Spacetime

In physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single 4‑dimensional continuum. Spacetime diagrams are useful in visualizing and understanding relativistic effects such as how different observers perceive where and when events occur.

Until the turn of the 20th century, the assumption had been that the three-dimensional geometry of the universe (its description in terms of locations, shapes, distances, and directions) was distinct from time (the measurement of when events occur within the universe). However, Albert Einstein's 1905 special theory of relativity postulated that the speed of light through empty space has one definite value—a constant—that is independent of the motion of the light source. Einstein's equations described important consequences of this fact: The distances and times between pairs of events vary when measured in different inertial frames of reference (separate vantage points that aren’t being subjected to g‑forces but have different velocities).

Einstein's theory was framed in terms of kinematics (the study of moving bodies), and showed how quantification of distances and times varied for measurements made in different reference frames. His theory was a breakthrough 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 Zurich—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 definition of a spacetime interval that combines distance and time. 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.

Minkowski's geometric interpretation of relativity was to prove vital to Einstein's development of his 1915 general theory of relativity, wherein he showed that spacetime becomes curved in the presence of mass or energy.

Introduction

Definitions

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Non-relativistic classical mechanics treats time as a universal quantity of measurement which is uniform throughout space and which is separate from space. Classical mechanics assumes that time has a constant rate of passage that is independent of the state of motion of an observer, or indeed of anything external.^[1] Furthermore, it assumes that space is Euclidean, which is to say, it assumes that space follows the geometry of common sense.^[2]

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, in addition, 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). Spacetime is thus four dimensional. An event is something that happens instantaneously at a single point in spacetime, represented by a set of coordinates x, y, z and t.

The word "event" used in relativity should not be confused with the use of the word "event" in normal conversation, where it might refer to an "event" as something such as a concert, sporting event, or a battle. These are not mathematical "events" in the way the word is used in relativity, because they have finite durations and extents. Unlike the analogies used to explain events, such as firecrackers or lightning bolts, mathematical events have zero duration and represent a single point in space.

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.^[3]^: 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.^[4] 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 km in space being equivalent to 1 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.^[5]

In special relativity, an "observer" will, in most cases, mean a frame of reference from which a set of objects or events are 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 a scientist is in control of a dense lattice of clocks, synchronized within her reference frame, that extends indefinitely throughout the three dimensions of space. Her location within the lattice is not important. She uses her latticework of clocks to determine the time and position of events taking place within its reach. The term observer refers to the entire ensemble of clocks associated with one inertial frame of reference.^[6]^: 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 delays 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.^[7]

History

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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 (a bright point at the center of a circular object's shadow due to diffraction) 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.^[8] Propagation of waves was then (wrongly) assumed to require the existence of a medium which waved: in the case of light waves, this was considered to be a hypothetical luminiferous aether.^{[note 2]} 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.^[9] 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.^[5]

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 were 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.^[10]^{: 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."^[11]

An important example is Henri Poincaré,^[12]^[13]^{: 73–80, 93–95} who in 1898 argued that the simultaneity of two events is a matter of convention.^[14]^{[note 3]} 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 4]} 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^[15] 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 space-time by defining various four vectors, namely four-position, four-velocity, and four-force.^[16]^[17] 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".^[17] Furthermore, even as late as 1909, Poincaré continued to believe in the dynamical interpretation of the Lorentz transform.^[10]^{: 163–174}. For these and other reasons, most historians of science argue that Poincaré did not invent what is now called special relativity.^[13]^[10]

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.^[13]^[10] While his results are mathematically equivalent to those of Lorentz and Poincaré, it was Einstein who 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. Einstein performed his analyses in terms of kinematics (the study of moving bodies without reference to forces) rather than dynamics. 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. 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 early results in developing general relativity. While it would appear that he did not at first think geometrically about spacetime,^[18]^: 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:^[19]

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.^[20]

A little more than a year before his death, Minkowski introduced his geometric interpretation of spacetime to the public on November 5, 1907 in a lecture to the Göttingen Mathematical society with the title, The Relativity Principle (Das Relativitatsprinzip). 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.^[19] On December 21, 1907, Minkowski spoke again to the Göttingen scientific society, and on September 21, 1908, Minkowski presented his famous talk, Space and Time (Raum und Zeit),^[21] to the German Society of Scientists and Physicians.^{[note 5]}

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. 1‑3), and included a remarkable demonstration that the concept of the invariant interval (discussed shortly), along with the empirical observation that the speed of light is finite, allows derivation of the entirety of special relativity.^{[note 6]}

Einstein, for his part, was initially dismissive of the 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.^[10]^{: 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

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In three-dimensions, the distance between two points can be defined using the Pythagorean theorem:

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 it measured by two different observers when one of the observers is moving, because of the Lorentz contraction. The situation is ever 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 time and distance carefully will find the same spacetime interval between any two events. Suppose an observer measures two events as being separated by a time $t$ and a spatial distance $x$ . Then the spacetime interval $s^{2}$ between the two events that are separated by a distance $\Delta x$ in space and a duration $\Delta t$ in time is:

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

(or for three space dimensions,

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

)

The constant $c$ , the speed of light, converts the units used to measure time (seconds) into units used to measure distance (meters).

Note on nomenclature: 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 Δx, etc. We are always concerned with the space and time differences between events. Since there is no preferred origin, fixed values of the coordinates have no real meaning.

The equation above is similar to the Pythagorean theorem, except with a minus sign between the $c^{2}t^{2}$ and the $x^{2}$ terms. Note also that 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.^[18]^: 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.

Note on nomenclature: There are two sign conventions in use in the relativity literature:

s^{2}=c^{2}t^{2}-x^{2}-y^{2}-z^{2}

and

s^{2}=-c^{2}t^{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

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In comparing measurements made by relatively moving observers in different reference frames, it is useful to work with the frames in a standard configuration. 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′.^[3]^: 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 ΔPQR forms a right triangle. Since OP = OQ = OR, the angle between x′ and x must also be θ.^[3]^{: 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

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In Fig. 2-4, event O is centered at the origin of the 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, since as illustrated in Fig. 2‑5, if another dimension is added to the diagram, the appearance would be that of two cones meeting with their apexes at the origin and centered around the time axis, one cone extending into the future, the other into the past. Since the spacetime interval is an invariant, all observers will assign the same events to the light cone of a given event. This statement is equivalent to the second postulate of special relativity: all observers measure the speed of light to be $c$ .^[18]^: 220

The light cone divides spacetime into separate regions. The interior of the future light cone consists of events that are separated from the origin event by more time than there is space: these events comprise the timelike future of the origin event. Likewise, the timelike past comprises the interior events of the past light cone. Since in timelike intervals, the separation in time is greater than the separation in space, timelike intervals are positive. The region exterior to the light cone consists of events that are separated from the origin event by more space than there is time. These events comprise the spacelike "Elsewhere" of the origin event. Events on the light cone itself are said to be lightlike (or null) separated from the origin. Because of the invariance of spacetime, all observers will agree on this division of spacetime.^[18]^: 220

The light cone has an essential role in defining the concept of causality. Signals cannot travel faster than the speed of light. In Fig. 2‑4, it is possible for a slower-than-or-equal-to-light-speed signal to travel from the position and time of O, to the position and time of D. It is hence possible for event O to have a causal influence on event D. The future light cone contains all of the events that could be causally influenced by O. Likewise, it is possible for a slower-than-or-equal-to-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 of the events that could have a causal influence on O. In contrast, in the Elsewhere region, event O cannot affect or be affected by event C, nor can event O affect or be affected by event B. There is no causal relationship between O and any events in the Elsewhere region.^[22]

Relativity of simultaneity

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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.^[23]

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 = 0.3 c, the events appear to occur in the order C, B, A. From the reference frame of an observer moving at v = −0.5 c, 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 = ${\textstyle {\sqrt {-s^{2}}}.}$ 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 = ${\textstyle {\sqrt {s^{2}}}}$ .^[18]^{: 220–221}

Invariant hyperbola

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In ordinary Euclidean space, the set of points that are equidistant from an origin form a circle (in two dimensions) or a sphere (in three dimensions). In Minkowski spacetime, the points at some constant spacetime interval from the origin form a curve given by the equation

c^{2}t^{2}-x^{2}={\text{some constant}}

The example above is the equation of a hyperbola in an x–ct spacetime diagram, which is termed an invariant hyperbola.

In Fig. 2-7a, the magenta hyperbolae connect events of equal spacelike separation from the origin, while the green hyperbolae connect events of equal timelike separation from the origin.

Note on nomenclature: The magenta hyperbolae, which cross the x axis, are termed timelike (not spacelike) hyperbolae because all intervals along the hyperbola are timelike intervals. Because of that, these hyperbolae represent actual paths that can be traversed by accerating particles in spacetime. On the other hand, the green hyperbolae, which cross the ct axis, are termed spacelike hyperbolae because all intervals along the hyperbolae are spacelike intervals.

Fig. 2‑7b shows that when viewed in an extra dimension of space, the timelike invariant hyperbolae generate hyperboloids of one sheet, while the spacelike invariant hyperbolae generate hyperboloids of two sheets.

Time dilation and length contraction

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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 0.3c, the elapsed time measured by the observer is 5.24 meters (1.75×10⁻⁸ s), while for a clock traveling at 0.7c, 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 than they would have without time dilation.^[18]^{: 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.^[3]^: 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.

Mutual time dilation and the twin paradox

Mutual time dilation

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Mutual time dilation and length contraction tend to strike beginners as inherently self-contradictory concepts. The worry is that if observer A measures observer B's clocks as running slowly, simply because B is moving at speed v relative to A, then the principle of relativity requires that observer B likewise measures A's clocks as running slowly. This is an important question that "goes to the heart of understanding special relativity."^[18]^: 198

Basically, A and B are performing two different measurements.

In order to measure the rate of ticking of one of B's clocks, A must use two of his own clocks, the first to record the time where B's clock first ticked at the first location of B, and second to record the time where B's clock emitted its second tick at the next location of B. Observer A needs two clocks because B is moving, so a grand total of three clocks are involved in the measurement. A's two clocks must be synchronized in A's frame. Conversely, B requires two clocks synchronized in her frame to record the ticks of A's clocks at the locations where A's clocks emitted their ticks. Therefore, A and B are performing their measurements with different sets of three clocks each. Since they are not doing the same measurement with the same clocks, there is no inherent necessity that the measurements be reciprocally "consistent" such that, if one observer measures the other's clock to be slow, the other observer measures the one's clock to be fast.^[18]^{: 198–199}

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 7]} 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.

Fig. 2-10 reinforces previous discussions about mutual time dilation. In this figure, Events A and C are separated from event O by equal timelike intervals. From the unprimed frame, events A and B are measured as simultaneous, but more time has passed for the unprimed observer than has passed for the primed observer. From the primed frame, events C and D are measured as simultaneous, but more time has passed for the primed observer than has passed for the unprimed observer. Each observer measures the clocks of the other observer as running more slowly.^[3]^: 124

Please note the importance of the word "measure". An observer's state of motion cannot affect an observed object, but it can affect the observer's observations of the object.

In Fig. 2-10, each line drawn parallel to the x axis represents a line of simultaneity for the unprimed observer. All events on that line have the same time value of ct. Likewise, each line drawn parallel to the x′ axis represents a line of simultaneity for the primed observer. All events on that line have the same time value of ct′.

Twin paradox

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Elementary introductions to special relativity often illustrate the differences between Galilean relativity and special relativity by posing a series of supposed "paradoxes". All paradoxes are, in reality, merely ill-posed or misunderstood 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.^[24]

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.^[18]^: 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.^[25]

Deeper analysis is needed before we can understand why these distinctions should result in a difference in the twins' ages. Consider the spacetime diagram of Fig. 2‑11. This 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 the evaluation the integral of the proper times along the curve (i.e. the path integral) to calculate the total amount of proper time experienced by the traveling twin.^[25]

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

For the rest of this discussion, we adopt Weiss's nomenclature, designating the stay-at-home twin as Terence and the traveling twin as Stella.^[25]

We had previously noted that Stella is not in an inertial frame. Given this fact, it is sometimes stated that full resolution of the twin paradox requires general relativity. This is not true.^[25]

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.^[25] 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.^[25]

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. We had previously noted that 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 Terrence 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. Terrence 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 Terrence has aged more than Stella when they are back together.^[25] As will be discussed in the forthcoming section Curvature of time, 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.^[18]^: 16

Gravitation

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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.^[18]^: 221 A few of these phenomena are described in the later sections of this article.

Basic mathematics of spacetime

Galilean transformations

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A basic goal is to be able to compare measurements made by observers in relative motion. Say we have 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′). Since we are dealing 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:^[26]^[27]

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.^[28]^: 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, assume that frame S′ is moving at velocity v with respect to frame S. Within frame S′, observer O′ measures an object moving with velocity u′. What is its velocity u with respect to frame S? Since x = ut, x′ = x − vt, and t = t′, we can write 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.

Example: Terence and Stella are at a 100 meter race. Terence is an official at the starting blocks, while Stella is a participant. At t = t′ = 0, Stella begins running at a speed of 9 m/s. At 5 s into the race, Terence, in his unprimed coordinate system, observes their mother, situated 45 m downfield from the starting blocks and 10 m to the left (45 m, 10 m, 0 m, 5 s), waving at Stella. Stella, in her primed coordinate system, observes their mother waving at her at x′ = x − vt = 45 m − 9 m/s × 5 s = 0 m, y′ = y = 10 m, i.e. she observes their mother as waving from directly to her left (0 m, 10 m, 0 m, 5 s).

Relativistic composition of velocities

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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 = &beta′. 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:^[28]^: 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.^[28]^: 49

Time dilation and length contraction revisited

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We had previously discussed, in qualitative terms, time dilation and length contraction. It is straightforward to obtain quantitative expressions for these effects. 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, we see in the literature 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 $\gamma$ :^[29]

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

We note that 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. Next, we note that 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

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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.

To transform the coordinates of an event from one frame to another in special relativity, we use the Lorentz transformations.

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, the v²/c² and vx/c² terms approach zero, and the Lorentz transformations approximate to the Galilean transformations.

As noted before, when we write $t'=\gamma (t-vx/c^{2}),$ $x'=\gamma (x-vt)$ and so forth, we most often really mean $\Delta t'=\gamma (\Delta t-v\Delta x/c^{2}),$ $\Delta x'=\gamma (\Delta x-v\Delta t)$ and so forth. Although, for brevity, we write the Lorentz transformation equations without deltas, it should be understood that x means Δx, etc. We are, in general, always concerned with the space and time differences between events.

Note on nomenclature: 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.^[30]^: 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′ = $\gamma$ (t − vx/c²) = 519.62 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.^[31]

The derivation given here and illustrated in Fig. 3‑5 is based on one presented by Bais^[28]^: 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. How do we determine w′ and x′ in terms of w and x? (Or the other way around, of course.)

It is easier at first to derive the inverse Lorentz transformation.

We start by noting that 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.^[30]^: 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.
We know that 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 = βγ'x′
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 we tacitly assumed while performing 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.^[28]^: 67 All inertial observers will agree on what constitutes accelerating and non-accelerating motion.^[30]^: 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.^[18]^: 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

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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. Fig. 3‑6a illustrates a Galilean spacetime diagram of the scenario where the source is moving directly away from a receiver that is stationary with respect to the medium. The source travels at a speed of v_s for a velocity parameter of β_s. With the source moving, the wavelength is affected, resulting in a frequency change as perceived by the observer. In the figure, the green arrows depict the transmission of each pulse at constant speed c through the medium. If the period T₀ of the pulses is 1 per unit of time, the received period is T = 1+β_s, so that the observed frequency f is given by

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

Fig. 3‑6b illustrates a Galilean spacetime diagram of the scenario where the receiver is moving directly away from a stationary source at a speed of v_r for a velocity parameter of β_r. With the source stationary, the wavelength is not changed, but the transmission velocity of the waves relative to the observer is decreased. In the diagram inset, we see that period T = 1+a. The variable "a" is calculated by noting the similar triangle relationship (1+a)/a = a/(a−β_r), so that T = 1/(1−β_r) and the observed frequency f is given by

f=(1-\beta _{r})f_{0}={\frac {c-v_{r}}{c}}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. 3‑7 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 = γw'. Since the slope of the green light ray is −1, T = w+βw = γw'(1+β). Hence, the relativistic Doppler effect is given by^[28]^: 58–59

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

Transverse Doppler effect

Suppose that a source, moving in a straight line, is at its closest point to the receiver. 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:^[30]^: 94–96

Fig. 3-8a. If a source, moving in a straight line, is crossing the receiver's field of view, what is the frequency measurement when the source is at its closest approach to the receiver?
Fig. 3-8b. If a source is moving in a straight line, what is the frequency measurement when the receiver sees the source as being closest to it?
Fig. 3-8c. If receiver is moving in a circle around the source, what frequency does the receiver measure?
Fig. 3-8d. If the source is moving in a circle around the receiver, what frequency does the receiver measure?

In scenario (a), when the source is closest to the receiver, the light hitting the receiver actually comes from a direction where the source had been some time back, and it has a significant longitudinal component, making an analysis from the frame of the receiver tricky. It is easier to make the analysis from S', the frame of the source. 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}}}

Scenario (b) is best analyzed from S, the frame of the receiver. 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, 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.)^[30]^: 94–96 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 8]}

Energy and momentum

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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 = mv. 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. 3‑9a, 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)' = γmc and p' = −βγmc, since the red axes are rescaled by gamma. Fig. 3‑9b 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.^[28]^: 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' = (1 − β)γE in a moving frame. This result can by derived by inspection of Fig. 3‑10 or by application of the Lorentz transformations, and is consistent with the analysis of Doppler effect given previously.^[28]^: 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, $\gamma$ approaches 1, so the spatial component of the relativistic momentum βγmc = γmv 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 = γm.
Likewise, comparing the time component of the relativistic momentum with that of the photon, γmc = m_relc = E/c, so that Einstein arrived at the relationship E = m_relc². 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 γmc² 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.^[28]^: 90–92^[30]^{: 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,^[32] 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.^[33] "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 that we had earlier established, E = ±pc.^[28]^: 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.

^[30]^{: 129–130, 180}

Conservation laws

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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.^[34] 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 p₁ = m₁v₁ and p₂ = m₂v₂ collide to produce a single particle of conserved mass m = m₁ + m₂ traveling at the center of mass velocity of the original system, v_cm = (m₁v₁ + m₂v₂)/(m₁ + m₂). The total momentum p = p₁ + p₂ is conserved.

Fig. 3‑11 illustrates the inelastic collision of two particles from a relativistic perspective. The time components E₁/c and E₂/c add up to total E/c of the resultant vector, meaning that energy is conserved. Likewise, the space components p₁ and p₂ 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 > m₁ + m₂.^[28]^: 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.^[30]^{: 134–138}

Choice of reference frames

The freedom to choose any frame in which to perform an analysis allows us to pick one which may be particularly convenient. For instance, whereas in conventional spacetime diagrams, the equivalence between the rest and moving frames is not immediately evident, if one chooses a third reference frame between the resting and moving frames with the two other frames moving in opposite directions with equal speed (the median frame), the result, often called a Loedel diagram (although it was independently developed by multiple authors), has equal units of length and time for both axes.^[35]^[36]^[37]^[38]

Figure 3-12.
(above) Lab Frame.
(right) Center of Momentum Frame

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. 3‑12 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.^[30]^{: 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 = m₁ + m₂. 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.^[30]^{: 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' = (v − u)/(1 − vu/c²), 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. As we have discussed in the previous section on four-momentum, this second option was what he chose.^[28]^: 104

Figure 3-13a. Energy-momentum diagram for decay of a charged pion

Figure 3-13b. 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.^[30]^: 127

Example: Because of the equivalence of mass and energy, elementary particle masses are customarily stated in energy units, where 1 MeV = 1×10⁶ 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. 3‑13a 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_ν = 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,^[39] so Fig. 3‑13b presents instead a graphing calculator solution. The energy of the neutrino is 29.79 MeV, and the energy of the muon is 33.91 − 29.79 = 4.12 MeV. Interestingly, most of the energy is carried off by the near-zero-mass neutrino.

Beyond the basics

Rapidity

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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.^[6]^: 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.^[40] 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^[30]^: 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;^[6]^: 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.^[30]^: 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.^[6]^[41] ^{[note 9]}

4‑vectors

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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 = (A₀, A₁, A₂, A₃) is a "4-vector" if its component A_i transform between frames according 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 Δt, Δx, Δy, Δz) under Lorentz transformations as well as rotations.^[24]^: 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 $E^{2}-p^{2}c^{2}.$ ^[30]^{: 178–181}^[24]^: 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 a rapidly accelerating particle as observed from a stationary frame.

Figure 4-3b. The momentarily comoving reference frames along the trajectory of a rapidly 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, as stated before in the earlier discussion of the transverse Doppler effect, 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: As discussed in the section on Energy and momentum,

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.^[30]^{: 178–181}^[24]^: 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. As noted in the previous discussion of energy and momentum conservation, 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.^[30]^: 186

Acceleration

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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.^[42]

Properly handling accelerating frames does requires 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.^[42]

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 10]} Will the string break?

The main article for this section recounts how, 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.^[30]^{: 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.^[30]^{: 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.^[30]^{: 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 of the spacelike line segment $A'B''$ turns out to be greater than the length of the spacelike line segment $AB$ .

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:^[43]

{\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}}

Accelerated observer with horizon

In the text accompanying Fig. 2‑7, we had noted that 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.^[28]^{: 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).^[28]^{: 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.^[28]^{: 110–113}

Introduction to curved spacetime

Basic propositions

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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 11]} 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.^[6]^{: 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.^[6]^{: 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.^[6]^{: 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."^[44]^: 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.^[45]^{: 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.^[45]^{: 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.^[45]^{: 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.

Rather than this section attempting to offer a (yet another) relatively non-mathematical presentation about general relativity, the reader is referred to the featured Wikipedia articles Introduction to general relativity and General relativity.

Instead, the focus in this section will be to explore a handful of elementary scenarios that serve to give somewhat of the flavor of general relativity.

Curvature of time

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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.^[24]^{: 118–126}

Shortly after the publication of the general theory in 1916, a number of scientists pointed out that general relativity predicts the existence of gravitational redshift. Einstein himself suggested 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 = (2gh)^1/2, so that its total energy E, as measured by an observer on the ground, is m = ½mv²/c² = m + mgh/c². (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'.^[24]^{: 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+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).^[46]

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 a 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.^[18]^: 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 note that 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.^[18]^: 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.^[47]^{: 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:^[18]^{: 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

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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.^[18]^{: 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.^[48] 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.^[49]

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.^[18]^{: 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.^[18]^{: 234–238}

The story of the 1919 Eddington eclipse expedition and Einstein's rise to fame is well told elsewhere.^[50]

Sources of spacetime curvature

Click here for a brief section summary

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 ≠ j terms (blue) represent shear stresses.^[51]

One important conclusion to be derived from the equations is that, colloquially speaking, gravity itself creates gravity.^{[note 12]} 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.^[18]^: 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. As we have discussed earlier in the section on Energy and 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.^[52]

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.^[53]) 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 << 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.

File:Galaxies-AGN-Inner-Structure.svg

Figure 5-8. Relativistic jet.

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.^[18]^{: 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.^[54]^[55]

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.^[18]^{: 243, 280}

The stress terms become highly significant when performing calculations such as hydrodynamic simulations of core-collapse supernovae.^[56]

Experimental verification

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,^[57] 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 didn't contribute as a source of gravity.

All that is well and good, but are there any direct, quantitative experimental or observational measurements that confirm that these terms contribute to gravity with the correct strength?

• Active, passive, and inertial mass

Before discussing the experimental evidence regarding these other sources of gravity, we need first to discuss Bondi's distinctions 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.^[58]

$m_{p}$ is the same as what we have earlier termed gravitational mass ( $m_{g}$ ) in our discussion of the equivalence principle in the Basic propositions section.

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 $m_{p}$ .^[58]

• 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).^[59]

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.^[60]

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.^[61]

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 1×10⁻¹².^[62]

• Gravitomagnetism

The existence of gravitomagnetism was proven by Gravity Probe B (GP-B), a satellite-based mission which launched on 20 April 2004.^[63] 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,^[64] while the geodetic effect was confirmed to better than 0.5%.^[65]^[66]

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,^[67] although there has been some disagreement on the accuracy of this result.^[68]

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.^[69]^[70]

Privileged character of 3+1 spacetime

The anthropic principle, also known as the observation selection effect, is the hypothesis that the range of possible observations that could be made about the universe is limited by the fact that observations are only possible in the type of universe that is capable of developing intelligent life. Proponents of the anthropic principle argue that it explains why the universe has the age and the fundamental physical constants necessary to accommodate intelligent life. If either had been significantly different, no one would have been around to make observations. Anthropic reasoning has been used to address the question as to why certain measured physical constants take the values that they do, rather than some other arbitrary values, and to explain a perception that the universe appears to be finely tuned for the existence of life.

There are many different formulations of the anthropic principle. Philosopher Nick Bostrom counts thirty, but the underlying principles can be divided into "weak" and "strong" forms, depending on the types of cosmological claims they entail. ^[71]

Definition and basis

The principle was formulated as a response to a series of observations that the laws of nature and parameters of the universe have values that are consistent with conditions for life as it is known rather than values that would not be consistent with life on Earth. The anthropic principle states that this is an a posteriori necessity, because if life were impossible, no living entity would be there to observe it, and thus it would not be known. That is, it must be possible to observe some universe, and hence, the laws and constants of any such universe must accommodate that possibility.

The term anthropic in "anthropic principle" has been argued^[72] to be a misnomer.^{[note 13]} While singling out the currently observable kind of carbon-based life, none of the finely tuned phenomena require human life or some kind of carbon chauvinism.^[73]^[74] Any form of life or any form of heavy atom, stone, star, or galaxy would do; nothing specifically human or anthropic is involved.^[75]

The anthropic principle has given rise to some confusion and controversy, partly because the phrase has been applied to several distinct ideas. All versions of the principle have been accused of discouraging the search for a deeper physical understanding of the universe. The anthropic principle is often criticized for lacking falsifiability and therefore its critics may point out that the anthropic principle is a non-scientific concept, even though the weak anthropic principle, "conditions that are observed in the universe must allow the observer to exist",^[76] is "easy" to support in mathematics and philosophy (i.e., it is a tautology or truism). However, building a substantive argument based on a tautological foundation is problematic. Stronger variants of the anthropic principle are not tautologies and thus make claims considered controversial by some and that are contingent upon empirical verification.^[77]

Anthropic observations

In 1961, Robert Dicke noted that the age of the universe, as seen by living observers, cannot be random.^[78] Instead, biological factors constrain the universe to be more or less in a "golden age", neither too young nor too old.^[79] If the universe were one tenth as old as its present age, there would not have been sufficient time to build up appreciable levels of metallicity (levels of elements besides hydrogen and helium) especially carbon, by nucleosynthesis. Small rocky planets did not yet exist. If the universe were 10 times older than it actually is, most stars would be too old to remain on the main sequence and would have turned into white dwarfs, aside from the dimmest red dwarfs, and stable planetary systems would have already come to an end. Thus, Dicke explained the coincidence between large dimensionless numbers constructed from the constants of physics and the age of the universe, a coincidence that inspired Dirac's varying-G theory.

Dicke later reasoned that the density of matter in the universe must be almost exactly the critical density needed to prevent the Big Crunch (the "Dicke coincidences" argument). The most recent measurements may suggest that the observed density of baryonic matter, and some theoretical predictions of the amount of dark matter, account for about 30% of this critical density, with the rest contributed by a cosmological constant. Steven Weinberg^[80] gave an anthropic explanation for this fact: he noted that the cosmological constant has a remarkably low value, some 120 orders of magnitude smaller than the value particle physics predicts (this has been described as the "worst prediction in physics").^[81] However, if the cosmological constant were only several orders of magnitude larger than its observed value, the universe would suffer catastrophic inflation, which would preclude the formation of stars, and hence life.

The observed values of the dimensionless physical constants (such as the fine-structure constant) governing the four fundamental interactions are balanced as if fine-tuned to permit the formation of commonly found matter and subsequently the emergence of life.^[82] A slight increase in the strong interaction (up to 50% for some authors^[83]) would bind the dineutron and the diproton and convert all hydrogen in the early universe to helium;^[84] likewise, an increase in the weak interaction also would convert all hydrogen to helium. Water, as well as sufficiently long-lived stable stars, both essential for the emergence of life as it is known, would not exist.^[85] More generally, small changes in the relative strengths of the four fundamental interactions can greatly affect the universe's age, structure, and capacity for life.

Origin

The phrase "anthropic principle" first appeared in Brandon Carter's contribution to a 1973 Kraków symposium honouring Copernicus's 500th birthday. Carter, a theoretical astrophysicist, articulated the Anthropic Principle in reaction to the Copernican Principle, which states that humans do not occupy a privileged position in the Universe. Carter said: "Although our situation is not necessarily central, it is inevitably privileged to some extent."^[86] Specifically, Carter disagreed with using the Copernican principle to justify the Perfect Cosmological Principle, which states that all large regions and times in the universe must be statistically identical. The latter principle underlies the steady-state theory, which had recently been falsified by the 1965 discovery of the cosmic microwave background radiation. This discovery was unequivocal evidence that the universe has changed radically over time (for example, via the Big Bang).^{[citation needed]}

Carter defined two forms of the anthropic principle, a "weak" one which referred only to anthropic selection of privileged spacetime locations in the universe, and a more controversial "strong" form that addressed the values of the fundamental constants of physics.

Roger Penrose explained the weak form as follows:

The argument can be used to explain why the conditions happen to be just right for the existence of (intelligent) life on the Earth at the present time. For if they were not just right, then we should not have found ourselves to be here now, but somewhere else, at some other appropriate time. This principle was used very effectively by Brandon Carter and Robert Dicke to resolve an issue that had puzzled physicists for a good many years. The issue concerned various striking numerical relations that are observed to hold between the physical constants (the gravitational constant, the mass of the proton, the age of the universe, etc.). A puzzling aspect of this was that some of the relations hold only at the present epoch in the Earth's history, so we appear, coincidentally, to be living at a very special time (give or take a few million years!). This was later explained, by Carter and Dicke, by the fact that this epoch coincided with the lifetime of what are called main-sequence stars, such as the Sun. At any other epoch, the argument ran, there would be no intelligent life around to measure the physical constants in question—so the coincidence had to hold, simply because there would be intelligent life around only at the particular time that the coincidence did hold!
— The Emperor's New Mind, chapter 10

One reason this is plausible is that there are many other places and times in which humans could have evolved. But when applying the strong principle, there is only one universe, with one set of fundamental parameters, so what exactly is the point being made? Carter offers two possibilities: First, humans can use their own existence to make "predictions" about the parameters. But second, "as a last resort", humans can convert these predictions into explanations by assuming that there is more than one universe, in fact a large and possibly infinite collection of universes, something that is now called the multiverse ("world ensemble" was Carter's term), in which the parameters (and perhaps the laws of physics) vary across universes. The strong principle then becomes an example of a selection effect, exactly analogous to the weak principle. Postulating a multiverse is certainly a radical step, but taking it could provide at least a partial answer to a question seemingly out of the reach of normal science: "Why do the fundamental laws of physics take the particular form we observe and not another?"

Since Carter's 1973 paper, the term anthropic principle has been extended to cover a number of ideas that differ in important ways from his. Particular confusion was caused by the 1986 book The Anthropic Cosmological Principle by John D. Barrow and Frank Tipler,^[87] which distinguished between a "weak" and "strong" anthropic principle in a way very different from Carter's, as discussed in the next section.

Carter was not the first to invoke some form of the anthropic principle. In fact, the evolutionary biologist Alfred Russel Wallace anticipated the anthropic principle as long ago as 1904: "Such a vast and complex universe as that which we know exists around us, may have been absolutely required [...] in order to produce a world that should be precisely adapted in every detail for the orderly development of life culminating in man."^[88] In 1957, Robert Dicke wrote: "The age of the Universe 'now' is not random but conditioned by biological factors [...] [changes in the values of the fundamental constants of physics] would preclude the existence of man to consider the problem."^[89]

Ludwig Boltzmann may have been one of the first in modern science to use anthropic reasoning. Prior to knowledge of the Big Bang Boltzmann's thermodynamic concepts painted a picture of a universe that had inexplicably low entropy. Boltzmann suggested several explanations, one of which relied on fluctuations that could produce pockets of low entropy or Boltzmann universes. While most of the universe is featureless in this model, to Boltzmann, it is unremarkable that humanity happens to inhabit a Boltzmann universe, as that is the only place where intelligent life could be.^[90]^[91]

Variants

Weak anthropic principle (WAP) (Carter): "... our location in the universe is necessarily privileged to the extent of being compatible with our existence as observers."^[86] For Carter, "location" refers to our location in time as well as space.

Strong anthropic principle (SAP) (Carter): "[T]he universe (and hence the fundamental parameters on which it depends) must be such as to admit the creation of observers within it at some stage. To paraphrase Descartes, cogito ergo mundus talis est."
The Latin tag ("I think, therefore the world is such [as it is]") makes it clear that "must" indicates a deduction from the fact of our existence; the statement is thus a truism.

In their 1986 book, The anthropic cosmological principle, John Barrow and Frank Tipler depart from Carter and define the WAP and SAP as follows:^[92]^[93]

Weak anthropic principle (WAP) (Barrow and Tipler): "The observed values of all physical and cosmological quantities are not equally probable but they take on values restricted by the requirement that there exist sites where carbon-based life can evolve and by the requirements that the universe be old enough for it to have already done so."^[94]
Unlike Carter they restrict the principle to carbon-based life, rather than just "observers". A more important difference is that they apply the WAP to the fundamental physical constants, such as the fine-structure constant, the number of spacetime dimensions, and the cosmological constant—topics that fall under Carter's SAP.

Strong anthropic principle (SAP) (Barrow and Tipler): "The Universe must have those properties which allow life to develop within it at some stage in its history."^[95]
This looks very similar to Carter's SAP, but unlike the case with Carter's SAP, the "must" is an imperative, as shown by the following three possible elaborations of the SAP, each proposed by Barrow and Tipler:^[96]

"There exists one possible Universe 'designed' with the goal of generating and sustaining 'observers'."
This can be seen as simply the classic design argument restated in the garb of contemporary cosmology. It implies that the purpose of the universe is to give rise to intelligent life, with the laws of nature and their fundamental physical constants set to ensure that life emerges and evolves.
"Observers are necessary to bring the Universe into being."
Barrow and Tipler believe that this is a valid conclusion from quantum mechanics, as John Archibald Wheeler has suggested, especially via his idea that information is the fundamental reality (see It from bit) and his Participatory anthropic principle (PAP) which is an interpretation of quantum mechanics associated with the ideas of John von Neumann and Eugene Wigner.
"An ensemble of other different universes is necessary for the existence of our Universe."
By contrast, Carter merely says that an ensemble of universes is necessary for the SAP to count as an explanation.

The philosophers John Leslie^[97] and Nick Bostrom^[91] reject the Barrow and Tipler SAP as a fundamental misreading of Carter. For Bostrom, Carter's anthropic principle just warns us to make allowance for anthropic bias—that is, the bias created by anthropic selection effects (which Bostrom calls "observation" selection effects)—the necessity for observers to exist in order to get a result. He writes:

Many 'anthropic principles' are simply confused. Some, especially those drawing inspiration from Brandon Carter's seminal papers, are sound, but... they are too weak to do any real scientific work. In particular, I argue that existing methodology does not permit any observational consequences to be derived from contemporary cosmological theories, though these theories quite plainly can be and are being tested empirically by astronomers. What is needed to bridge this methodological gap is a more adequate formulation of how observation selection effects are to be taken into account.
— Anthropic bias, Introduction^[98]

Strong self-sampling assumption (SSSA) (Bostrom): "Each observer-moment should reason as if it were randomly selected from the class of all observer-moments in its reference class."
Analysing an observer's experience into a sequence of "observer-moments" helps avoid certain paradoxes; but the main ambiguity is the selection of the appropriate "reference class": for Carter's WAP this might correspond to all real or potential observer-moments in our universe; for the SAP, to all in the multiverse. Bostrom's mathematical development shows that choosing either too broad or too narrow a reference class leads to counter-intuitive results, but he is not able to prescribe an ideal choice.

According to Jürgen Schmidhuber, the anthropic principle essentially just says that the conditional probability of finding yourself in a universe compatible with your existence is always 1. It does not allow for any additional nontrivial predictions such as "gravity won't change tomorrow". To gain more predictive power, additional assumptions on the prior distribution of alternative universes are necessary.^[99]^[100]

Playwright and novelist Michael Frayn describes a form of the strong anthropic principle in his 2006 book The Human Touch, which explores what he characterises as "the central oddity of the Universe":

It's this simple paradox. The Universe is very old and very large. Humankind, by comparison, is only a tiny disturbance in one small corner of it – and a very recent one. Yet the Universe is only very large and very old because we are here to say it is... And yet, of course, we all know perfectly well that it is what it is whether we are here or not.^[101]

Character of anthropic reasoning

Carter chose to focus on a tautological aspect of his ideas, which has resulted in much confusion. In fact, anthropic reasoning interests scientists because of something that is only implicit in the above formal definitions, namely that humans should give serious consideration to there being other universes with different values of the "fundamental parameters"—that is, the dimensionless physical constants and initial conditions for the Big Bang. Carter and others have argued that life would not be possible in most such universes. In other words, the universe humans live in is fine tuned to permit life. Collins & Hawking (1973) characterized Carter's then-unpublished big idea as the postulate that "there is not one universe but a whole infinite ensemble of universes with all possible initial conditions".^[102] If this is granted, the anthropic principle provides a plausible explanation for the fine tuning of our universe: the "typical" universe is not fine-tuned, but given enough universes, a small fraction will be capable of supporting intelligent life. Ours must be one of these, and so the observed fine tuning should be no cause for wonder.

Although philosophers have discussed related concepts for centuries, in the early 1970s the only genuine physical theory yielding a multiverse of sorts was the many-worlds interpretation of quantum mechanics. This would allow variation in initial conditions, but not in the truly fundamental constants. Since that time a number of mechanisms for producing a multiverse have been suggested: see the review by Max Tegmark.^[103] An important development in the 1980s was the combination of inflation theory with the hypothesis that some parameters are determined by symmetry breaking in the early universe, which allows parameters previously thought of as "fundamental constants" to vary over very large distances, thus eroding the distinction between Carter's weak and strong principles. At the beginning of the 21st century, the string landscape emerged as a mechanism for varying essentially all the constants, including the number of spatial dimensions.^{[note 14]}

The anthropic idea that fundamental parameters are selected from a multitude of different possibilities (each actual in some universe or other) contrasts with the traditional hope of physicists for a theory of everything having no free parameters. As Albert Einstein said: "What really interests me is whether God had any choice in the creation of the world." In 2002, some proponents of the leading candidate for a "theory of everything", string theory, proclaimed "the end of the anthropic principle"^[104] since there would be no free parameters to select. In 2003, however, Leonard Susskind stated: "... it seems plausible that the landscape is unimaginably large and diverse. This is the behavior that gives credence to the anthropic principle."^[105]

The modern form of a design argument is put forth by intelligent design. Proponents of intelligent design often cite the fine-tuning observations that (in part) preceded the formulation of the anthropic principle by Carter as a proof of an intelligent designer. Opponents of intelligent design are not limited to those who hypothesize that other universes exist; they may also argue, anti-anthropically, that the universe is less fine-tuned than often claimed, or that accepting fine tuning as a brute fact is less astonishing than the idea of an intelligent creator. Furthermore, even accepting fine tuning, Sober (2005)^[106] and Ikeda and Jefferys,^[107]^[108] argue that the anthropic principle as conventionally stated actually undermines intelligent design.

Paul Davies's book The Goldilocks Enigma (2006) reviews the current state of the fine-tuning debate in detail, and concludes by enumerating the following responses to that debate:^[79]^{: 261–267}

The absurd universe: Our universe just happens to be the way it is.
The unique universe: There is a deep underlying unity in physics that necessitates the Universe being the way it is. A Theory of Everything will explain why the various features of the Universe must have exactly the values that have been recorded.
The multiverse: Multiple universes exist, having all possible combinations of characteristics, and humans inevitably find themselves within a universe that allows us to exist.
Intelligent design: A creator designed the Universe with the purpose of supporting complexity and the emergence of intelligence.
The life principle: There is an underlying principle that constrains the Universe to evolve towards life and mind.
The self-explaining universe: A closed explanatory or causal loop: "perhaps only universes with a capacity for consciousness can exist". This is Wheeler's participatory anthropic principle (PAP).
The fake universe: Humans live inside a virtual reality simulation.

Omitted here is Lee Smolin's model of cosmological natural selection, also known as fecund universes, which proposes that universes have "offspring" that are more plentiful if they resemble our universe. Also see Gardner (2005).^[109]

Clearly each of these hypotheses resolve some aspects of the puzzle, while leaving others unanswered. Followers of Carter would admit only option 3 as an anthropic explanation, whereas 3 through 6 are covered by different versions of Barrow and Tipler's SAP (which would also include 7 if it is considered a variant of 4, as in Tipler 1994).

The anthropic principle, at least as Carter conceived it, can be applied on scales much smaller than the whole universe. For example, Carter (1983)^[110] inverted the usual line of reasoning and pointed out that when interpreting the evolutionary record, one must take into account cosmological and astrophysical considerations. With this in mind, Carter concluded that given the best estimates of the age of the universe, the evolutionary chain culminating in Homo sapiens probably admits only one or two low probability links.

Observational evidence

No possible observational evidence bears on Carter's WAP, as it is merely advice to the scientist and asserts nothing debatable. The obvious test of Barrow's SAP, which says that the universe is "required" to support life, is to find evidence of life in universes other than ours. Any other universe is, by most definitions, unobservable (otherwise it would be included in our portion of this universe^{[undue weight? – discuss]}). Thus, in principle Barrow's SAP cannot be falsified by observing a universe in which an observer cannot exist.

Philosopher John Leslie^[111] states that the Carter SAP (with multiverse) predicts the following:

Physical theory will evolve so as to strengthen the hypothesis that early phase transitions occur probabilistically rather than deterministically, in which case there will be no deep physical reason for the values of fundamental constants;
Various theories for generating multiple universes will prove robust;
Evidence that the universe is fine tuned will continue to accumulate;
No life with a non-carbon chemistry will be discovered;
Mathematical studies of galaxy formation will confirm that it is sensitive to the rate of expansion of the universe.

Hogan^[112] has emphasised that it would be very strange if all fundamental constants were strictly determined, since this would leave us with no ready explanation for apparent fine tuning. In fact, humans might have to resort to something akin to Barrow and Tipler's SAP: there would be no option for such a universe not to support life.

Probabilistic predictions of parameter values can be made given:

a particular multiverse with a "measure", i.e. a well defined "density of universes" (so, for parameter X, one can calculate the prior probability P(X₀) dX that X is in the range X₀ < X < X₀ + dX), and
an estimate of the number of observers in each universe, N(X) (e.g., this might be taken as proportional to the number of stars in the universe).

The probability of observing value X is then proportional to N(X) P(X). A generic feature of an analysis of this nature is that the expected values of the fundamental physical constants should not be "over-tuned", i.e. if there is some perfectly tuned predicted value (e.g. zero), the observed value need be no closer to that predicted value than what is required to make life possible. The small but finite value of the cosmological constant can be regarded as a successful prediction in this sense.

One thing that would not count as evidence for the anthropic principle is evidence that the Earth or the Solar System occupied a privileged position in the universe, in violation of the Copernican principle (for possible counterevidence to this principle, see Copernican principle), unless there was some reason to think that that position was a necessary condition for our existence as observers.

Applications of the principle

The nucleosynthesis of carbon-12

Fred Hoyle may have invoked anthropic reasoning to predict an astrophysical phenomenon. He is said to have reasoned, from the prevalence on Earth of life forms whose chemistry was based on carbon-12 nuclei, that there must be an undiscovered resonance in the carbon-12 nucleus facilitating its synthesis in stellar interiors via the triple-alpha process. He then calculated the energy of this undiscovered resonance to be 7.6 million electronvolts.^[113]^[114] Willie Fowler's research group soon found this resonance, and its measured energy was close to Hoyle's prediction.

However, in 2010 Helge Kragh argued that Hoyle did not use anthropic reasoning in making his prediction, since he made his prediction in 1953 and anthropic reasoning did not come into prominence until 1980. He called this an "anthropic myth", saying that Hoyle and others made an after-the-fact connection between carbon and life decades after the discovery of the resonance.

An investigation of the historical circumstances of the prediction and its subsequent experimental confirmation shows that Hoyle and his contemporaries did not associate the level in the carbon nucleus with life at all.^[115]

Cosmic inflation

Don Page criticized the entire theory of cosmic inflation as follows.^[116] He emphasized that initial conditions that made possible a thermodynamic arrow of time in a universe with a Big Bang origin, must include the assumption that at the initial singularity, the entropy of the universe was low and therefore extremely improbable. Paul Davies rebutted this criticism by invoking an inflationary version of the anthropic principle.^[117] While Davies accepted the premise that the initial state of the visible universe (which filled a microscopic amount of space before inflating) had to possess a very low entropy value—due to random quantum fluctuations—to account for the observed thermodynamic arrow of time, he deemed this fact an advantage for the theory. That the tiny patch of space from which our observable universe grew had to be extremely orderly, to allow the post-inflation universe to have an arrow of time, makes it unnecessary to adopt any "ad hoc" hypotheses about the initial entropy state, hypotheses other Big Bang theories require.

String theory

String theory predicts a large number of possible universes, called the "backgrounds" or "vacua". The set of these vacua is often called the "multiverse" or "anthropic landscape" or "string landscape". Leonard Susskind has argued that the existence of a large number of vacua puts anthropic reasoning on firm ground: only universes whose properties are such as to allow observers to exist are observed, while a possibly much larger set of universes lacking such properties go unnoticed.^[105]

Steven Weinberg^[118] believes the anthropic principle may be appropriated by cosmologists committed to nontheism, and refers to that principle as a "turning point" in modern science because applying it to the string landscape "may explain how the constants of nature that we observe can take values suitable for life without being fine-tuned by a benevolent creator". Others—most notably David Gross but also Lubos Motl, Peter Woit, and Lee Smolin—argue that this is not predictive. Max Tegmark,^[119] Mario Livio, and Martin Rees^[120] argue that only some aspects of a physical theory need be observable and/or testable for the theory to be accepted, and that many well-accepted theories are far from completely testable at present.

Jürgen Schmidhuber (2000–2002) points out that Ray Solomonoff's theory of universal inductive inference and its extensions already provide a framework for maximizing our confidence in any theory, given a limited sequence of physical observations, and some prior distribution on the set of possible explanations of the universe.

Zhi-Wei Wang and Samuel L. Braunstein proved that life's existence in the universe depends on various fundamental constants. It suggests that without a complete understanding of these constants, one might incorrectly perceive the universe as being intelligently designed for life. This perspective challenges the view that our universe is unique in its ability to support life.^[121]

Dimensions of spacetime

There are two kinds of dimensions: spatial (bidirectional) and temporal (unidirectional).^[123] 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".^[124] 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 15]}

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.^[125] 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.^[126] 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.^[127]

Max Tegmark expands on the preceding argument in the following anthropic manner.^[128] 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.)^[128] 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.^[128] 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.^[129]

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.^[130]^[131]

Metaphysical interpretations

Some of the metaphysical disputes and speculations include, for example, attempts to back Pierre Teilhard de Chardin's earlier interpretation of the universe as being Christ centered (compare Omega Point), expressing a creatio evolutiva instead the elder notion of creatio continua.^[132] From a strictly secular, humanist perspective, it allows as well to put human beings back in the center, an anthropogenic shift in cosmology.^[132] Karl W. Giberson^[133] has laconically stated that

What emerges is the suggestion that cosmology may at last be in possession of some raw material for a postmodern creation myth.
— Karl W. Giberson

William Sims Bainbridge disagreed with de Chardin's optimism about a future Omega point at the end of history, arguing that logically, humans are trapped at the Omicron point, in the middle of the Greek alphabet rather than advancing to the end, because the universe does not need to have any characteristics that would support our further technical progress, if the anthropic principle merely requires it to be suitable for our evolution to this point.^[134]

The anthropic cosmological principle

A thorough extant study of the anthropic principle is the book The anthropic cosmological principle by John D. Barrow, a cosmologist, and Frank J. Tipler, a cosmologist and mathematical physicist. This book sets out in detail the many known anthropic coincidences and constraints, including many found by its authors. While the book is primarily a work of theoretical astrophysics, it also touches on quantum physics, chemistry, and earth science. An entire chapter argues that Homo sapiens is, with high probability, the only intelligent species in the Milky Way.

The book begins with an extensive review of many topics in the history of ideas the authors deem relevant to the anthropic principle, because the authors believe that principle has important antecedents in the notions of teleology and intelligent design. They discuss the writings of Fichte, Hegel, Bergson, and Alfred North Whitehead, and the Omega Point cosmology of Teilhard de Chardin. Barrow and Tipler carefully distinguish teleological reasoning from eutaxiological reasoning; the former asserts that order must have a consequent purpose; the latter asserts more modestly that order must have a planned cause. They attribute this important but nearly always overlooked distinction to an obscure 1883 book by L. E. Hicks.^[135]

Seeing little sense in a principle requiring intelligent life to emerge while remaining indifferent to the possibility of its eventual extinction, Barrow and Tipler propose the final anthropic principle (FAP): Intelligent information-processing must come into existence in the universe, and, once it comes into existence, it will never die out.^[136]

Barrow and Tipler submit that the FAP is both a valid physical statement and "closely connected with moral values". FAP places strong constraints on the structure of the universe, constraints developed further in Tipler's The Physics of Immortality.^[137] One such constraint is that the universe must end in a Big Crunch, which seems unlikely in view of the tentative conclusions drawn since 1998 about dark energy, based on observations of very distant supernovas.

In his review^[138] of Barrow and Tipler, Martin Gardner ridiculed the FAP by quoting the last two sentences of their book as defining a completely ridiculous anthropic principle (CRAP):

At the instant the Omega Point is reached, life will have gained control of all matter and forces not only in a single universe, but in all universes whose existence is logically possible; life will have spread into all spatial regions in all universes which could logically exist, and will have stored an infinite amount of information, including all bits of knowledge that it is logically possible to know. And this is the end.^[139]

Reception and controversies

Carter has frequently expressed regret for his own choice of the word "anthropic", because it conveys the misleading impression that the principle involves humans in particular, to the exclusion of non-human intelligence more broadly.^[140] Others^[141] have criticised the word "principle" as being too grandiose to describe straightforward applications of selection effects.

A common criticism of Carter's SAP is that it is an easy deus ex machina that discourages searches for physical explanations. To quote Penrose again: "It tends to be invoked by theorists whenever they do not have a good enough theory to explain the observed facts."^[142]

Carter's SAP and Barrow and Tipler's WAP have been dismissed as truisms or trivial tautologies—that is, statements true solely by virtue of their logical form and not because a substantive claim is made and supported by observation of reality. As such, they are criticized as an elaborate way of saying, "If things were different, they would be different",^{[citation needed]} which is a valid statement, but does not make a claim of some factual alternative over another.

Critics of the Barrow and Tipler SAP claim that it is neither testable nor falsifiable, and thus is not a scientific statement but rather a philosophical one. The same criticism has been leveled against the hypothesis of a multiverse, although some argue^[143] that it does make falsifiable predictions. A modified version of this criticism is that humanity understands so little about the emergence of life, especially intelligent life, that it is effectively impossible to calculate the number of observers in each universe. Also, the prior distribution of universes as a function of the fundamental constants is easily modified to get any desired result.^[144]

Many criticisms focus on versions of the strong anthropic principle, such as Barrow and Tipler's anthropic cosmological principle, which are teleological notions that tend to describe the existence of life as a necessary prerequisite for the observable constants of physics. Similarly, Stephen Jay Gould,^[145]^[146] Michael Shermer,^[147] and others claim that the stronger versions of the anthropic principle seem to reverse known causes and effects. Gould compared the claim that the universe is fine-tuned for the benefit of our kind of life to saying that sausages were made long and narrow so that they could fit into modern hotdog buns, or saying that ships had been invented to house barnacles.^{[citation needed]} These critics cite the vast physical, fossil, genetic, and other biological evidence consistent with life having been fine-tuned through natural selection to adapt to the physical and geophysical environment in which life exists. Life appears to have adapted to the universe, and not vice versa.

Some applications of the anthropic principle have been criticized as an argument by lack of imagination, for tacitly assuming that carbon compounds and water are the only possible chemistry of life (sometimes called "carbon chauvinism", see also alternative biochemistry).^[148] The range of fundamental physical constants consistent with the evolution of carbon-based life may also be wider than those who advocate a fine-tuned universe have argued.^[149] For instance, Harnik et al.^[150] propose a Weakless Universe in which the weak nuclear force is eliminated. They show that this has no significant effect on the other fundamental interactions, provided some adjustments are made in how those interactions work. However, if some of the fine-tuned details of our universe were violated, that would rule out complex structures of any kind—stars, planets, galaxies, etc.

Lee Smolin has offered a theory designed to improve on the lack of imagination that anthropic principles have been accused of. He puts forth his fecund universes theory, which assumes universes have "offspring" through the creation of black holes whose offspring universes have values of physical constants that depend on those of the mother universe.^[151]

The philosophers of cosmology John Earman,^[152] Ernan McMullin,^[153] and Jesús Mosterín contend that "in its weak version, the anthropic principle is a mere tautology, which does not allow us to explain anything or to predict anything that we did not already know. In its strong version, it is a gratuitous speculation".^[154] A further criticism by Mosterín concerns the flawed "anthropic" inference from the assumption of an infinity of worlds to the existence of one like ours:

The suggestion that an infinity of objects characterized by certain numbers or properties implies the existence among them of objects with any combination of those numbers or characteristics [...] is mistaken. An infinity does not imply at all that any arrangement is present or repeated. [...] The assumption that all possible worlds are realized in an infinite universe is equivalent to the assertion that any infinite set of numbers contains all numbers (or at least all Gödel numbers of the [defining] sequences), which is obviously false.

Notes

^ To save screen space and data usage, Wikipedia mobile collapses sections when displaying on a phone screen. The target of an internal link does not exist unless the section where it is located has been expanded at least once and cached.
^ 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. Galison (2003), op cit.
^ 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. Galison (2003), op cit.
^ The geometry of Minkowski spacetime is closely connected to certain variants of sphere geometry (such as Lie sphere geometry or Conformal geometry) developed in the 19th century. For instance, the Lorentz transformation is a special case of spherical wave transformations. In particular, as pointed out by Poincaré (1912) and others, it is simply isomorphic to the Laguerre group which transforms spheres into spheres and planes into planes. The isomorphism between the Möbius group (which is isomorphic to the group of isometries in hyperbolic R3) and the Lorentz group is also well known.
^ (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." Minkowski (1909), op cit.
^ In a Cartesian plane, ordinary rotation leaves a circle unchanged. In spacetime, hyperbolic rotation preserves the hyperbolic metric.
^ 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. See Carroll, (op. cit., pp. 112–113)
^ "Anthropic" means "of or pertaining to mankind or humans".
^ Strictly speaking, the number of non-compact dimensions; see String theory.
^ 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.

Footnotes

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^ Tegmark (1998) op. cit.
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^ Tegmark, Max (1 April 1997). "On the dimensionality of spacetime". Classical and Quantum Gravity. 14 (4): L69–L75. arXiv:gr-qc/9702052. Bibcode:1997CQGra..14L..69T. doi:10.1088/0264-9381/14/4/002. ISSN 0264-9381. S2CID 250904081.
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^ ^a ^b ^c 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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^ ^a ^b Johann. Dorschner. und. Ralph. Neuhäuser. Evolution. des. Kosmos. und. der. Punkt. omega, in Nikolaus Knoepffler, H. James Birx, Teilhard de Chardin, V&R unipress GmbH, 2005, p. 109 ff
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^ Bainbridge, William Sims, 1997, "The Omicron point: Sociological application of the anthropic theory, in Chaos and complexity in sociology: Myths, models and theory, Raymond E. Eve, Sara Horsfall, and Mary E. Lee, editors, pp. 91–101. Thousand Oaks, California: Sage publications. ISBN 0-7619-0890-0
^ Hicks, L. E. (1883). A critique of design arguments. New York: Scribner's.
^ Barrow and Tipler 1986: 23
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^ Gardner, M., "WAP, SAP, PAP, and FAP", The New York review of books 23, No. 8 (May 8, 1986): 22–25.
^ Barrow and Tipler 1986: 677
^ e.g. Carter (2004) op. cit.
^ e.g. message from Martin Rees presented at the Kavli-CERCA conference (see video in External links)
^ Penrose, R. (1989). "10". The Emperor's new mind. Oxford University Press. ISBN 978-0-19-851973-7.
^ Are parallel universes unscientific nonsense? Insider tips for criticizing the multiverse Tegmark, Max. February 4, 2014.
^ Starkman, G. D.; Trotta, R. (2006). "Why anthropic reasoning cannot predict Λ". Physical Review Letters. 97 (20): 201301. arXiv:astro-ph/0607227. Bibcode:2006PhRvL..97t1301S. doi:10.1103/PhysRevLett.97.201301. PMID 17155671. S2CID 27409290. See also this news story.
^ Gould, Stephen Jay (1998). "Clear thinking in the sciences". Lectures at Harvard University.
^ Gould, Stephen Jay (2002). Why people believe weird things: Pseudoscience, superstition, and other confusions of our time. W.H. Freeman. ISBN 978-0-7167-3090-3.
^ Shermer, Michael (2007). Why Darwin matters. Macmillan. ISBN 978-0-8050-8121-3.
^ e.g. Carr, B. J.; Rees, M. J. (1979). "The anthropic principle and the structure of the physical world". Nature. 278 (5705): 605–612. Bibcode:1979Natur.278..605C. doi:10.1038/278605a0. S2CID 4363262.
^ Stenger, Victor J. (2000). Timeless reality: Symmetry, simplicity, and multiple universes. Prometheus books. ISBN 978-1-57392-859-5.
^ Harnik, R.; Kribs, G.; Perez, G. (2006). "A Universe without weak interactions". Physical Review. D74 (3): 035006. arXiv:hep-ph/0604027. Bibcode:2006PhRvD..74c5006H. doi:10.1103/PhysRevD.74.035006. S2CID 14340180.
^ Lee Smolin (2001). Tyson, Neil deGrasse; Soter, Steve (eds.). Cosmic horizons: Astronomy at the cutting edge. The New Press. pp. 148–152. ISBN 978-1-56584-602-9.
^ Earman John (1987). "The SAP also rises: A critical examination of the anthropic principle". American Philosophical Quarterly. 24: 307–317.
^ McMullin, Ernan. (1994). "Fine-tuning the Universe?" In M. Shale & G. Shields (ed.), Science, technology, and religious ideas, Lanham: University Press of America.
^ Mosterín, Jesús. (2005). Op. cit.

References

Barrow, John D.; Tipler, Frank J. (1986). The Anthropic Cosmological Principle (1st ed.). Oxford University Press. ISBN 978-0-19-282147-8. LCCN 87028148.
Bostrom, N. (2002), Anthropic bias: Observation selection effects in science and philosophy, Routledge, ISBN 978-0-415-93858-7 5 chapters available online. Archived 2006-08-26 at the Wayback Machine
Cirkovic, M. M. (2002). "On the first anthropic argument in astrobiology". Earth, Moon, and Planets. 91 (4): 243–254. arXiv:astro-ph/0306185. Bibcode:2002EM&P...91..243C. doi:10.1023/A:1026266630823. S2CID 17341587.
Cirkovic, M. M. (2004). "The anthropic principle and the duration of the cosmological past". Astronomical and Astrophysical Transactions. 23 (6): 567–597. arXiv:astro-ph/0505005. Bibcode:2004A&AT...23..567C. doi:10.1080/10556790412331335327. S2CID 6068309.
Conway Morris, Simon (2003). Life's solution: Inevitable humans in a lonely Universe. Cambridge University Press.
Craig, William Lane (1987). "Critical review of The anthropic cosmological principle". International Philosophical Quarterly. 27 (4): 437–47. doi:10.5840/ipq198727433.
Hawking, Stephen W. (1988). A brief history of time. New York: Bantam books. p. 174. ISBN 978-0-553-34614-5.
Stenger, Victor J. (1999), "Anthropic design", The skeptical inquirer 23 (August 31, 1999): 40–43
Mosterín, Jesús (2005). "Anthropic explanations in cosmology". In P. Háyek, L. Valdés and D. Westerstahl (ed.), Logic, methodology and philosophy of science, Proceedings of the 12th international congress of the LMPS. London: King's college publications, pp. 441–473. ISBN 1-904987-21-4.
Taylor; Stuart Ross (1998). Destiny or chance: Our Solar System and its place in the cosmos. Cambridge University Press. ISBN 978-0-521-78521-1.
Tegmark, Max (1997). "On the dimensionality of spacetime". 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. A simple anthropic argument for why there are 3 spatial and 1 temporal dimensions.
Tipler, F. J. (2003). "Intelligent life in cosmology". International Journal of Astrobiology. 2 (2): 141–48. arXiv:0704.0058. Bibcode:2003IJAsB...2..141T. doi:10.1017/S1473550403001526. S2CID 119283361.
Walker, M. A. & Cirkovic, M. M. (2006). "Anthropic reasoning, naturalism and the contemporary design argument". International Studies in the Philosophy of Science. 20 (3): 285–307. CiteSeerX 10.1.1.212.2588. doi:10.1080/02698590600960945. S2CID 8804703. Shows that some of the common criticisms of anthropic principle based on its relationship with numerology or the theological design argument are wrong.
Ward, P. D. & Brownlee, D. (2000). Rare Earth: Why complex life is uncommon in the Universe. Springer Verlag. ISBN 978-0-387-98701-9.
Vilenkin, Alex (2006). Many worlds in one: The search for other universes. Hill and Wang. ISBN 978-0-8090-9523-0.

External links

Nick Bostrom: web site devoted to the anthropic principle.
Friederich, Simon. Fine-tuning, review article of the discussion about fine-tuning, highlighting the role of the anthropic principles.
Gijsbers, Victor. (2000). Theistic anthropic principle refuted – Positive atheism magazine.
Chown, Marcus, Anything Goes, New scientist, 6 June 1998. On Max Tegmark's work.
Stephen Hawking, Steven Weinberg, Alexander Vilenkin, David Gross and Lawrence Krauss: Debate on anthropic reasoning Archived 2020-04-10 at the Wayback Machine Kavli-CERCA conference video archive.
Sober, Elliott R. 2009, "Absence of evidence and evidence of absence – Evidential transitivity in connection with fossils, fishing, fine-tuning, and firing squads.^{[permanent dead link]}" Philosophical Studies, 2009, 143: 63–90.
"Anthropic coincidence" – The anthropic controversy as a segue to Lee Smolin's theory of cosmological natural selection.
Leonard Susskind and Lee Smolin debate the anthropic principle.
Debate among scientists on arxiv.org.
Evolutionary probability and fine tuning
Benevolent design and the anthropic principle at MathPages
Critical review of "The privileged planet"
The anthropic principle – a review.
Berger, Daniel, 2002, "An impertinent résumé of the Anthropic cosmological principle. Archived 2016-03-04 at the Wayback Machine" A critique of Barrow & Tipler.
Jürgen Schmidhuber: Papers on algorithmic theories of everything and the anthropic principle's lack of predictive power.
Paul Davies: Cosmic jackpot – Interview about the anthropic principle (starts at 40 min), 15 May 2007.

Quantized spacetime

In general relativity, spacetime is assumed to be smooth and continuous—and not just in the mathematical sense. In the theory of quantum mechanics, there is an inherent discreteness present in physics. In attempting to reconcile these two theories, it is sometimes postulated that spacetime should be quantized at the very smallest scales. Current theory is focused on the nature of spacetime at the Planck scale. Causal sets, loop quantum gravity, string theory, causal dynamical triangulation, and black hole thermodynamics all predict a quantized spacetime with agreement on the order of magnitude. Loop quantum gravity makes precise predictions about the geometry of spacetime at the Planck scale.^[1]

Spin networks provide a language to describe quantum geometry of space. Spin foam does the same job on spacetime. A spin network is a one-dimensional graph, together with labels on its vertices and edges which encodes aspects of a spatial geometry.^[2]

Mathematics of spacetimes

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").^[3] 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.

Introduction

Riemannian geometry was first put forward in generality by Bernhard Riemann in the 19th century. It deals with a broad range of geometries whose metric properties vary from point to point, including the standard types of non-Euclidean geometry.

Every smooth manifold admits a Riemannian metric, which often helps to solve problems of differential topology. It also serves as an entry level for the more complicated structure of pseudo-Riemannian manifolds, which (in four dimensions) are the main objects of the theory of general relativity. Other generalizations of Riemannian geometry include Finsler geometry.

There exists a close analogy of differential geometry with the mathematical structure of defects in regular crystals. Dislocations and disclinations produce torsions and curvature.^[4]^[5]

The following articles provide some useful introductory material:

Classical theorems

What follows is an incomplete list of the most classical theorems in Riemannian geometry. The choice is made depending on its importance and elegance of formulation. Most of the results can be found in the classic monograph by Jeff Cheeger and D. Ebin (see below).

The formulations given are far from being very exact or the most general. This list is oriented to those who already know the basic definitions and want to know what these definitions are about.

General theorems

Gauss–Bonnet theorem The integral of the Gauss curvature on a compact 2-dimensional Riemannian manifold is equal to 2πχ(M) where χ(M) denotes the Euler characteristic of M. This theorem has a generalization to any compact even-dimensional Riemannian manifold, see generalized Gauss-Bonnet theorem.
Nash embedding theorems. They state that every Riemannian manifold can be isometrically embedded in a Euclidean space Rⁿ.

Geometry in large

In all of the following theorems we assume some local behavior of the space (usually formulated using curvature assumption) to derive some information about the global structure of the space, including either some information on the topological type of the manifold or on the behavior of points at "sufficiently large" distances.

Pinched sectional curvature

Sphere theorem. If M is a simply connected compact n-dimensional Riemannian manifold with sectional curvature strictly pinched between 1/4 and 1 then M is diffeomorphic to a sphere.
Cheeger's finiteness theorem. Given constants C, D and V, there are only finitely many (up to diffeomorphism) compact n-dimensional Riemannian manifolds with sectional curvature |K| ≤ C, diameter ≤ D and volume ≥ V.
Gromov's almost flat manifolds. There is an ε_n > 0 such that if an n-dimensional Riemannian manifold has a metric with sectional curvature |K| ≤ ε_n and diameter ≤ 1 then its finite cover is diffeomorphic to a nil manifold.

Sectional curvature bounded below

Cheeger–Gromoll's soul theorem. If M is a non-compact complete non-negatively curved n-dimensional Riemannian manifold, then M contains a compact, totally geodesic submanifold S such that M is diffeomorphic to the normal bundle of S (S is called the soul of M.) In particular, if M has strictly positive curvature everywhere, then it is diffeomorphic to Rⁿ. G. Perelman in 1994 gave an astonishingly elegant/short proof of the Soul Conjecture: M is diffeomorphic to Rⁿ if it has positive curvature at only one point.
Gromov's Betti number theorem. There is a constant C = C(n) such that if M is a compact connected n-dimensional Riemannian manifold with positive sectional curvature then the sum of its Betti numbers is at most C.
Grove–Petersen's finiteness theorem. Given constants C, D and V, there are only finitely many homotopy types of compact n-dimensional Riemannian manifolds with sectional curvature K ≥ C, diameter ≤ D and volume ≥ V.

Sectional curvature bounded above

The Cartan–Hadamard theorem states that a complete simply connected Riemannian manifold M with nonpositive sectional curvature is diffeomorphic to the Euclidean space Rⁿ with n = dim M via the exponential map at any point. It implies that any two points of a simply connected complete Riemannian manifold with nonpositive sectional curvature are joined by a unique geodesic.
The geodesic flow of any compact Riemannian manifold with negative sectional curvature is ergodic.
If M is a complete Riemannian manifold with sectional curvature bounded above by a strictly negative constant k then it is a CAT(k) space. Consequently, its fundamental group Γ = $π$ ₁(M) is Gromov hyperbolic. This has many implications for the structure of the fundamental group:

it is finitely presented;
the word problem for Γ has a positive solution;
the group Γ has finite virtual cohomological dimension;
it contains only finitely many conjugacy classes of elements of finite order;
the abelian subgroups of Γ are virtually cyclic, so that it does not contain a subgroup isomorphic to Z×Z.

Ricci curvature bounded below

Myers theorem. If a complete Riemannian manifold has positive Ricci curvature then its fundamental group is finite.
Bochner's formula. If a compact Riemannian n-manifold has non-negative Ricci curvature, then its first Betti number is at most n, with equality if and only if the Riemannian manifold is a flat torus.
Splitting theorem. If a complete n-dimensional Riemannian manifold has nonnegative Ricci curvature and a straight line (i.e. a geodesic that minimizes distance on each interval) then it is isometric to a direct product of the real line and a complete (n-1)-dimensional Riemannian manifold that has nonnegative Ricci curvature.
Bishop–Gromov inequality. The volume of a metric ball of radius r in a complete n-dimensional Riemannian manifold with positive Ricci curvature has volume at most that of the volume of a ball of the same radius r in Euclidean space.
Gromov's compactness theorem. The set of all Riemannian manifolds with positive Ricci curvature and diameter at most D is pre-compact in the Gromov-Hausdorff metric.

Negative Ricci curvature

The isometry group of a compact Riemannian manifold with negative Ricci curvature is discrete.
Any smooth manifold of dimension n ≥ 3 admits a Riemannian metric with negative Ricci curvature.^[6] (This is not true for surfaces.)

Positive scalar curvature

The n-dimensional torus does not admit a metric with positive scalar curvature.
If the injectivity radius of a compact n-dimensional Riemannian manifold is ≥ π then the average scalar curvature is at most n(n-1).

Notes

^ Hooft, G. T. (2001). "Obstacles on the way towards the quantisation of space, time and matter—And possible resolutions" (PDF). Studies in History and Philosophy of Science Part B: Studies in History and Philosophy of Modern Physics. 32 (2): 157–180. Retrieved 14 April 2017.
^ Baez, John C. "An Introduction to Spin Foam Models of Quantum Gravity and BF Theory". Retrieved 14 April 2017.
^ maths.tcd.ie
^ Kleinert, Hagen (1989), Gauge Fields in Condensed Matter Vol II, World Scientific, pp. 743–1440
^ Kleinert, Hagen (2008), Multivalued Fields in Condensed Matter, Electromagnetism, and Gravitation (PDF), World Scientific, pp. 1–496, Bibcode:2008mfcm.book.....K
^ Joachim Lohkamp has shown (Annals of Mathematics, 1994) that any manifold of dimension greater than two admits a metric of negative Ricci curvature.

References

Books

Berger, Marcel (2000), Riemannian Geometry During the Second Half of the Twentieth Century, University Lecture Series, vol. 17, Rhode Island: American Mathematical Society, ISBN 0-8218-2052-4. (Provides a historical review and survey, including hundreds of references.)
Cheeger, Jeff; Ebin, David G. (2008), Comparison theorems in Riemannian geometry, Providence, RI: AMS Chelsea Publishing; Revised reprint of the 1975 original.
Gallot, Sylvestre; Hulin, Dominique; Lafontaine, Jacques (2004), Riemannian geometry, Universitext (3rd ed.), Berlin: Springer-Verlag.
Jost, Jürgen (2002), Riemannian Geometry and Geometric Analysis, Berlin: Springer-Verlag, ISBN 3-540-42627-2.
Petersen, Peter (2006), Riemannian Geometry, Berlin: Springer-Verlag, ISBN 0-387-98212-4

From Riemann to Differential Geometry and Relativity (Lizhen Ji, Athanase Papadopoulos, and Sumio Yamada, Eds.) Springer, 2017, XXXIV, 647 p. ISBN 978-3-319-60039-0

Papers

Brendle, Simon; Schoen, Richard M. (2008), "Classification of manifolds with weakly 1/4-pinched curvatures", Acta Math, 200: 1–13, arXiv:0705.3963, Bibcode:2007arXiv0705.3963B, doi:10.1007/s11511-008-0022-7, S2CID 15463483

External links

Riemannian geometry by V. A. Toponogov at the Encyclopedia of Mathematics
Weisstein, Eric W., "Riemannian Geometry", MathWorld

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.^[1]

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.^[1]

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.^[1]

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.^[1]

Topology

The assumptions contained in the definition of a spacetime are usually justified by the following considerations.

The connectedness assumption serves two main purposes. First, different observers making measurements (represented by coordinate charts) should be able to compare their observations on the non-empty intersection of the charts. If the connectedness assumption were dropped, this would not be possible. Second, for a manifold, the properties of connectedness and path-connectedness are equivalent, and one requires the existence of paths (in particular, geodesics) in the spacetime to represent the motion of particles and radiation.^[2]

Every spacetime is paracompact. This property, allied with the smoothness of the spacetime, gives rise to a smooth linear connection, an important structure in general relativity.^[2] Some important theorems on constructing spacetimes from compact and non-compact manifolds include the following:

A compact manifold can be turned into a spacetime if, and only if, its Euler characteristic is 0. (Proof idea: the existence of a Lorentzian metric is shown to be equivalent to the existence of a nonvanishing vector field.)
Any non-compact 4-manifold can be turned into a spacetime.^[3]

Foreshadowings of the spacetime concept

In myth and literature

Incas regarded space and time as a single concept, referred to as pacha (Quechua: pacha, Aymara: pacha).^[4]^[5] The peoples of the Andes maintain a similar understanding.^[6]

Arthur Schopenhauer wrote in 1813: "...the representation of coexistence is impossible in Time alone; it depends, for its completion, upon the representation of Space; because, in mere Time, all things follow one another, and in mere Space all things are side by side; it is accordingly only by the combination of Time and Space that the representation of coexistence arises."^[7]

The idea of a unified spacetime is stated by Edgar Allan Poe in his essay on cosmology titled Eureka (1848) that "Space and duration are one". In 1895, in his novel The Time Machine, H. G. Wells wrote, "There is no difference between time and any of the three dimensions of space except that our consciousness moves along it", and that "any real body must have extension in four directions: it must have Length, Breadth, Thickness, and Duration".

In math and physics

In Encyclopedie, published in 1754, under the term dimension Jean le Rond d'Alembert speculated that duration (time) might be considered a fourth dimension if the idea was not too novel.^[8]

Another early venture was by Joseph Louis Lagrange in his Theory of Analytic Functions (1797, 1813). He said, "One may view mechanics as a geometry of four dimensions, and mechanical analysis as an extension of geometric analysis".^[9]

Concerning his development of the theory of quaternions in 1843, William Rowan Hamilton wrote, "Time is said to have only one dimension, and space to have three dimensions. ... The mathematical quaternion partakes of both these elements; in technical language it may be said to be 'time plus space', or 'space plus time': and in this sense it has, or at least involves a reference to, four dimensions."^[10]

Various authors have credited W. K. Clifford with having anticipated the concept of curved spacetime as far back as 1870,^[11]^[12] although the actual extent to which Clifford anticipated spacetime curvature is debatable: in an 1876 publication, Clifford speculated that "curvature of space is what really happens in that phenomenon which we call the motion of matter", i.e. he wrote on the possibility of curved space without mentioning time.^[13]

Section summaries

Introduction summary

^Definitions (click here to return to main)

In classical mechanics, time is separate from space. In special relativity, time and space are fused together into a single 4-dimensional "manifold" called spacetime.
The technical term "manifold" and the great speed of light imply that at ordinary speeds, there is little that humans might observe which is noticeably different from what they would observe if the world followed the geometry of "common sense."
Things that happen in spacetime are called "events". Events are idealized, four-dimensional points. There is no such thing as an event in motion.
The path of a particle in spacetime traces out a succession of events, which is called the particle's "world line".
In special relativity, to "observe" or "measure" an event means to ascertain its position and time against a hypothetical infinite latticework of synchronized clocks. To "observe" an event is not the same as to "see" an event.

^History (click here to return to main)

To mid-1800s scientists, the wave nature of light implied a medium that waved. Much research was directed to elucidate the properties of this hypothetical medium, called the "luminiferous aether". Experiments provided contradictory results. For example, stellar aberration implied no coupling between matter and the aether, while the Michelson–Morley experiment demanded complete coupling between matter and the aether.
FitzGerald and Lorentz independently proposed the length contraction hypothesis, a desperate ad hoc proposal that particles of matter, when traveling through the aether, are physically compressed in their direction of travel.
Henri Poincaré was to come closer than any other of Einstein's predecessors to arriving at what is currently known as the special theory of relativity.
"The special theory of relativity ... was ripe for discovery in 1905."
Einstein's theory of special relativity (1905), which was based on kinematics and a careful examination of the meaning of measurement, was the first to completely explain the experimental difficulties associated with measurements of light. It represented not merely a theory of electrodynamics, but a fundamental re-conception of the nature of space and time.
Having been scooped by Einstein, Hermann Minkowski spent several years developing his own interpretation of relativity. Between 1907 to 1908, he presented his geometric interpretation of special relativity, which has come to be known as Minkowski space, or spacetime.

Spacetime in special relativity summary

^Spacetime interval (click here to return to main)

Time by itself and length by itself are not invariants, since observers in relative motion will disagree on the time between events or the distance between events.
On the other hand, observers in relative motion will agree on the measure of a particular combination of distance and time called the "spacetime interval."
Spacetime intervals can be positive, negative or zero. Particles moving at the speed of light have zero spacetime intervals and do not age.
Spacetime diagrams are typically drawn with only a single space and a single time coordinate. The time axis is scaled by $c$ so that the space and time coordinates have the same units (meters).

^Reference frames (click here to return to main)

To simplify analyses of two reference frames in relative motion, Galilean (i.e. conventional 3-space) diagrams of the frames may be set in a standard configuration with aligned axes whose origins coincide when t = 0.
A spacetime diagram in standard configuration is typically drawn with only a single space and a single time coordinate. The "unprimed frame" will have orthogonal x and ct axes. The axes of the "primed frame" will share a common origin with the unprimed axes, but its x' and ct' axes will be inclined by equal and opposite angles from the x and ct axes.
Although the axes of the unprimed frame are orthogonal and the axes of the primed frame are inclined, the frames are actually equivalent. The asymmetry is due to unavoidable mapping distortions, and should be considered no stranger than the mapping distortions that occur, say, when mapping a spherical Earth onto a flat map.

^Light cone (click here to return to main)

On a spacetime diagram, two 45° diagonal lines crossing the origin represent light signals to and from the origin. In a diagram with an extra space direction, the diagonal lines form a "light cone".
The light cone divides spacetime into a "timelike future" (separated from the origin by more time than space), a "timelike past", and an "elsewhere" region (separated from the origin by a "spacelike" interval with more space than time).
Events in the future and past light cones are causally related to the origin. Events in the elsewhere region do not have a causal relationship with the origin.

^Relativity of simultaneity (click here to return to main)

If two events are timelike separated (causally related), then their before-after ordering is fixed for all observers.
If two events are spacelike separated (non-causally related), then different observers with different relative motions may have reverse judgments on which event occurred before the other.
Simultaneous events are necessarily spacelike separated.
The spacetime interval between two simultaneous events gives the "proper distance". The spacetime interval measured along a world line gives the "proper time".

^Invariant hyperbola (click here to return to main)

In a plane, the set of points equidistant from the origin form a circle.
In a spacetime diagram, a set of points at a fixed spacetime interval from the origin forms an invariant hyperbola.
The loci of points at constant spacelike and timelike intervals from the origin form timelike and spacelike invariant hyperbolae.

^Time dilation and length contraction (click here to return to main)

If frame S' is in relative motion to frame S, its ct' axis is tilted with respect to ct.
Because of this tilt, one light-second on the ct' axis maps to greater than one light-second on the ct axis. Likewise, one light-second on the ct axis maps to greater than one light-second on the ct' axis. Each observer measures the other's clocks as running slow.
The world sheet of a rod one light-second in length aligned parallel to the x' axis projects to less than one light-second on the x axis. Likewise, the world sheet of a rod one light-second in length aligned parallel to the x axis projects to less than one light-second on the x' axis. Each observer measures the other's rulers as being foreshortened.

^Mutual time dilation and the twin paradox (click here to return to main)

^Mutual time dilation (click here to return to main)

To beginners, mutual time dilation seems self-contradictory because two observers in relative motion will each measure the other's clock as running more slowly.
Careful consideration of how time measurements are performed reveals that there is no inherent necessity for the two observers' measurements to be reciprocally "consistent."
In order to measure the rate of ticking of one of B's clocks, observer A must use two of his own clocks to record the time where B's clock made a first tick, and the time where B's clock made a second tick, so that a grand total of three clocks are involved in the measurement. Conversely, observer B uses three clocks to measure the rate of ticking of one of A's clocks. A and B are not doing the same measurement with the same clocks.

^Twin paradox (click here to return to main)

In the twin paradox, one twin makes a journey into space in a high-speed rocket, returning home to find that the twin who remained on Earth has aged more.
The twin paradox is not a paradox because the twins' paths through spacetime are not equivalent.
Throughout both the outbound and the inbound legs of the traveling twin's journey, she measures the stay-at-home twin's clocks as running slower than her own. But during the turnaround, a shift takes place in the events of the stay-at-home twin's world line that the traveling twin considers to be simultaneous with her own.

^Gravitation (click here to return to main)

In the absence of gravity, spacetime is flat, is uniform throughout, and serves as nothing more than a static background for the events that take place in it.
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.

Basic mathematics of spacetime summary

^Galilean transformations (click here to return to main)

A basic goal is to be able to compare measurements made by observers in relative motion.
Transformation between Galilean frames is linear. Given that two coordinate systems are in standard configuration, the coordinate transformation in the x-axis is simply

x'=x-vt

Velocities are simply additive. If frame S' is moving at velocity v with respect to frame S, and within frame S', observer O' measures an object moving with velocity u', then

u'=u-v

or

u=u'+v

^Relativistic composition of velocities (click here to return to main)

The relativistic composition of velocities is more complex than the Galilean composition of velocities:

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

In the low speed limit, the overall result is indistinguishable from the Galilean formula.
The sum of two velocities cannot be greater than the speed of light.

^Time dilation and length contraction revisited (click here to return to main)

The Lorentz factor, gamma $\gamma ,$ appears very frequently in relativity. Given $\beta =v/c,$

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

$\gamma$ is the time dilation factor, while $1/\gamma$ is the length contraction factor.
The Lorentz factor is undefined for $v\geq c.$

^Lorentz transformations (click here to return to main)

The Lorentz transformations combine expressions for time dilation, length contraction, and relativity of simultaneity into a unified set of expressions for mapping measurements between two inertial reference frames.
Given two coordinate systems in standard configuration, the transformation equations for the $t$ and $x$ axes are:

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

There have been many alternative derivations of the Lorentz transformations since Einstein's original work in 1905.
The Lorentz transformations have a mathematical property called linearity. Because of this: (i) Spacetime looks the same everywhere. (ii) There is no preferred frame. (iii) If two Lorentz transformations are applied sequentially, the result is also a Lorentz transformation.

^Doppler effect (click here to return to main)

The formulas for classical Doppler effect depend on whether it is the source or the receiver that is moving with respect to the medium.
In relativity, there is no distinction between a source moving away from the receiver or a receiver moving away from the source. For the longitudinal Doppler effect, a single formula holds for both scenarios:

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

Transverse Doppler shift is a relativistic effect that has no classical analog. Although there are subtleties involved, the basic scenarios can be analyzed by simple time dilation arguments.

^Energy and momentum (click here to return to main)

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.
The relativistic energy-momentum vector has terms for energy and for spatial momentum. 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})

Consideration of the interrelationships between the various components of the relativistic momentum vector led Einstein to his famous $E=mc^{2}$ equation as well as to his concept of relativistic mass.

^Conservation laws (click here to return to main)

The conservation laws arise from fundamental symmetries of nature.
Classical conservation of mass does not hold true in relativity. Since mass and energy are interconvertible, conservation of mass is replaced by conservation of mass-energy.
For analysis of energy and momentum problems involving interacting particles, the most convenient frame is usually the "center-of-momentum" frame.
Newtonian momenta, calculated as $p=mv,$ fail to behave properly under Lorentzian transformation. The relativistic conservation law for energy and momentum replaces the three classical conservation laws for energy, momentum and mass.

Beyond the basics summary

^Rapidity (click here to return to main)

The formulas to perform Lorentz transformation and relativistic composition of velocities are nonlinear, making them more complex than the corresponding Galilean formulas. This nonlinearity is an artifact of our choice of parameters.
The natural functions for expressing the relationships between different frames are the hyperbolic functions. In a spacetime diagram, the velocity parameter $\beta$ is the analog of slope. The rapidity, φ, is defined by

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

Many expressions in special relativity take on a considerably simpler form when expressed in terms of rapidity. For example, the relativistic composition of velocities becomes simply $\phi =\phi _{1}+\phi _{2}.$
The Lorentz boost in the x direction becomes a hyperbolic rotation:

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

.

^4‑vectors (click here to return to main)

General relativity requires knowledge of tensors, which are linear maps between objects like the 4-vectors that belong to the spacetime of relativity. Knowledge of 4-vectors is a prerequisite to understanding tensors.
A 4-tuple, A = (A₀, A₁, A₂, A₃) is a "4-vector" if its component A_i transform between frames according the Lorentz transformation. The last three components of a 4-vector must be a standard vector in three-dimensional space. 4-vectors exhibit closure under linear combination, inner-product invariance, and invariance of the magnitude of a vector.
Examples of 4-vectors include the displacement 4-vector, the velocity 4-vector, the energy-momentum 4-vector, and the acceleration 4-vector.
The use of momentarily comoving reference frames enables special relativity to deal with accelerating particles.
Physical laws must be valid in all frames, but the laws of classical mechanics with their time-dependent 3-vectors fail to behave properly under Lorentz transformation. Valid physical laws must be formulated as equations connecting objects from spacetime like scalars and 4-vectors via tensors of suitable rank.

^Acceleration (click here to return to main)

It is a common misconception that special relativity is unable to handle accelerating objects or accelerating reference frames. Special relativity handles such situations quite well. It is only when gravitation is significant that general relativity is required.
The Dewan–Beran–Bell 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. The issues become almost trivial when analyzed with the aid of spacetime diagrams.
Certain special relativity problem setups can lead to insight about phenomena normally associated with general relativity, such as event horizons.

Introduction to curved spacetime summary

^Basic propositions (click here to return to main)

General relativity asserts that "action-at-a-distance" does not exist. The motions of a satellite orbiting the Earth are not dictated by distant forces exerted by the Earth, Moon and Sun. Rather, the satellite is always following a straight line in its local inertial frame.
Each particle's local frame varies from point to point as a result of the curvature of spacetime.
General relativity is based on two central propositions: (1) The laws of physics cannot depend on what coordinate system one uses. (2) In any sufficiently small region of space, the effects of gravitation are the same as those from acceleration. This second proposition is the equivalence principle.

^Curvature of time (click here to return to main)

Gravitational fields make it impossible to construct a global inertial frame, as is required by special relativity.
A photon climbing in Earth's gravitational field loses energy and is redshifted.
The gravitational redshift implies that gravity makes time run slower. This amounts to a statement that time is curved.
The prediction of curved time is not unique to general relativity. Rather, it is predicted by any theory of gravitation that respects the principle of equivalence.
Newtonian gravitation is a theory of curved time. General relativity is a theory of curved time and curved space.

^Curvature of space (click here to return to main)

Curvature of time completely accounts for all Newtonian gravitational effects.
There are curvature terms for the spatial components of the invariant interval as well, but the effects on planetary orbits and the like are tiny. This is because the speeds of planets and satellites in their orbits are very much slower than the speed of light.
Nevertheless, Urbain Le Verrier, in 1859, was able to demonstrate discrepancies in the orbit of Mercury from that predicted by Newton's laws.
Einstein showed that this discrepancy, the anomalous precession of Mercury, is explained by the spatial terms in the curvature of spacetime.
For light, the spatial terms in the invariant interval are comparable in magnitude to the temporal term, so the effects of the curvature of space are comparable to the effects of the curvature of time.
The famous 1919 Eddington eclipse expedition showed that the bending of light around the Sun includes a component explained by the curvature of space.

^Sources of spacetime curvature (click here to return to main)

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: Mass-energy density, momentum density, pressure, and shear stress.
Gravity itself is a source of gravity.
Moving or rotating masses can generate gravitomagnetic fields analogous to the magnetic fields generated by moving charges.
Pressure as a source of gravity leads to dramatic differences between the predictions of general relativity versus those of Newtonian gravitation.
Experiment has verified the ability of pressure and momentum to act as sources of spacetime curvature. Only stress has eluded experimental verification as a source of spacetime curvature, although mathematical consistency of the Einstein field equations demands that it acts so.

Notes

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.

References

^ ^a ^b ^c ^d Bär, Christian; Fredenhagen, Klaus (2009). "Lorentzian Manifolds". Quantum Field Theory on Curved Spacetimes: Concepts and Mathematical Foundations. Dordrecht: Springer. pp. 39–58. ISBN 9783642027796. Archived from the original (PDF) on 13 April 2017. Retrieved 14 April 2017.
^ ^a ^b Gray, R. Dale (2015). Manifolds, Groups, Bundles, and Spacetime. Lulu.com. pp. 103–126. ISBN 9781329408258. Retrieved 14 April 2017.
^ Israel, S.W.; Hawking, W. (1989). General Relativity: An Einstein Centenary Survey (1st ed.). Cambridge: Cambridge University Press. p. 219. ISBN 0521299284.
^ "Pacha: un concepto andino de espacio y tiempo. | Manga Quespi | Revista Española de Antropología Americana" (PDF). Revistas.ucm.es. Retrieved 17 December 2016.
^ Steele, Paul R.; Allen, Catherine J. (2004). Handbook of Inca Mythology. Santa Barbara, California: ABC-Clio. p. 86. ISBN 1576073548.
^ Ardener, Shirley (1997). Women and space: ground rules and social maps (2nd ed.). Oxford: Berg. p. 36. ISBN 0854967281.
^ Schopenhauer, Arthur (1813). On the Fourfold Root of the Principle of Sufficient Reason. Read Books Ltd. Retrieved 21 June 2017.
^ "ARTFL Encyclopédie Search Results". artflsrv02.uchicago.edu. Retrieved 17 December 2016.
^ "Archibald : Time as a fourth dimension". Projecteuclid.org. 3 July 2007. Retrieved 17 December 2016.
^ Graves, Robert Perceval (1889). Life of Sir William Rowan Hamilton. Hodges, Figgis, & Company. p. 635. Retrieved 21 June 2017.
^ Bell, Eric Temple (1945). The Development of Mathematics. McGraw-Hill Inc. pp. 359–360. ISBN 9780070043305.
^ Weyl, Hermann (1923). Raum Zeit Materie. Berlin: Springer-Verlag. p. 101.
^ Clifford, William Kingdon (1876). "On the Space-Theory of Matter". Proceedings of the Cambridge philosophical society. 2: 157–158.

External links

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.

[1] To save screen space and data usage, Wikipedia mobile collapses sections when displaying on a phone screen. The target of an internal link does not exist unless the section where it is located has been expanded at least once and cached.

[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. Galison (2003), op cit.

[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. Galison (2003), op cit.

[26] The geometry of Minkowski spacetime is closely connected to certain variants of sphere geometry (such as Lie sphere geometry or Conformal geometry) developed in the 19th century. For instance, the Lorentz transformation is a special case of spherical wave transformations. In particular, as pointed out by Poincaré (1912) and others, it is simply isomorphic to the Laguerre group which transforms spheres into spheres and planes into planes. The isomorphism between the Möbius group (which is isomorphic to the group of isometries in hyperbolic R3) and the Lorentz group is also well known.

[27] (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." Minkowski (1909), op cit.

[30] In a Cartesian plane, ordinary rotation leaves a circle unchanged. In spacetime, hyperbolic rotation preserves the hyperbolic metric.

[39] 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.

[50] 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.

[52] 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.

[54] 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."

[64] 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. See Carroll, (op. cit., pp. 112–113)

[86] "Anthropic" means "of or pertaining to mankind or humans".

[118] Strictly speaking, the number of non-compact dimensions; see String theory.

[140] 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.

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v t e Creationism
General	Biblical cosmology Creation myth Creationism Creationist museum Genesis creation narrative History of creationism
Book of Genesis	Adam and Eve Allegorical interpretations of Genesis Flood geology Genesis flood narrative Noah Noah's Ark Tower of Babel Ussher chronology
Types	Creation science Gap creationism Hindu creationism Intelligent design Islamic creationism Neo-creationism Old Earth creationism Progressive creationism Theistic evolution Young Earth creationism
Controversies	Outline Creation–evolution controversy Creation and evolution in public education Edwards v. Aguillard Kansas evolution hearings Kitzmiller v. Dover Area School District Teach the Controversy
Related	Anthropic principle Biblical inerrancy Biblical literalism Created kind Nephilim Omphalos hypothesis
Category:Creationism

Authority control databases
National	Spain France BnF data Germany Israel United States Japan Czech Republic
Other	IdRef

v t e Riemannian geometry (Glossary)
Basic concepts	Curvature tensor Scalar Ricci Sectional Exponential map Geodesic Inner product Metric tensor Levi-Civita connection Covariant derivative Signature Raising and lowering indices/Musical isomorphism Parallel transport Riemannian manifold Pseudo-Riemannian manifold Riemannian volume form
Types of manifolds	Hermitian Hyperbolic Kähler Kenmotsu
Main results	Fundamental theorem of Riemannian geometry Gauss's lemma Gauss–Bonnet theorem Hopf–Rinow theorem Nash embedding theorem Ricci flow Schur's lemma
Generalizations	Finsler Hilbert Sub-Riemannian
Applications	General relativity Geometrization conjecture Poincaré conjecture Uniformization theorem

Authority control databases
National	Spain France BnF data Germany Israel United States Japan Czech Republic
Other	IdRef

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
Dimensions by number	Zero One Two Three Four Five Six Seven Eight n-dimensions
See also	Hyperspace Codimension
Category

Introduction

Definitions

History

Spacetime in special relativity

Spacetime interval

Reference frames

Light cone

Relativity of simultaneity

Invariant hyperbola

Time dilation and length contraction

Mutual time dilation and the twin paradox

Mutual time dilation

Twin paradox

Gravitation

Basic mathematics of spacetime

Galilean transformations

Relativistic composition of velocities

Time dilation and length contraction revisited

Lorentz transformations

Deriving the Lorentz transformations

Linearity of the Lorentz transformations

Doppler effect

Longitudinal Doppler effect

Transverse Doppler effect

Energy and momentum

Extending momentum to four dimensions

Momentum of light

Mass-energy relationship

Four-momentum

Conservation laws

Total momentum

Choice of reference frames

Energy and momentum conservation

Beyond the basics

Rapidity

4‑vectors

Definition of 4-vectors

Properties of 4-vectors

Examples of 4-vectors

4-vectors and physical law

Acceleration

Dewan–Beran–Bell spaceship paradox

Accelerated observer with horizon

Introduction to curved spacetime

Basic propositions

Curvature of time

Curvature of space

Sources of spacetime curvature

Energy-momentum

Pressure and stress

Experimental verification

• Active, passive, and inertial mass

• Pressure as a gravitational source

• Gravitomagnetism

Privileged character of 3+1 spacetime

Definition and basis

Anthropic observations

Origin

Variants

Character of anthropic reasoning

Observational evidence

Applications of the principle

The nucleosynthesis of carbon-12

Cosmic inflation

String theory

Dimensions of spacetime

Metaphysical interpretations

The anthropic cosmological principle

Reception and controversies

See also

Notes

Footnotes

References

External links

Quantized spacetime

Mathematics of spacetimes

Riemannian geometry

Introduction

Classical theorems

General theorems