motion during the time it takes its light to reach an observer. The source would be time-dilated relative to the receiver, but the redshift implied by this time dilation would be offset by a blueshift due to the longitudinal component of the relative motion between the receiver and the apparent position of the source. (b) It is much easier if, instead, we analyze the scenario from the frame of the source. An observer situated at the source knows, from the problem statement, that the receiver is at its closest point to him. That means that the receiver has no longitudinal component of motion to complicate the analysis. Since the receiver's clocks are time-dilated relative to the source, the light that the receiver receives is therefore blue-shifted by a factor of gamma.

[50] Not all experiments characterize the effect in terms of a redshift. For example, the Kündig experiment was set up to measure transverse blueshift using a Mössbauer source setup at the center of a centrifuge rotor and an absorber at the rim.

[57] Rapidity arises naturally as a coordinates on the pure boost generators inside the Lie algebra algebra of the Lorentz group. Likewise, rotation angles arise naturally as coordinates (modulo 2 $π$ ) on the pure rotation generators in the Lie algebra. (Together they coordinatize the whole Lie algebra.) A notable difference is that the resulting rotations are periodic in the rotation angle, while the resulting boosts are not periodic in rapidity (but rather one-to-one). The similarity between boosts and rotations is formal resemblance.

[59] In relativity theory, proper acceleration is the physical acceleration (i.e., measurable acceleration as by an accelerometer) experienced by an object. It is thus acceleration relative to a free-fall, or inertial, observer who is momentarily at rest relative to the object being measured.

[61] Newton himself was acutely aware of the inherent difficulties with these assumptions, but as a practical matter, making these assumptions was the only way that he could make progress. In 1692, he wrote to his friend Richard Bentley: "That Gravity should be innate, inherent and essential to Matter, so that one body may act upon another at a distance thro' a Vacuum, without the Mediation of any thing else, by and through which their Action and Force may be conveyed from one to another, is to me so great an Absurdity that I believe no Man who has in philosophical Matters a competent Faculty of thinking can ever fall into it."

[71] More precisely, the gravitational field couples to itself. In Newtonian gravity, the potential due to two point masses is simply the sum of the potentials of the two masses, but this does not apply to GR. This can be thought of as the result of the equivalence principle: If gravitation did not couple to itself, two particles bound by their mutual gravitational attraction would not have the same inertial mass (due to negative binding energy) as their gravitational mass.^[51]^{: 112–113}

[93] D'Inverno writes that Lieber's book inspired him, as a young teenager, to take up relativity as his life's work. He warns, however: "This is a very bizarre book in appearance. The book is not set out in the usual way but rather as though it were concrete poetry. Moreover, it is interspersed by surrealist drawings by Hugh Lieber involving the symbols from the text. I must confess that at first sight the book looks rather cranky, but it is not." ^[76]^: 11

[94] Lieber, as did Einstein, preferred to use subscripted $dx_{i}\,{\text{'s}}$ rather than superscripted $dx^{i}\,{\text{'s}}$ in the tensor formulas. Current practice is to use superscripts to emphasize that the $dx^{i}\,{\text{'s}}$ are displacement vectors that transform as contravariant vectors. Also, we have preferred to use $\Gamma _{\mu \nu }^{\lambda }$ rather than the outmoded $\{\mu \nu ,\lambda \}$ notation for the Christoffel symbol. Furthermore, Lieber used $B_{\sigma \tau \rho }^{\alpha }$ rather than the current more commonly used $R_{\sigma \rho \tau }^{\alpha }=-R_{\sigma \tau \rho }^{\alpha }$ for the Riemann-Christoffel curvature tensor. Note that some textbook authors have adopted a definition of the curvature tensor that is of reverse sign to the definition adopted here.

[98] An important theorem states that if a tensor equation is true in one system of coordinates, then it is true in all systems, whether they be Cartesian, cylindrical, spherical, rotated or in relative motion, etc. This theorem provides a powerful method of proof for a tensor equation: It needs only be proven to be true in one coordinate system (chosen for its ease of calculation) to be true for all.^[78]^: 45–46

[101] Note: Certain superficially plausible manipulations in tensor calculus, performed by mistaken analogy with common algebraic manipulations, are in fact incorrect, as can be shown by expanding the terms following the notational rules that have been given. Contrast the following identities with the similar-looking but incorrect non-identities:^[81]^: 3
$a_{ij}(x_{j}+y_{j})\equiv a_{ij}x_{j}+a_{ij}y_{j}$

$a_{ij}(x_{i}+y_{j})\not \equiv a_{ij}x_{i}+a_{ij}y_{j}\quad$ NO!

$a_{ij}x_{i}y_{j}\equiv a_{ij}y_{j}x_{i}$

$a_{ij}x_{i}x_{j}\equiv a_{ji}x_{i}x_{j}$

$a_{ij}x_{i}y_{j}\not \equiv a_{ij}y_{i}x_{j}\quad$ NO!

$(a_{ij}+a_{ji})x_{i}x_{j}\equiv 2a_{ij}x_{i}x_{j}$

$(a_{ij}+a_{ji})x_{i}y_{j}\not \equiv 2a_{ij}x_{i}y_{j}\quad$ NO!

$(a_{ij}-a_{ji})x_{i}x_{j}\equiv 0$

[102] Although one should be careful about accidentally misapplying concepts of single-variable calculus to multivariable calculus, the product rule in multivariable calculus looks almost identical to the rule in single-variable calculus: ${\frac {\partial }{\partial x}}(uv)=u{\frac {\partial v}{\partial x}}+v{\frac {\partial u}{\partial x}}$

[103] Although this rearrangement of terms in the product is legitimate, various other manipulations that are common when working with full derivatives are not. In particular, one may not treat partial derivatives like fractions. Partial derivatives must be treated as complete entities whose numerators and denominators cannot be separated. So we should never pull them apart like ${\frac {\partial f}{\partial t}}=kxt^{2}\;\implies \;\partial f=kxt^{2}\partial t.$ Never do this. With full derivatives, this is permissible because full derivatives represent the ratio of two differentials. But there are no such things as partial differentials. $\partial f$ and $\partial t$ do not separately exist.

[104] Except at "singular" points in space, which are points where matter is located.

[105] It is sufficient to prove the Quotient Theorem true for a particular case, since it will be evident that the argument is of general application. For example, suppose $X_{\gamma \delta ...}^{\alpha \beta ...}A_{\alpha }$ is known to be a contravariant vector for all choices of the covariant vector $A_{\alpha }.$ Since $X_{\gamma \delta ...}^{\alpha \beta ...}A_{\alpha }$ is a contravariant vector, it follows the pattern of (D3):
${\bar {X}}_{\gamma \delta ...}^{\alpha \beta ...}{\bar {A}}_{\alpha }={\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\nu }}}X_{\gamma \delta ...}^{\alpha \nu ...}A_{\alpha }$
Since we are given that $A_{\alpha }$ is a covariant vector,
${\bar {A}}_{\alpha }={\frac {\partial x^{\mu }}{\partial {\bar {x}}^{\alpha }}}A_{\mu }\quad$ or $\quad A_{\alpha }={\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}{\bar {A}}_{\mu }$
Substituting,
${\bar {X}}_{\gamma \delta ...}^{\alpha \beta ...}{\bar {A}}_{\alpha }={\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\nu }}}X_{\gamma \delta ...}^{\alpha \nu ...}{\frac {\partial {\bar {x}}^{\mu }}{\partial x^{\alpha }}}{\bar {A}}_{\mu }$
Swapping the dummy indices $\alpha$ and $\mu$ on the right-hand side, then rearranging, we get
$\left[{\bar {X}}_{\gamma \delta ...}^{\alpha \beta ...}-{\frac {\partial {\bar {x}}^{\alpha }}{\partial x^{\mu }}}{\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\nu }}}X_{\gamma \delta ...}^{\mu \nu ...}\right]{\bar {A}}_{\alpha }=0$
${\bar {A}}_{\alpha }$ would not generally be zero, therefore
${\bar {X}}_{\gamma \delta ...}^{\alpha \beta ...}={\frac {\partial {\bar {x}}^{\alpha }}{\partial x^{\mu }}}{\frac {\partial {\bar {x}}^{\beta }}{\partial x^{\nu }}}X_{\gamma \delta ...}^{\mu \nu ...}$
Comparison with (D4) shows that $X_{\gamma \delta ...}^{\mu \nu ...}$ transforms as a contravariant tensor of rank two.^[75]^{: 312–314}^[80]^: 94–95

[106] Einstein introduced a powerful comma notation for the partial derivative of a function. He would simplify the appearance of (K1) as follows:^[79]^{: 149, 157} $\Gamma _{\mu \nu }^{\lambda }={\frac {1}{2}}g^{\lambda \alpha }\left(g_{\mu \alpha ,\nu }+g_{\nu \alpha ,\mu }-g_{\mu \nu ,\alpha }\right)$ We won't use this notation, but it is frequently found in the literature.

[107] Especially in the older literature, one often sees covariant tensors of rank one referred to as "covectors", while contravariant tensors of rank one are referred to simply as "vectors".

[110] The precise consequences of a finite speed of light depend on the mechanism assumed to underlie Newtonian gravitation. Laplace was considering a mechanism whereby gravity is caused by "the impulse of a fluid directed towards the centre of the attracting body". In an alternative mechanistic theory, the Earth would always be pulled toward the optical position of the Sun, which is displaced forward from its geometric position due to aberration. This would cause a pull ahead of the Earth, which would cause the orbit of the Earth to rapidly spiral outward. In reality, however, any finite speed of gravity would result in the violation of conservation of energy and conservation of angular momentum. Gravitational wave astronomers have confirmed that the speed of gravity equals c to a high degree of accuracy. The seeming paradox between the measured finite speed of gravity and the stability of the Earth's orbit is resolved by general relativity.

[111] In the older literature, the recommended pronunciation is often given as "nabla square"

[112] ${\frac {\partial }{\partial r}}\ln {\sqrt {-g}}={\frac {\partial }{\partial r}}(\ln {\sqrt {e^{\lambda +\nu }r^{4}\sin ^{2}\theta }})=\,$ ${\tfrac {1}{2}}\lambda '+{\tfrac {1}{2}}\nu '+{\frac {2}{r}}$
$\;{\frac {\partial ^{2}}{\partial r^{2}}}\ln {\sqrt {-g}}=\,$ ${\tfrac {1}{2}}\lambda ''+{\tfrac {1}{2}}\nu ''-{\frac {2}{r^{2}}}$

[113] The constant $m$ is the mass of the central particle in relativistic units. It has dimensions of length and is often called the geometric mass. The identification of $m$ with geometric mass is often expressed as a boundary condition argument, for instance in Adler (2021),^[78]^{: 125–129} but in actuality, as explained in D'Inverno (1992),^[76]^{: 186–190} the field equations force this interpretation.

[114] Very basic treatments of the subject may be found in D'Inverno (1992)^[76]^{: 82–83, 99–101} and in Lawden (2002).^[80]^{: 114–117}

[131] This is because the law of gravitation (or any other inverse-square law) follows from the concept of flux and the proportional relationship of flux density and field strength. If N = 3, then 3-dimensional solid objects have surface areas proportional to the square of their size in any selected spatial dimension. In particular, a sphere of radius r has a surface area of 4πr². More generally, in a space of N dimensions, the strength of the gravitational attraction between two bodies separated by a distance of r would be inversely proportional to r^N−1.

[62] Different reporters viewing the scenarios presented in this figure interpret the scenarios differently depending on their knowledge of the situation. (i) A first reporter, at the center of mass of particles 2 and 3 but unaware of the large mass 1, concludes that a force of repulsion exists between the particles in scenario A while a force of attraction exists between the particles in scenario B. (ii) A second reporter, aware of the large mass 1, smiles at the first reporter's naiveté. This second reporter knows that in reality, the apparent forces between particles 2 and 3 really represent tidal effects resulting from their differential attraction by mass 1. (iii) A third reporter, trained in general relativity, knows that there are, in fact, no forces at all acting between the three objects. Rather, all three objects move along geodesics in spacetime.

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[134] Tangherlini, F. R. (1963). "Schwarzschild field in n dimensions and the dimensionality of space problem". Nuovo Cimento. 27 (3): 636–651. Bibcode:1963NCim...27..636T. doi:10.1007/BF02784569. S2CID 119683293.

[Anthropic_principle_tegmark-dim-135] Tegmark, Max (April 1997). "On the dimensionality of spacetime" (PDF). Classical and Quantum Gravity. 14 (4): L69–L75. arXiv:gr-qc/9702052. Bibcode:1997CQGra..14L..69T. doi:10.1088/0264-9381/14/4/002. S2CID 15694111. Retrieved 16 December 2006.

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[137] Scargill, J. H. C. (26 February 2020). "Existence of life in 2 + 1 dimensions". Physical Review Research. 2 (1): 013217. arXiv:1906.05336. Bibcode:2020PhRvR...2a3217S. doi:10.1103/PhysRevResearch.2.013217. S2CID 211734117.

[138] "Life could exist in a 2D universe (according to physics, anyway)". technologyreview.com. Retrieved 16 June 2021.

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v t e Dimension
Dimensional spaces	Vector space Euclidean space Affine space Projective space Free module Manifold Algebraic variety Spacetime
Other dimensions	Krull Lebesgue covering Inductive Hausdorff Minkowski Fractal Degrees of freedom
Polytopes and shapes	Hyperplane Hypersurface Hypercube Hyperrectangle Demihypercube Hypersphere Cross-polytope Simplex Hyperpyramid
Number systems	Hypercomplex numbers Cayley–Dickson construction
Dimensions by number	Zero One Two Three Four Five Six Seven Eight n-dimensions
See also	Hyperspace Codimension
Category

v t e Time measurement and standards
Chronometry Orders of magnitude Metrology
International standards	Coordinated Universal Time offset UT ΔT DUT1 International Earth Rotation and Reference Systems Service ISO 31-1 ISO 8601 International Atomic Time 12-hour clock 24-hour clock Barycentric Coordinate Time Barycentric Dynamical Time Civil time Daylight saving time Geocentric Coordinate Time International Date Line IERS Reference Meridian Leap second Solar time Terrestrial Time Time zone 180th meridian
Obsolete standards	Ephemeris time Greenwich Mean Time Prime meridian
Time in physics	Absolute space and time Spacetime Chronon Continuous signal Coordinate time Cosmological decade Discrete time and continuous time Proper time Theory of relativity Time dilation Gravitational time dilation Time domain Time-translation symmetry T-symmetry
Horology	Clock Astrarium Atomic clock Complication History of timekeeping devices Hourglass Marine chronometer Marine sandglass Radio clock Watch stopwatch Water clock Sundial Dialing scales Equation of time History of sundials Sundial markup schema
Calendar	Gregorian Hebrew Hindu Holocene Islamic (lunar Hijri) Julian Solar Hijri Astronomical Dominical letter Epact Equinox Intercalation Julian day Leap year Lunar Lunisolar Solar Solstice Tropical year Weekday determination Weekday names
Archaeology and geology	Chronological dating Geologic time scale International Commission on Stratigraphy
Astronomical chronology	Galactic year Nuclear timescale Precession Sidereal time
Other units of time	Instant Flick Shake Jiffy Second Minute Moment Hour Day Week Fortnight Month Year Olympiad Lustrum Decade Century Saeculum Millennium
Related topics	Chronology Duration music Mental chronometry Decimal time Metric time System time Time metrology Time value of money Timekeeper

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