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Until the turn of the 20th century, the assumption had been that the 3D 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 precisely described important consequences of this fact: both the shape of space and the measurement of time simultaneously change for observers who have different relative velocities, which is to say, for observers who have different "inertial frames of reference."
Einstein's theory was framed in terms of kinematics (the study of moving bodies), and showed how measurements of space and time varied for observers 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—what mathematicians call a 4‑dimensional "manifold." 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 among observers, the spacetime interval is independent of the inertial frame of reference in which they are recorded. The resultant "spacetime" came to be known as Minkowski space.
- 1 Introduction
- 2 Foreshadowings of the spacetime concept
- 3 Spacetime in special relativity
- 4 Spacetime in general relativity
- 5 Quantized spacetime
- 6 Mathematics of spacetimes
- 7 See also
- 8 Notes
- 9 References
- 10 Further reading
- 11 External links
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. Furthermore, it assumes that space is Euclidean, which is to say, it assumes that space follows the geometry of common sense.
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. An extremely large scale factor, (conventionally called the "speed of light") relates distances measured in space with distances measured in time. The magnitude of this scale factor (nearly km in space being equivalent to 1 s 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 300,000 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.
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".:105
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, 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. 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.
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. 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. The famous Michelson–Morley experiment of 1887 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. 3).
George Francis FitzGerald in 1883 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.: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. 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.:163–174
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.: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.:151–152 The spacetime of special relativity has since come to be known as Minkowski spacetime.
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 and a spatial distance . Then the spacetime interval between the two events is given by
The equation above is simply an expression of the Pythagorean theorem, except with a minus sign between the and the terms. Note also that the spacetime interval is the quantity , not 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 as a distinct symbol in itself, rather than the square of something.:217
Because of the minus sign, the spacetime interval between two events can be zero. Spacetime intervals are zero when . 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. 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 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.
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 the upper left of Fig. 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' 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'.:107
Fig. 5 (right) 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
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. 5 (lower left) 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. 5 (right) in the frame of observer O. The light paths have slope = 4 and −1 so that ΔPQR forms a right triangle. Since OP = OQ = OR, the angle between x' and x must also be θ.:113–118
In Fig. 6, 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. 7, 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 .:220
The light cone divides spacetime into separate regions. The interior of the future light cone consists of events that are separated from the origin event by more time than there is space: these events comprise the "timelike future" of the origin event. Likewise, the "timelike past" comprises the interior events of the past light cone. Since in timelike intervals, the separation in time is greater than the separation in space, timelike intervals are negative. 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. Because of the invariance of spacetime, all observers will agree on this division of spacetime.: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. 6, 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.
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. 6 was drawn from the reference frame of an observer moving at . 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.
Fig. 8 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. 5. The three events (A, B, C) are simultaneous from the reference frame of an observer moving at . From the reference frame of an observer moving at , the events appear to occur in the order C, B, A. From the reference frame of an observer moving at , 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.:220–221
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
The example above is the equation of a hyperbola in an x–ct spacetime diagram, which is termed an "invariant hyperbola".
In Fig. 9 (right), 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. Fig. 9 (left) shows that when viewed in an extra dimension of space, the spacelike invariant hyperbolae generate hyperboloids of one sheet, while the timelike invariant hyperbolae generate hyperboloids of two sheets.
Time dilation and length contraction
Fig. 10 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.:220–221 The measurement of time dilation by two observers in different inertial reference frames is mutual. If observer O measures the clocks of observer O' as running slower in his frame, observer O' in turn will measure the clocks of observer O as running slower.
Length contraction, like time dilation, is a manifestation of the relativity of simultaneity. Measurement of length requires measurement of the spacetime interval between two events that are simultaneous in one's frame of reference. But events that are simultaneous in one frame of reference are, in general, not simultaneous in other frames of reference.
Fig. 11 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 intersection of their world lines with the x axis yields the foreshortened length OC.: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 mere optical illusion, 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, there is a fundamental difference between measurement versus 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. A sphere in motion retains the appearance of a sphere, although images on the surface of the sphere will appear distorted. Fig. 12 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 2]
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.
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.":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.:198–199
In regards to mutual length contraction, Fig. 11 illustrates that the primed and unprimed frames are mutually rotated by a hyperbolic angle (analogous to ordinary angles in Euclidean geometry).[Note 3] Because of this rotation, the projection of a primed meter-stick onto the unprimed x-axis is foreshortened, while the projection of an unprimed meter-stick onto the primed x'-axis is likewise foreshortened.
Fig. 13 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 appear 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 appear 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.:124
In Fig. 13, 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 sees 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 mutual time dilation argument presented above by avoiding the requirement for a third clock.: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.
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. 14. 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.
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.
Complications arise if the twin paradox is analyzed from Stella's point of view. 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. 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.
Figs. 8 and 13 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. 14, 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.
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.:221 A few of these phenomena are described in the later sections of this article.
Foreshadowings of the spacetime concept
In myth and literature
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
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".
The ancient idea of the cosmos gradually was described mathematically with differential equations, differential geometry, and abstract algebra. These mathematical articulations blossomed in the nineteenth century as electrical technology stimulated men like Michael Faraday and James Clerk Maxwell to describe the reciprocal relations of electric and magnetic fields. Daniel Siegel phrased Maxwell's role in relativity as follows:
[...] the idea of the propagation of forces at the velocity of light through the electromagnetic field as described by Maxwell's equations—rather than instantaneously at a distance—formed the necessary basis for relativity theory.:189
[Maxwell] was not able to create the theory that he envisaged except by giving up the use of any model, and by extending by means of analogy the abstract system of electrodynamics to displacement currents.
In Siegel's estimation, "this very abstract view of the electromagnetic fields, involving no visualizable picture of what is going on out there in the field, is Maxwell's legacy.":191
Describing the behaviour of electric fields and magnetic fields led Maxwell to view the combination as an electromagnetic field. These fields have a value at every point of spacetime. It is the intermingling of electric and magnetic manifestations, described by Maxwell's equations, that give spacetime its structure. In particular, the rate of motion of an observer determines the electric and magnetic profiles of the electromagnetic field. The propagation of the field is determined by the electromagnetic wave equation, which requires spacetime for description.:1–30
Various authors have credited W. K. Clifford with having anticipated the spacetime concept in general relativity as far back as 1870, although the actual extent to which Clifford anticipated spacetime 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 speculated on the possibility of curved space, not curved spacetime.
Spacetime in special relativity
The geometry of spacetime in special relativity is described by the Minkowski metric on R4. This spacetime is called Minkowski space. The Minkowski metric is usually denoted by and can be written as a four-by-four matrix:
where the Landau–Lifshitz time-like convention is being used. A basic assumption of relativity is that coordinate transformations must leave spacetime intervals invariant. Intervals are invariant under Lorentz transformations. This invariance property leads to the use of four-vectors (and other tensors) in describing physics.:1–30
Strictly speaking, one can also consider events in Newtonian physics as a single spacetime. This is Galilean–Newtonian relativity, and the coordinate systems are related by Galilean transformations. However, since these preserve spatial and temporal distances independently, such a spacetime can always be decomposed into spatial coordinates plus temporal coordinates, which is not possible for general spacetimes.
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.: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.
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.
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.
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.
Mathematics of spacetimes
For physical reasons, a spacetime continuum is mathematically defined as a four-dimensional, smooth, connected Lorentzian manifold . This means the smooth Lorentz metric has signature . 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 are used. Moreover, for simplicity's sake, units of measurement are usually chosen such that the speed of light is equal to 1.
A reference frame (observer) can be identified with one of these coordinate charts; any such observer can describe any event . Another reference frame may be identified by a second coordinate chart about . Two observers (one in each reference frame) may describe the same event but obtain different descriptions.
Usually, many overlapping coordinate charts are needed to cover a manifold. Given two coordinate charts, one containing (representing an observer) and another containing (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.
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 ). In general, they will disagree about the exact location and timing of this impact, i.e., they will have different 4-tuples (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.
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.
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. 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.
Often in relativity, spacetimes that have some form of symmetry are studied. As well as helping to classify spacetimes, these symmetries usually serve as a simplifying assumption in specialized work. Some of the most popular ones include:
The causal structure of a spacetime describes causal relationships between pairs of points in the spacetime based on the existence of certain types of curves joining the points.
- Basic introduction to the mathematics of curved spacetime
- Global spacetime structure
- Metric space
- Philosophy of space and time
- There are two sign conventions in use in the relativity literature:
s2 = c2t2 − x2 − y2 − z2
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. Since it is unclear which sign convention is in greater use, one should simply accept the fact that these variations exist.
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