# Spacetime

(Redirected from Space time continuum)

In physics, spacetime is any mathematical model that fuses three-dimensional 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.

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, expanding upon Einstein's work, 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 later development of general relativity, where he showed that spacetime becomes curved in the presence of mass or energy. The spacetime of general relativity is an example of what mathematicians call a 4‑dimensional manifold.

## Introduction

### Definitions

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.

Mathematically, spacetime is a manifold, which is to say, it is a topological space that locally resembles Euclidean space near each point. By analogy, at small enough scales, a globe appears flat.[3] An extremely large scale factor, ${\displaystyle c}$ (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.[4]

Things that happen in spacetime are called events. An event is something that happens instantaneously at a single point in spacetime, represented by a set of coordinates x, y, z and t. Events have neither duration in time nor extent in space. The tokens typically used in popular expositions of relativity to represent events—sparks, firecrackers, lightning bolts and the like—are not events because they have finite durations and extents. Unlike the analogies used to explain events, mathematical events, since they have no duration, have no speed and cannot be in motion.

On the other hand, 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.[5]:105

Figure 1-1. Special relativity uses the term observer in a manner distinct from the ordinary English meaning.

In special relativity, an observer is 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 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 An ideal observer experiences no time delays between the firing of an event and its recording. In real life, there will typically be delays between the emission of a signal and its detection. However, 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.

Physicists distinguish between what one measures or observes (after one has factored out signal propagation delays), versus what one sees (what one visualizes 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

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

By the mid-1800s, various experiments such as the observation of the Arago spot and differential measurements of the speed of light in air versus water were considered to have proven the wave nature of light as opposed to corpuscular theory.[8] Waves implied the existence of a medium which waved, but attempts to measure the properties of the hypothetical luminiferous aether implied by these experiments provided 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 (Fig. 1‑3).[4]

Figure 1-3. (top) The direction of stellar aberration differs at different times of the year. (bottom) Aether dragging—a theory once advanced to explain the inability to detect the effects of aether flow on the speed of light—is incompatible with stellar aberration. [Click here for additional details 1]

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, and it predicted various physical effects which might be observable.[10]:163–174 For example, most physicists believed that Lorentz contraction would be detectable by such experiments as the Trouton–Noble experiment or the Experiments of Rayleigh and Brace.[11]:64 However, these gave negative results, and in his 1904 theory of the electron, Lorentz explained these negative results as an inevitable consequence of his transforms. Poincaré, correcting some errors in Lorentz's analysis, demonstrated the non-detectability of the aether, but continued to believe in the dynamical interpretation of the Lorentz transform for the rest of his life.[10]:163–174

Present day hindsight tends to place the culmination of the turn-of-the-20th-century crisis in physics on the Michelson and Morley experiment. But to Einstein, his ultimate inspiration may have been inconsistencies that he perceived in the manner in which Maxwell's theory of electromagnetism was being interpreted. Although in 1905, Einstein wrote of the Moving magnet and conductor problem as being generally understood as leading to asymmetries, these asymmetries do not appear to have been noted as a significant issue requiring explanation by any of the leading expositors of Maxwell's theory, including Hertz, Lorentz, and Maxwell himself.[11]:135–142

Einstein's theory of special relativity, introduced in 1905, provided a complete resolution to these and other major physics puzzles, and it made startling predictions that have since been repeatedly confirmed. Einstein performed his analyses in terms of kinematics (the study of moving bodies without reference to forces) rather than dynamics. It would appear that he did not at first think geometrically about spacetime. It was Einstein's former mathematics professor, Hermann Minkowski, who was to provide a geometric interpretation of special relativity.[12]:219

Einstein was initially dismissive of the geometric interpretation of special relativity, regarding it as überflüssige Gelehrsamkeit (superfluous learnedness). However, the geometric interpretation of relativity was to prove vital to Einstein's later development of general relativity, and in 1916, Einstein fully acknowledged his indebtedness to Minkowski, whose interpretation greatly facilitated the transition to general relativity.[10]:151–152 The spacetime of special relativity has since come to be known as Minkowski spacetime.

### Spacetime 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 ${\displaystyle t}$ and a spatial distance ${\displaystyle x}$. Then the spacetime interval between the two events is given by

${\displaystyle s^{2}=c^{2}t^{2}-x^{2}}$   (or for three space dimensions, ${\displaystyle s^{2}=c^{2}t^{2}-x^{2}-y^{2}-z^{2}}$)[note 2]
Figure 1-4. Spacetime diagram illustrating two photons, A and B, originating at the same event, and a slower-than-light-speed object, C.

The equation above is simply an expression of the Pythagorean theorem, except with a minus sign between the ${\displaystyle c^{2}t^{2}}$ and the ${\displaystyle x^{2}}$ terms. Note also that the spacetime interval is the quantity ${\displaystyle s^{2},}$ not ${\displaystyle s}$ 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 ${\displaystyle s^{2}}$ as a distinct symbol in itself, rather than the square of something.[12]:217

Because of the minus sign, the spacetime interval between two events can be zero. Spacetime intervals are zero when ${\displaystyle x=\pm ct}$. In other words, the spacetime interval is zero for something moving at the speed of light. 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. 1‑4 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 ${\displaystyle c}$ 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.

### Reference frames

Figure 1-5. Galilean diagram of two frames of reference in standard configuration.
Figure 1-6. (a) Galilean diagram of two frames of reference in standard configuration. (b) spacetime diagram of two frames of reference. (c) spacetime diagram showing the path of a reflected light pulse.

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. 1‑5, 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'.[5]:107

Fig. 1‑6a redraws Fig. 1‑5 in a different orientation. Fig. 1‑6b 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

${\displaystyle \tan \ \theta =v/c}$

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. 1‑6c 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. 1‑6b 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.[5]: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

Figure 1-7. The light cone centered on an event divides the rest of spacetime into the future, the past, and "elsewhere".

In Fig. 1-7, 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. 1‑8, 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 ${\displaystyle c}$.[12]:220

Figure 1-8. Light cone in 2D space plus a time dimension.

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.[12]: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. 1‑7, 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.[13]

### Relativity of simultaneity

Figure 1-9. Animation illustrating relativity of simultaneity.

All observers will agree that for any given event, an event within the given event's future light cone occurs after the given event. Likewise, for any given event, an event within the given event's past light cone occurs before the given event. The before-after relationship observed for timelike-separated events remains unchanged no matter what the reference frame of the observer, i.e. no matter how the observer may be moving. The situation is quite different for spacelike-separated events. Fig. 1‑7 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.[14]

Fig. 1-9 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. 1‑6. 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.

The spacetime interval gives the same distance that an observer would measure if the events being measured were simultaneous to the observer. The spacetime interval hence provides a measure of proper distance, i.e. the true distance in spacetime. Likewise, the 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. The spacetime interval hence provides a measure of proper time.[12]:220–221

### Invariant hyperbola

Figure 1-10. (a) Families of invariant hyperbolae. (b) Hyperboloids of two sheets and one sheet.

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

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

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

In Fig. 1-10a, 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. 1‑10b 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

Figure 1-11. The invariant hyperbola comprises the points that can be reached from the origin in a fixed proper time by clocks traveling at different speeds.

Fig. 1-11 illustrates the invariant hyperbola for all events that can be reached from the origin in a proper time of 5 meters (approximately 1.67×10−8s). 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 (1.75×10−8s), while for a clock traveling at 0.7c, the elapsed time measured by the observer is 7.00 meters (2.34×10−8s). 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.[12]: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.

Figure 1-12. In this spacetime diagram, the 1 m length of the moving rod, as measured in the primed frame, is the foreshortened distance OC when projected onto the unprimed frame.

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. 1-12 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.[5]: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.

### Measurement versus visual appearance

Time dilation and length contraction are not optical illusions, but genuine effects. Measurements of these effects are not an artifact of Doppler shift, nor are they the result of neglecting to take into account the time it takes light to travel from an event to an observer.

That being said, scientists make a fundamental distinction between measurement or observation on the one hand, versus visual appearance, or what one sees.

Figure 1-13. Comparison of the measured length contraction of a cube versus its visual appearance.

For many years, the distinction between the two had not been generally appreciated. For example, it had generally been thought that a length contracted object passing by an observer would in fact actually be seen as length contracted. In 1959, James Terrell and Roger Penrose independently pointed out that differential time lag effects in signals reaching the observer from the different parts of a moving object result in a fast moving object's visual appearance being quite different from its measured shape. For example, a receding object would appear contracted, an approaching object would appear elongated, and a passing object would have a skew appearance that has been likened to a rotation.[15][16][17][18] A sphere in motion retains the appearance of a sphere, although images on the surface of the sphere will appear distorted.[19]

Figure 1-14. Galaxy M87 streams out a black-hole-powered jet of electrons and other sub-atomic particles traveling at nearly the speed of light.

Fig. 1‑13 illustrates a cube viewed from a distance of four times the length of its sides. At high speeds, the sides of the cube that are perpendicular to the direction of motion appear hyperbolic in shape. The cube is actually not rotated. Rather, light from the rear of the cube takes longer to reach one's eyes compared with light from the front, during which time the cube has moved to the right. This illusion has come to be known as Terrell rotation or the Terrell–Penrose effect.[note 3]

Another example where visual appearance is at odds with measurement comes from the observation of apparent superluminal motion in various radio galaxies, BL Lac objects, quasars, and other astronomical objects that eject relativistic-speed jets of matter at narrow angles with respect to the viewer. An optical illusion results giving the appearance of faster than light travel.[20][21][22] In Fig. 1‑14, galaxy M87 streams out a high-speed jet of subatomic particles almost directly towards us, but Penrose–Terrell rotation causes the jet to appear to be moving laterally in the same manner that the appearance of the cube in Fig. 1‑13 has been stretched out.[23]

### Mutual time dilation and the twin paradox

#### Mutual time dilation

Mutual time dilation and length contraction tend to strike beginners as inherently self-contradictory concepts. 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."[12]: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.[12]:198–199

In regards to mutual length contraction, Fig. 1‑12 illustrates that the primed and unprimed frames are mutually rotated by a hyperbolic angle (analogous to ordinary angles in Euclidean geometry).[note 4] 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.

Figure 1-15. Mutual time dilation

Fig. 1-15 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.[5]: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. 1-15, 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'.

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.[12]:207 Nevertheless, the twin paradox is not a 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.[24]

Figure 1-16. Spacetime explanation of the twin paradox

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. 1‑16. 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.[24]

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.[24]

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.[24] Analyzed in Stella's frame, she is motionless for the entire trip. When she fires her rockets for the turnaround, she experiences a pseudo-gravitational field that exactly cancels the force of her rockets, so she remains motionless.[24]

Figs. 1-9 and 1-16 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. 1‑16, 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.[24]

### Gravitation

This introductory section has focused on the spacetime of special relativity, since it is the easiest to describe. Minkowski spacetime is flat, takes no account of gravity, is uniform throughout, and serves as nothing more than a static background for the events that take place in it. The presence of gravity greatly complicates the description of spacetime. In general relativity, spacetime is no longer a static background, but actively interacts with the physical systems that it contains. Spacetime curves in the presence of matter, can propagate waves, bends light, and exhibits a host of other phenomena.[12]: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:[25][26]

${\displaystyle x'=x-vt}$
${\displaystyle y'=y}$
${\displaystyle z'=z}$
${\displaystyle t'=t.}$
Figure 2-1. Galilean Spacetime and composition of velocities.

Fig. 2-1 illustrates that in Newton's theory, time is universal, not the velocity of light.[27]: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

${\displaystyle u'=u-v}$  or  ${\displaystyle u=u'+v}$

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 - 9×5 = 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

Figure 2-2. Relativistic composition of velocities

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

${\displaystyle \beta =v/c}$

Fig. 2-2a 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. 2‑2b:

1. From the platform, the composite speed of the bullet is given by u = c(s + r)/(a + b).
2. 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.
3. The ratios of corresponding sides of the two yellow triangles are constant, so that r/a = b/s = n/m = β'. So b = u's/c and r = u'a/c.
4. 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:[27]:42–48
${\displaystyle u={v+u' \over 1+(vu'/c^{2})}.}$

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.[27]:49

### Time dilation and length contraction revisited

Figure 2-3. Spacetime diagrams illustrating time dilation and length contraction

We had previously discussed, in qualitative terms, time dilation and length contraction. It is straightforward to obtain quantitative expressions for these effects. Fig. 2‑3 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 :

${\displaystyle \mathrm {T} =ct}$ and ${\displaystyle w=ct}$ are common.
One also sees very frequently the use of the convention ${\displaystyle c=1.}$
Figure 2-4. Lorentz factor as a function of velocity

In Fig. 2-3a, segments OA and OK represent equal spacetime intervals. Time dilation is represented by the ratio OB/OK. The invariant hyperbola has the equation w = x2 + k2 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 ${\displaystyle \gamma }$:[28]

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

We note that if v is greater than or equal to c, the expression for ${\displaystyle \gamma }$ 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. 2-3b, segments OA and OK represent equal spacetime intervals. Length contraction is represented by the ratio OB/OK. The invariant hyperbola has the equation x = w2 + k2 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

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

### Lorentz transformations

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

To compare one set of coordinates to another in special relativity, we use the Lorentz transformations.

The Lorentz factor appears in the Lorentz transformations:

{\displaystyle {\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:

{\displaystyle {\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.

When we write ${\displaystyle t'=\gamma (t-vx/c^{2}),}$ ${\displaystyle x'=\gamma (x-vt)}$ and so forth, we most often really mean ${\displaystyle \Delta t'=\gamma (\Delta t-v\Delta x/c^{2}),}$ ${\displaystyle \Delta x'=\gamma (\Delta x-v\Delta t)}$ 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. Since there is no preferred origin, fixed values of the coordinates have no real meaning.

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.[29]: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' = ${\displaystyle \gamma }$(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 77.9x106 km). 1).

#### Deriving the Lorentz transformations

Figure 2-5. Derivation of Lorentz Transformation

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.[30]

The derivation given here and illustrated in Fig. 2‑5 is based on one presented by Bais[27]: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.

1. 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.[29]:27–28
2. From the drawing, w = a + b and x = r + s
3. From previous results using similar triangles, we know that s/a = b/r = v/c = β.
4. We know that because of time dilation, a = γw'
5. Substituting equation (4) into s/a = β yields s = γw'β.
6. Length contraction and similar triangles give us r = γx' and b = βr = βγx'
7. Substituting the expressions for s, a, r and b into the equations in Step 2 immediately yield
${\displaystyle w=\gamma w'+\beta \gamma x'}$
${\displaystyle x=\gamma x'+\beta \gamma w'}$

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.[27]:67 All inertial observers will agree on what constitutes accelerating and non-accelerating motion.[29]: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.[12]:190

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

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

### Doppler effect

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

#### Longitudinal Doppler effect

Figure 2-6. Galilean spacetime diagrams of classical Doppler effect. (a) Source moving away from the receiver. (b) Receiver moving away from the source.

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. 2‑6a 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 vs for a velocity parameter of βs. 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+βs, so that the observed frequency f is given by

${\displaystyle f={\frac {1}{1+\beta _{s}}}f_{0}={\frac {c}{c+v_{s}}}f_{0}}$
Figure 2-7. Spacetime diagram of relativistic Doppler effect

Fig. 2‑6b illustrates a Galilean spacetime diagram of the scenario where the receiver is moving directly away from a stationary source at a speed of vr for a velocity parameter of βr. 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−βr), so that T = 1/(1−βr) and the observed frequency f is given by

${\displaystyle f=(1-\beta _{r})f_{0}={\frac {c-v_{r}}{c}}f_{0}}$

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. 2‑7 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). Hence, the relativistic Doppler effect is given by[27]:58–59

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

#### Transverse Doppler effect

Figure 2-8. Transverse Doppler effect scenarios

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:[29]:94–96

• Fig. 2-8a. 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. 2-8b. 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. 2-8c. If receiver is moving in a circle around the source, what frequency does the receiver measure?
• Fig. 2-8d. 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

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

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

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

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 ${\displaystyle \gamma }$, 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.)[29]: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 5]

### Energy and momentum

#### Four-momentum

Figure 2-9. Relativistic spacetime momentum vector

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. 2‑9a, 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 components of the four-momentum 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. 2‑9b 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.[27]:84–87

#### Momentum of light

Figure 2-10. Energy and momentum of light in different inertial frames

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. 2‑10 or by application of the Lorentz transformations, and is consistent with the analysis of Doppler effect given previously.[27]: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, ${\displaystyle \gamma }$ 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 mrel = γm.
• Likewise, comparing the time component of the relativistic momentum with that of the photon, γmc = mrelc = E/c, so that Einstein arrived at the relationship E = mrelc2. 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 the time component of the relativistic momentum vector at low velocity. A series expansion of γm yields

${\displaystyle \gamma m={\frac {m}{\sqrt {1-\beta ^{2}}}}\approx m+{\frac {1}{2}}m\beta ^{2}...}$

In other words, at low velocity, mrel = m + ½ mv2/c2. The second term is just an expression for the kinetic energy of the particle divided by c2. Mass indeed appears to be another form of energy.[27]:90–92[29]:129–130,180

The concept of relativistic mass that Einstein introduced in 1905, mrel, 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, 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.[31] "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 (E, pc). Expressed as a formula,

${\displaystyle E^{2}-p^{2}c^{2}=m^{2}c^{4}}$

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.[27]:90–92

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

${\displaystyle P\equiv (E/c,{\vec {p}})=(E/c,p_{x},p_{y},p_{z})}$ or alternatively,
${\displaystyle P\equiv (E,{\vec {p}})=(E,p_{x},p_{y},p_{z})}$ using the convention that ${\displaystyle c=1.}$[29]:129–130,180

### Conservation laws

In physics, conservation laws state that certain particular measurable properties of an isolated physical system do not change as the system evolves over time. In 1915, Emmy Noether discovered that underlying each conservation law is a fundamental symmetry of nature.[32] 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

Figure 2-11. Relativistic conservation of 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, vcm = (m1v1 + m2v2)/(m1 + m2). The total momentum p = p1 + p2 is conserved.

Fig. 2‑11 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.[27]: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.[29]: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.[33][34][35][36]

Figure 2-12.
(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. 2‑12 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.[29]: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.[29]: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.[27]:104

Figure 2-13a. Energy-momentum diagram for decay of a charged pion
Figure 2-13b. 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.[29]: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. 2‑13a 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 Eν = 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,[37] so Fig. 2‑13b 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.

### Acceleration

It is a common misconception that special relativity is applicable only to inertial frames, and that it is unable to handle accelerating objects or accelerating reference frames. Actually, accelerating objects can generally be analyzed without needing to deal with accelerating frames at all. It is only when gravitation is significant that general relativity is required.[38]

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.[38]

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

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.[39]

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. 2‑14, 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. 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 argument lead to opposite conclusions. Both lines of argument presented below are flawed, even though one of the conclusions happens to be the correct conclusion.[29]: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.[29]: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.[29]:106,120–122

Figure 2-15. The blue lines represent the world lines of two observers A and B who accelerate in the same direction with the same constant magnitude acceleration. At A' and B', the observers stop accelerating. The dotted line is a line of simultaneity for either observer after acceleration stops.

A spacetime diagram (Fig. 2‑15) makes the correct solution to this paradox almost immediately evident. Two observers in Minkowski spacetime accelerate with constant magnitude ${\displaystyle k}$ acceleration for proper time ${\displaystyle \sigma }$ (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 ${\displaystyle A'B''}$ turns out to be greater than the length of the spacelike line segment ${\displaystyle AB}$.

The length increase can be calculated with the help of the Lorentz transformation. If, as illustrated in Fig. 2‑15, the acceleration is finished, the ships will remain at a constant offset in some frame ${\displaystyle S'.}$ If ${\displaystyle x_{A}}$ and ${\displaystyle x_{B}=x_{A}+L}$ are the ships' positions in ${\displaystyle S,}$ the positions in frame ${\displaystyle S'}$ are:[40]

{\displaystyle {\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

Figure 2-16. Accelerated relativistic observer with horizon. Another well-drawn illustration of the same topic may be viewed here.

In the text accompanying Fig. 1‑10, 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. 2‑16 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 ${\displaystyle \beta }$ approaches a limit of one as ${\displaystyle ct}$ increases. Likewise, ${\displaystyle \gamma }$ approaches infinity.

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

1. We remember that ${\displaystyle \beta =ct/x.}$
2. Since ${\displaystyle c^{2}t^{2}-x^{2}=s^{2},}$ we conclude that ${\displaystyle \beta (ct)=ct/{\sqrt {c^{2}t^{2}-s^{2}}}.}$
3. ${\displaystyle \gamma =1/{\sqrt {1-\beta ^{2}}}=}$ ${\displaystyle {\sqrt {c^{2}t^{2}-s^{2}}}/s}$
4. From the relativistic force law, ${\displaystyle F=dp/dt=}$${\displaystyle dpc/d(ct)=d(\beta \gamma mc^{2})/d(ct).}$
5. Substituting ${\displaystyle \beta (ct)}$ from step 2 and the expression for ${\displaystyle \gamma }$ from step 3 yields ${\displaystyle F=mc^{2}/s,}$ which is a constant expression.[27]:110–113

Fig. 2‑16 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).[27]: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.[27]:110–113

## Beyond the basics

### Rapidity

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Figure 3-1a. 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 3-1b. 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 3-2. 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. 3‑1a 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. 3‑1b shows a unit hyperbola with sinh(a) and cosh(a), where a is likewise interpreted as twice the tinted area.[41] Fig. 3‑2 presents plots of the sinh, cosh, and tanh functions.

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

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

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

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

In a spacetime diagram, the velocity parameter ${\displaystyle \beta }$ is the analog of slope. The rapidity, φ, is defined by[29]:96–99

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

where

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

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

${\displaystyle \beta ={\frac {\beta _{1}+\beta _{2}}{1+\beta _{1}\beta _{2}}}=}$ ${\displaystyle {\frac {\tanh \phi _{1}+\tanh \phi _{2}}{1+\tanh \phi _{1}\tanh \phi _{2}}}=}$ ${\displaystyle \tanh(\phi _{1}+\phi _{2}),}$

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

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

${\displaystyle \gamma ={\frac {1}{\sqrt {1-\beta ^{2}}}}={\frac {1}{\sqrt {1-\tanh ^{2}\phi }}}=\cosh \phi ,}$
${\displaystyle \gamma \beta ={\frac {\beta }{\sqrt {1-\beta ^{2}}}}={\frac {\tanh \phi }{\sqrt {1-\tanh ^{2}\phi }}}=\sinh \phi .}$

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

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

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

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

In other words, Lorentz boosts represent hyperbolic rotations in Minkowski spacetime.[29]: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][42] [note 6]

### 4‑vectors

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

#### Definition of 4-vectors

A 4-tuple, A = (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

{\displaystyle {\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.[39]: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, ${\displaystyle {\vec {A}}}$ and ${\displaystyle {\vec {B}}}$ are 3-vectors:
${\displaystyle A\cdot B\equiv }$ ${\displaystyle A_{0}B_{0}-A_{1}B_{1}-A_{2}B_{2}-A_{3}B_{3}\equiv }$ ${\displaystyle A_{0}B_{0}-{\vec {A}}\cdot {\vec {B}}}$
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 ${\displaystyle A\cdot B=0.}$ 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 ${\displaystyle A\cdot A=0,}$ while a zero vector is one whose components are all zero. Special cases illustrating the invariance of the norm include the invariant interval ${\displaystyle c^{2}t^{2}-x^{2}}$ and the invariant length of the relativistic momentum vector ${\displaystyle E^{2}-p^{2}c^{2}.}$[29]:178–181[39]: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).
${\displaystyle dS\equiv (dt,dx,dy,dz)}$
• Velocity 4-vector: This results when the displacement 4-vector is divided by ${\displaystyle d\tau }$, where ${\displaystyle d\tau }$ is the proper time between the two events that yield dt, dx, dy, and dz.
${\displaystyle V\equiv {\frac {dS}{d\tau }}={\frac {(dt,dx,dy,dz)}{dt/\gamma }}=}$ ${\displaystyle \gamma \left(1,{\frac {dx}{dt}},{\frac {dy}{dt}},{\frac {dz}{dt}}\right)=}$ ${\displaystyle (\gamma ,\gamma {\vec {v}})}$
Figure 3-3a. The momentarily comoving reference frames of a rapidly accelerating particle as observed from a stationary frame.
Figure 3-3b. 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, ${\displaystyle d\tau =0}$ 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: This results from multiplying the velocity 4-vector by the invariant mass m:[6]:111–121
${\displaystyle P\equiv mV=(\gamma m,\gamma m{\vec {v}})=(E,{\vec {p}})}$
As indicated before, there are varying treatments for the energy-momentum 4-vector so that one may also see it expressed as ${\displaystyle (E/c,{\vec {p}})}$ or ${\displaystyle (E,{\vec {p}}c).}$ 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 ${\displaystyle \tau .}$
${\displaystyle A\equiv {\frac {dV}{d\tau }}=}$ ${\displaystyle {\frac {d}{d\tau }}(\gamma ,\gamma {\vec {v}})=}$ ${\displaystyle \gamma \left({\frac {d\gamma }{dt}},{\frac {d(\gamma {\vec {v}})}{dt}}\right)}$
• Force 4-vector: This is the derivative of the momentum 4-vector with respect to ${\displaystyle \tau .}$
${\displaystyle F\equiv {\frac {dP}{d\tau }}=}$ ${\displaystyle \gamma \left({\frac {dE}{dt}},{\frac {d{\vec {p}}}{dt}}\right)=}$ ${\displaystyle \gamma \left({\frac {dE}{dt}},{\vec {f}}\right)}$

As expected, the final components of the above 4-vectors are all standard 3-vectors corresponding to spatial 3-momentum, 3-force etc.[29]:178–181[39]: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.[29]:186

## 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 7] 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. 4‑1, 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."[43]: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.[44]:137–149
Figure 4-2. Equivalence principle
• 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. 4-2, 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.[44]:141–149
An alternative expression of the equivalence principle is to note that in Newton's universal law of gravitation, F = GMmg /r2 = mgg and in Newton's second law, F = m ia, there is no a priori reason why the gravitational mass mg should be equal to the inertial mass m i. The equivalence principle states that these two masses are identical.[44]: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

Figure 4-3. Einstein's argument suggesting gravitational redshift

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.[39]: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. 4‑3) 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'.[39]: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

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

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).[45]

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.[12]: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.[12]: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.[46]: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:[12]:229–232

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

### Curvature of space

The ${\displaystyle (1-2GM/(c^{2}r))}$ coefficient in front of ${\displaystyle (c\Delta t)^{2}}$ 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 ${\displaystyle G}$ and ${\displaystyle M}$, and because of the ${\displaystyle r}$ 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 ${\displaystyle (c\Delta t)^{2}}$ term dwarfs the spatial terms.[12]: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.[47] 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.

Figure 4-4. General relativity is a theory of curved time and curved space. Click here to animate

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.[48]

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.[12]:234–238

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

In Newton's gravitation, the ${\displaystyle (1-2GM/(c^{2}r))}$ coefficient in front of ${\displaystyle (c\Delta t)^{2}}$ predicts bending of light around a star. In general relativity, the ${\displaystyle (1+2GM/(c^{2}r))}$ coefficient in front of ${\displaystyle \left[(\Delta x)^{2}+(\Delta y)^{2}+(\Delta z)^{2}\right]}$ predicts a doubling of the total bending.[12]:234–238

The story of the 1919 Eddington eclipse expedition and Einstein's rise to fame is well told elsewhere.[49]

### Uniformly accelerated reference frame

Figure 4-5. Rindler chart, assuming c = 1, plotted on a Minkowski diagram. The dashed lines are the Rindler horizons.

Before reading this section, one should review the previous sections on the Dewan–Beran–Bell spaceship paradox and the Accelerated observer with horizon.

Technically, the uniformly accelerated reference frame that we consider in this section has nothing to do with general relativity. But this special relativity setup, called the Rindler coordinate system, is important because it provides insight into various phenomena of general relativity, including black holes.

#### Accelerated observer with horizon revisited

In Fig. 4‑5, each hyperbola represents the world line of a Rindler observer accelerating with a constant proper acceleration. (${\displaystyle c=1}$ in this graph.) If ${\displaystyle g_{i}}$ represents the acceleration of a particular observer, then ${\displaystyle g_{i}=1/X}$ or, if we include the c's, ${\displaystyle g_{i}=c^{2}/X.}$

If ${\displaystyle O}$ is the origin and ${\displaystyle A}$ is a constantly accelerating observer, the line ${\displaystyle OA}$ and the particle's ${\displaystyle x'}$ axis are the same line. Since the ${\displaystyle x'}$ axis is a line of simultaneity, this means that the origin event ${\displaystyle O}$ is always at the same "now" as the Rindler observer. The origin event ${\displaystyle O}$ is acting rather like the event horizon of a black hole. As viewed by a Rindler observer, time seems to stand still at ${\displaystyle O,}$ even as time seems to stand still at the edge of a black hole, as viewed by a distant observer.

The events on a particular hyperbola are all separated from the origin ${\displaystyle O}$ by a constant spacelike interval ${\displaystyle 1/g}$ or, if we include the c's, ${\displaystyle c^{2}/g.}$ Like Alice in Through the Looking Glass, the Rindler observer is constantly accelerating away from ${\displaystyle O,}$ but doesn't get any further away from it, as measured in his own frame.[29]:197–200

#### Rindler space

It is desirable to be able to create a frame of uniformly accelerating particles in which the distance between particles (as measured in any particle's MCRF) is constant. From analysis of the Dewan–Beran–Bell spaceship paradox, it is evident that it is all too easy to create an accelerating frame where the distances between particles do not remain constant.

The Equivalence Principle states that a frame in a gravitational field is equivalent to an accelerating frame. But to properly model the effects of gravity in an accelerating frame, the distances between particles should be constant as measured in the frame.

In Fig. 4‑5, consider a particle ${\displaystyle A}$ accelerating along one of the hyperbolae with proper acceleration ${\displaystyle g_{A}.}$ Consider a second particle ${\displaystyle B}$ accelerating along a second hyperbola with proper acceleration ${\displaystyle g_{B}.}$ Let ${\displaystyle a}$ and ${\displaystyle b}$ be the initial distances of ${\displaystyle A}$ and ${\displaystyle B}$ from the origin. Furthermore, let us require that both ${\displaystyle A}$ and ${\displaystyle B}$ lie along the same line of simultaneity. From the presentation given in the "horizon revisited" section, we conclude that the following must hold:

${\displaystyle g_{A}=1/a}$ and ${\displaystyle g_{B}=1/b}$

In other words, for all points along a line of simultaneity to be traveling along hyperbolae with a common origin, their proper accelerations must be inversely proportional to their initial distances from the origin.

The Rindler frame, therefore, matches the desired static frame, which is also often called "Rindler space". ${\displaystyle A}$ and ${\displaystyle B}$ are initially at a distance ${\displaystyle b-a}$ from each other, and this distance remains constant with time. A person walking around in this frame experiences a static world where the force of gravity is proportional to ${\displaystyle 1/z,}$ where ${\displaystyle z}$ is the distance to the apparent event horizon at the origin.[29]:197–200

## Spacetime in general relativity

In general relativity, it is assumed that spacetime is curved by the presence of matter (energy), this curvature being represented by the Riemann tensor. In special relativity, the Riemann tensor is identically zero, and so this concept of non-curvedness is sometimes expressed by the statement Minkowski spacetime is flat.[46]:1–30

The earlier discussed notions of time-like, light-like and space-like intervals in special relativity can similarly be used to classify one-dimensional curves through curved spacetime. A time-like curve can be understood as one where the interval between any two infinitesimally close events on the curve is time-like, and likewise for light-like and space-like curves. Technically the three types of curves are usually defined in terms of whether the tangent vector at each point on the curve is time-like, light-like or space-like. The world line of a slower-than-light object will always be a time-like curve, the world line of a massless particle such as a photon will be a light-like curve, and a space-like curve could be the world line of a hypothetical tachyon. In the local neighborhood of any event, time-like curves that pass through the event will remain inside that event's past and future light cones, light-like curves that pass through the event will be on the surface of the light cones, and space-like curves that pass through the event will be outside the light cones. One can also define the notion of a three-dimensional spacelike hypersurface, a continuous three-dimensional slice through the four-dimensional property with the property that every curve that is contained entirely within this hypersurface is a space-like curve.[50]

Many spacetime continua have physical interpretations which most physicists would consider bizarre or unsettling. For example, a compact spacetime has closed timelike curves, which violate our usual ideas of causality (that is, future events could affect past ones). For this reason, mathematical physicists usually consider only restricted subsets of all the possible spacetimes. One way to do this is to study "realistic" solutions of the equations of general relativity. Another way is to add some additional "physically reasonable" but still fairly general geometric restrictions and try to prove interesting things about the resulting spacetimes. The latter approach has led to some important results, most notably the Penrose–Hawking singularity theorems.[51]

## Privileged character of 3+1 spacetime

Properties of n+m-dimensional spacetimes

There are two kinds of dimensions, spatial (bidirectional) and temporal (unidirectional). 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. 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 says 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). 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 the strength of field. 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 area of 4πr ². 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 rN−1.

In 1920, Paul Ehrenfest showed that if there is only one time dimension and greater 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.[52] 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 ${\displaystyle 5+2k}$ spatial dimensions, where k is a whole number, then wave impulses become distorted. In 1922, Hermann Weyl showed that Maxwell's theory of electromagnetism works only with three dimensions of space and one of time.[53] 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.[54]

Max Tegmark expands on the preceding argument in the following anthropic manner.[55] 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.)

## 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.[56]

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.[57]

## Mathematics of spacetimes

### Riemannian geometry

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"). It is a very broad and abstract generalization of the differential geometry of surfaces in R3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dimensions. It enabled the formulation of Einstein's general theory of relativity, made profound impact on group theory and representation theory, as well as analysis, and spurred the development of algebraic and differential topology.

### Curved manifolds

For physical reasons, a spacetime continuum is mathematically defined as a four-dimensional, smooth, connected Lorentzian manifold ${\displaystyle (M,g)}$. This means the smooth Lorentz metric ${\displaystyle g}$ has signature ${\displaystyle (3,1)}$. 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 ${\displaystyle (x,y,z,t)}$ are used. Moreover, for simplicity's sake, units of measurement are usually chosen such that the speed of light ${\displaystyle c}$ is equal to 1.[58]

A reference frame (observer) can be identified with one of these coordinate charts; any such observer can describe any event ${\displaystyle p}$. Another reference frame may be identified by a second coordinate chart about ${\displaystyle p}$. Two observers (one in each reference frame) may describe the same event ${\displaystyle p}$ but obtain different descriptions.[58]

Usually, many overlapping coordinate charts are needed to cover a manifold. Given two coordinate charts, one containing ${\displaystyle p}$ (representing an observer) and another containing ${\displaystyle q}$ (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.[58]

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 ${\displaystyle p}$). In general, they will disagree about the exact location and timing of this impact, i.e., they will have different 4-tuples ${\displaystyle (x,y,z,t)}$ (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.[58]

### 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.[59]

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.[59] 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.[60]

## 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).[61][62] The peoples of the Andes maintain a similar understanding.[63]

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."[64]

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.[65]

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".[66]

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."[67]

Various authors have credited W. K. Clifford with having anticipated the concept of curved spacetime as far back as 1870,[68][69] 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.[70]

### Maxwell's contributions

Maxwell's equations were an essential inspiration for Einstein's development of special relativity. Possibly the most important aspect was their denial of instantaneous action at a distance. Rather, according to them, forces are propagated at the velocity of light through the electromagnetic field.[71]:189

Maxwell's original equations are based on the idea that light travels through a sea of molecular vortices known as the "luminiferous aether", and that the speed of light has to be respective to the reference frame of this aether. Measurements designed to measure the speed of the Earth through the aether conflicted with this notion, though.[note 8]

A more theoretical approach was suggested by Hendrik Lorentz along with George FitzGerald and Joseph Larmor. Both Larmor (1897) and Lorentz (1899, 1904) derived the Lorentz transformation (so named by Henri Poincaré) as one under which Maxwell's equations were invariant. Poincaré (1900) analyzed the coordination of moving clocks by exchanging light signals. He also established the mathematical group property of the Lorentz transformation (Poincaré 1905). Sometimes this transformation is called the FitzGerald–Lorentz transformation or even the FitzGerald–Lorentz–Einstein transformation.

Albert Einstein dismissed the notion of the aether as an unnecessary one, and he concluded that Maxwell's equations predicted the existence of a fixed speed of light, independent of the velocity of the observer. Hence, he used the Maxwell's equations as the starting point for his Special Theory of Relativity. In doing so, he established that the FitzGerald–Lorentz transformation is valid for all matter and space, and not just Maxwell's equations. Maxwell's equations played a key role in Einstein's groundbreaking scientific paper on special relativity (1905). For example, in the opening paragraph of his paper, he began his theory by noting that a description of an electric conductor moving with respect to a magnet must generate a consistent set of fields regardless of whether the force is calculated in the rest frame of the magnet or that of the conductor.[72]

The general theory of relativity has also had a close relationship with Maxwell's equations. For example, Theodor Kaluza and Oskar Klein in the 1920s showed that Maxwell's equations could be derived by extending general relativity into five physical dimensions. This strategy of using additional dimensions to unify different forces remains an active area of research in physics.

## Section summaries

### Introduction summary

• 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.
• 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.
• Einstein's theory of special relativity (1905), which was based on kinematics and a careful examination of the meaning of measurement, completely resolved the problems raised by the aforementioned experiments.
• Hermann Minkowski, in 1908, published a geometric interpretation of special relativity which has come to be known as Minkowski space, or spacetime.
• 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 ${\displaystyle c}$ so that the space and time coordinates have the same units (meters).
• 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.
• 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.
• 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".
• 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.
• 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.
• Because it takes time for light to travel from different points of an object to an observer, the visual appearance of an object traveling at high speed will not correspond to its measured dimensions.
• For example, a high speed object passing by an observer will not appear length contracted, but rotated. This illusion is known as "Terrell rotation".
• The apparent faster-than-light speeds of jets of matter ejected by certain black holes also represent optical illusion.
• 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.
• 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.
• 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

• 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
${\displaystyle x'=x-vt}$
• 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
${\displaystyle u'=u-v}$  or  ${\displaystyle u=u'+v}$
• The relativistic composition of velocities is more complex than the Galilean composition of velocities:
${\displaystyle u={v+u' \over 1+(vu'/c^{2})}.}$
• 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.
• The Lorentz factor, gamma ${\displaystyle \gamma ,}$ appears very frequently in relativity. Given ${\displaystyle \beta =v/c,}$
${\displaystyle \gamma ={\frac {1}{\sqrt {1-v^{2}/c^{2}}}}={\frac {1}{\sqrt {1-\beta ^{2}}}}}$
• ${\displaystyle \gamma }$ is the time dilation factor, while ${\displaystyle 1/\gamma }$ is the length contraction factor.
• The Lorentz factor is undefined for ${\displaystyle v\geq c.}$
• 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 ${\displaystyle t}$ and ${\displaystyle x}$ axes are:
{\displaystyle {\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.
• 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:
${\displaystyle f={\sqrt {\frac {1-\beta }{1+\beta }}}\,f_{0}.}$
• 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.
• 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
${\displaystyle P\equiv (E/c,{\vec {p}})=(E/c,p_{x},p_{y},p_{z})}$
• Consideration of the interrelationships between the various components of the relativistic momentum vector led Einstein to his famous ${\displaystyle E=mc^{2}}$ equation as well as to his concept of relativistic mass.
• 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 ${\displaystyle p=mv,}$ 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.
• 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.

### Beyond the basics summary

• 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 ${\displaystyle \beta }$ is the analog of slope. The rapidity, φ, is defined by
${\displaystyle \beta \equiv \tanh \phi \equiv {\frac {v}{c}}}$
• 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 ${\displaystyle \phi =\phi _{1}+\phi _{2}.}$
• The Lorentz boost in the x direction becomes a simple rotation:
${\displaystyle {\begin{pmatrix}ct'\\x'\end{pmatrix}}={\begin{pmatrix}\cosh \phi &-\sinh \phi \\-\sinh \phi &\cosh \phi \end{pmatrix}}{\begin{pmatrix}ct\\x\end{pmatrix}}}$.
• 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.

### Introduction to curved spacetime summary

• 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.
• 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 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.
• Certain special relativity setups are useful in that they provide insight into various aspects of general relativity. These setups include the Rindler chart, where each hyperbola represents the world line of an observer accelerating with a constant proper acceleration.
• To a Rindler observer, time seems to stand still at the origin, analogous to the manner in which time seems to stand still at the edge of a black hole.
• The Rindler observer is constantly accelerating away from the origin, but never gets any further from it, as measured in its own frame.
• A particular frame of the Rindler coordinate system, called "Rindler space", is useful because it is a uniformly accelerating frame where the distances between particles is constant.

## Notes

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.
2. ^ There are two sign conventions in use in the relativity literature: s2 = c2t2 − x2 − y2 − z2
and
s2 = −c2t2 + x2 + y2 + z2
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.
3. ^ Even though it has been many decades since Terrell and Penrose published their observations, popular writings continue to conflate measurement versus appearance. For example, Michio Kaku wrote in Einstein's Cosmos (W. W. Norton & Company, 2004. p. 65): "... imagine that the speed of light is only 20 miles per hour. If a car were to go down the street, it might look compressed in the direction of motion, being squeezed like an accordion down to perhaps 1 inch in length."
4. ^ In a Cartesian plane, ordinary rotation leaves a circle unchanged. In spacetime, hyperbolic rotation preserves the hyperbolic metric.
5. ^ 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.
6. ^ 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 2π) 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.
7. ^ 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."
8. ^ Experiments like the Michelson–Morley experiment in 1887 showed that the "aether" moved at the same speed as Earth. While other experiments, such as measurements of the aberration of light from the stars, showed that the ether is moving relative to the Earth.

1. ^ Stellar aberration arises when the motions of an observer have a component perpendicular to the path of star light. Fig. 1‑3 (bottom left) illustrates the observed effect of aberration: To center a star in the eyepiece, the telescope must be tilted into the direction of Earth's motion from the actual position of the star. Fig. 1‑3 (bottom right) shows a telescope immersed in the conjectured aether (in blue) and illustrates how aether dragging—if real—would make aberration disappear for earthbound telescopes; they could be pointed at any star's predicted location and it would always be centered in eyepieces.
2. ^ Different reporters viewing the scenarios presented in this figure interpret the scenarios differently depending on their knowledge of the situation. (i) A first reporter, at the center of mass of particles 2 and 3 but unaware of the large mass 1, concludes that a force of repulsion exists between the particles in scenario A while a force of attraction exists between the particles in scenario B. (ii) A second reporter, aware of the large mass 1, smiles at the first reporter's naiveté. This second reporter knows that in reality, the apparent forces between particles 2 and 3 really represent tidal effects resulting from their differential attraction by mass 1. (iii) A third reporter, trained in general relativity, knows that there are, in fact, no forces at all acting between the three objects. Rather, all three objects move along geodesics in spacetime.

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