General relativity |
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Special relativity |
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**Time dilation** is the difference in elapsed time as measured by two clocks, either due to a relative velocity between them (special relativity), or a difference in gravitational potential between their locations (general relativity). When unspecified, "time dilation" usually refers to the effect due to velocity.

- History
- Time dilation caused by a relative velocity
- Simple inference 2
- Reciprocity
- Experimental testing
- Proper time and Minkowski diagram
- Derivation and formulation
- Hyperbolic motion
- Clock hypothesis
- Time dilation caused by gravity or acceleration
- Experimental testing 2
- Combined effect of velocity and gravitational time dilation
- Experimental testing 3
- In popular culture
- See also
- Footnotes
- References
- Further reading
- External links

After compensating for varying signal delays resulting from the changing distance between an observer and a moving clock (i.e. Doppler effect), the observer will measure the moving clock as ticking slower than a clock that is at rest in the observer's own reference frame. In addition, a clock that is close to a massive body (and which therefore is at lower gravitational potential) will record less elapsed time than a clock situated farther from the said massive body (and which is at a higher gravitational potential).

These predictions of the theory of relativity have been repeatedly confirmed by experiment, and they are of practical concern, for instance in the operation of satellite navigation systems such as GPS and Galileo.^{ [1] }

Time dilation by the Lorentz factor was predicted by several authors at the turn of the 20th century.^{ [2] }^{ [3] } Joseph Larmor (1897), at least for electrons orbiting a nucleus, wrote that individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio: .^{ [4] } Emil Cohn (1904) specifically related this formula to the rate of clocks.^{ [5] } In the context of special relativity it was shown by Albert Einstein (1905) that this effect concerns the nature of time itself, and he was also the first to point out its reciprocity or symmetry.^{ [6] } Subsequently, Hermann Minkowski (1907) introduced the concept of proper time which further clarified the meaning of time dilation.^{ [7] }

Special relativity indicates that, for an observer in an inertial frame of reference, a clock that is moving relative to them will be measured to tick slower than a clock that is at rest in their frame of reference. This case is sometimes called special relativistic time dilation. The faster the relative velocity, the greater the time dilation between one another, with time slowing to a stop as one approaches the speed of light (299,792,458 m/s).

Theoretically, time dilation would make it possible for passengers in a fast-moving vehicle to advance further into the future in a short period of their own time. For sufficiently high speeds, the effect is dramatic. For example, one year of travel might correspond to ten years on Earth. Indeed, a constant 1 g acceleration would permit humans to travel through the entire known Universe in one human lifetime.^{ [9] }

With current technology severely limiting the velocity of space travel, however, the differences experienced in practice are minuscule: after 6 months on the International Space Station (ISS), orbiting Earth at a speed of about 7,700 m/s, an astronaut would have aged about 0.005 seconds less than those on Earth.^{ [10] } The cosmonauts Sergei Krikalev and Sergei Avdeyev both experienced time dilation of about 20 milliseconds compared to time that passed on Earth.^{ [11] }^{ [12] }

Time dilation can be inferred from the observed constancy of the speed of light in all reference frames dictated by the second postulate of special relativity.^{ [13] }^{ [14] }^{ [15] }^{ [16] }

This constancy of the speed of light means that, counter to intuition, the speeds of material objects and light are not additive. It is not possible to make the speed of light appear greater by moving towards or away from the light source.

Consider then, a simple vertical clock consisting of two mirrors A and B, between which a light pulse is bouncing. The separation of the mirrors is *L* and the clock ticks once each time the light pulse hits mirror A.

In the frame in which the clock is at rest (see left part of the diagram), the light pulse traces out a path of length 2*L* and the period of the clock is 2*L* divided by the speed of light:

From the frame of reference of a moving observer traveling at the speed *v* relative to the resting frame of the clock (right part of diagram), the light pulse is seen as tracing out a longer, angled path. Keeping the speed of light constant for all inertial observers requires a lengthening of the period of this clock from the moving observer's perspective. That is to say, as measured in a frame moving relative to the local clock, this clock will be running more slowly. Straightforward application of the Pythagorean theorem leads to the well-known prediction of special relativity:

The total time for the light pulse to trace its path is given by:

The length of the half path can be calculated as a function of known quantities as:

Elimination of the variables *D* and *L* from these three equations results in:

which expresses the fact that the moving observer's period of the clock is longer than the period in the frame of the clock itself.

Because all clocks that have a common period in the resting frame should have a common period when observed from the moving frame, all other clocks—mechanical, electronic, optical (such as an identical horizontal version of the clock in the example)—should exhibit the same velocity-dependent time dilation.^{ [17] }

Given a certain frame of reference, and the "stationary" observer described earlier, if a second observer accompanied the "moving" clock, each of the observers would perceive the other's clock as ticking at a *slower* rate than their own local clock, due to them both perceiving the other to be the one that is in motion relative to their own stationary frame of reference.

Common sense would dictate that, if the passage of time has slowed for a moving object, said object would observe the external world's time to be correspondingly sped up. Counterintuitively, special relativity predicts the opposite. When two observers are in motion relative to each other, each will measure the other's clock slowing down, in concordance with them being in motion relative to the observer's frame of reference.

While this seems self-contradictory, a similar oddity occurs in everyday life. If two persons A and B observe each other from a distance, B will appear small to A, but at the same time, A will appear small to B. Being familiar with the effects of perspective, there is no contradiction or paradox in this situation.^{ [18] }

The reciprocity of the phenomenon also leads to the so-called twin paradox where the aging of twins, one staying on Earth and the other embarking on space travel, is compared, and where the reciprocity suggests that both persons should have the same age when they reunite. On the contrary, at the end of the round-trip, the traveling twin will be younger than the sibling on Earth. The dilemma posed by the paradox can be explained by the fact that situation is not symmetric. The twin staying on Earth is in a single inertial frame, and the traveling twin is in two different inertial frames: one on the way out and another on the way back. See also Twin paradox#Role of acceleration.

- The stated purpose by Ives and Stilwell (1938, 1941) of these experiments was to verify the time dilation effect, predicted by Larmor–Lorentz ether theory, due to motion through the ether using Einstein's suggestion that Doppler effect in canal rays would provide a suitable experiment. These experiments measured the Doppler shift of the radiation emitted from cathode rays, when viewed from directly in front and from directly behind. The high and low frequencies detected were not the classically predicted values:The high and low frequencies of the radiation from the moving sources were measured as:
^{ [19] }as deduced by Einstein (1905) from the Lorentz transformation, when the source is running slow by the Lorentz factor. - Hasselkamp, Mondry, and Scharmann
^{ [20] }(1979) measured the Doppler shift from a source moving at right angles to the line of sight. The most general relationship between frequencies of the radiation from the moving sources is given by:as deduced by Einstein (1905).^{ [21] }For*ϕ*= 90° (cos*ϕ*= 0) this reduces to*f*_{detected}=*f*_{rest}γ. This lower frequency from the moving source can be attributed to the time dilation effect and is often called the transverse Doppler effect and was predicted by relativity. - In 2010 time dilation was observed at speeds of less than 10 metres per second using optical atomic clocks connected by 75 metres of optical fiber.
^{ [22] }

- A comparison of muon lifetimes at different speeds is possible. In the laboratory, slow muons are produced; and in the atmosphere, very fast-moving muons are introduced by cosmic rays. Taking the muon lifetime at rest as the laboratory value of 2.197 μs, the lifetime of a cosmic-ray-produced muon traveling at 98% of the speed of light is about five times longer, in agreement with observations. An example is Rossi and Hall (1941), who compared the population of cosmic-ray-produced muons at the top of a mountain to that observed at sea level.
^{ [23] } - The lifetime of particles produced in particle accelerators are longer due to time dilation. In such experiments, the "clock" is the time taken by processes leading to muon decay, and these processes take place in the moving muon at its own "clock rate", which is much slower than the laboratory clock. This is routinely taken into account in particle physics, and many dedicated measurements have been performed. For instance, in the muon storage ring at CERN the lifetime of muons circulating with γ = 29.327 was found to be dilated to 64.378 μs, confirming time dilation to an accuracy of 0.9 ± 0.4 parts per thousand.
^{ [24] }

In the Minkowski diagram from the first image on the right, clock C resting in inertial frame S′ meets clock A at *d* and clock B at *f* (both resting in S). All three clocks simultaneously start to tick in S. The worldline of A is the ct-axis, the worldline of B intersecting *f* is parallel to the ct-axis, and the worldline of C is the ct′-axis. All events simultaneous with *d* in S are on the x-axis, in S′ on the x′-axis.

The proper time between two events is indicated by a clock present at both events.^{ [25] } It is invariant, i.e., in all inertial frames it is agreed that this time is indicated by that clock. Interval *df* is, therefore, the proper time of clock C, and is shorter with respect to the coordinate times *ef=dg* of clocks B and A in S. Conversely, also proper time *ef* of B is shorter with respect to time *if* in S′, because event *e* was measured in S′ already at time *i* due to relativity of simultaneity, long before C started to tick.

From that it can be seen, that the proper time between two events indicated by an unaccelerated clock present at both events, compared with the synchronized coordinate time measured in all other inertial frames, is always the *minimal* time interval between those events. However, the interval between two events can also correspond to the proper time of accelerated clocks present at both events. Under all possible proper times between two events, the proper time of the unaccelerated clock is *maximal*, which is the solution to the twin paradox.^{ [25] }

In addition to the light clock used above, the formula for time dilation can be more generally derived from the temporal part of the Lorentz transformation.^{ [26] } Let there be two events at which the moving clock indicates and , thus:

Since the clock remains at rest in its inertial frame, it follows , thus the interval is given by:

where Δ*t* is the time interval between *two co-local events* (i.e. happening at the same place) for an observer in some inertial frame (e.g. ticks on their clock), known as the * proper time *, Δ`t′` is the time interval between those same events, as measured by another observer, inertially moving with velocity *v* with respect to the former observer, *v* is the relative velocity between the observer and the moving clock, *c* is the speed of light, and the Lorentz factor (conventionally denoted by the Greek letter gamma or γ) is:

Thus the duration of the clock cycle of a moving clock is found to be increased: it is measured to be "running slow". The range of such variances in ordinary life, where *v* ≪ *c*, even considering space travel, are not great enough to produce easily detectable time dilation effects and such vanishingly small effects can be safely ignored for most purposes. As an approximate threshold, time dilation may become important when an object approaches speeds on the order of 30,000 km/s (1/10 the speed of light).^{ [27] }

In special relativity, time dilation is most simply described in circumstances where relative velocity is unchanging. Nevertheless, the Lorentz equations allow one to calculate proper time and movement in space for the simple case of a spaceship which is applied with a force per unit mass, relative to some reference object in uniform (i.e. constant velocity) motion, equal to *g* throughout the period of measurement.

Let *t* be the time in an inertial frame subsequently called the rest frame. Let *x* be a spatial coordinate, and let the direction of the constant acceleration as well as the spaceship's velocity (relative to the rest frame) be parallel to the *x*-axis. Assuming the spaceship's position at time *t* = 0 being *x* = 0 and the velocity being *v*_{0} and defining the following abbreviation:

the following formulas hold:^{ [28] }

Position:

Velocity:

Proper time as function of coordinate time:

In the case where *v*(0) = *v*_{0} = 0 and *τ*(0) = *τ*_{0} = 0 the integral can be expressed as a logarithmic function or, equivalently, as an inverse hyperbolic function:

As functions of the proper time of the ship, the following formulae hold:^{ [29] }

Position:

Velocity:

Coordinate time as function of proper time:

The **clock hypothesis** is the assumption that the rate at which a clock is affected by time dilation does not depend on its acceleration but only on its instantaneous velocity. This is equivalent to stating that a clock moving along a path measures the proper time, defined by:

The clock hypothesis was implicitly (but not explicitly) included in Einstein's original 1905 formulation of special relativity. Since then, it has become a standard assumption and is usually included in the axioms of special relativity, especially in light of experimental verification up to very high accelerations in particle accelerators.^{ [30] }^{ [31] }

Gravitational time dilation is experienced by an observer that, at a certain altitude within a gravitational potential well, finds that their local clocks measure less elapsed time than identical clocks situated at higher altitude (and which are therefore at higher gravitational potential).

Gravitational time dilation is at play e.g. for ISS astronauts. While the astronauts' relative velocity slows down their time, the reduced gravitational influence at their location speeds it up, although to a lesser degree. Also, a climber's time is theoretically passing slightly faster at the top of a mountain compared to people at sea level. It has also been calculated that due to time dilation, the core of the Earth is 2.5 years younger than the crust.^{ [32] } "A clock used to time a full rotation of the Earth will measure the day to be approximately an extra 10 ns/day longer for every km of altitude above the reference geoid."^{ [33] } Travel to regions of space where extreme gravitational time dilation is taking place, such as near (but not beyond the event horizon of) a black hole, could yield time-shifting results analogous to those of near-lightspeed space travel.

Contrarily to velocity time dilation, in which both observers measure the other as aging slower (a reciprocal effect), gravitational time dilation is not reciprocal. This means that with gravitational time dilation both observers agree that the clock nearer the center of the gravitational field is slower in rate, and they agree on the ratio of the difference.

- In 1959, Robert Pound and Glen A. Rebka measured the very slight gravitational redshift in the frequency of light emitted at a lower height, where Earth's gravitational field is relatively more intense. The results were within 10% of the predictions of general relativity. In 1964, Pound and J. L. Snider measured a result within 1% of the value predicted by gravitational time dilation.
^{ [34] }(See Pound–Rebka experiment) - In 2010, gravitational time dilation was measured at the Earth's surface with a height difference of only one meter, using optical atomic clocks.
^{ [22] }

High-accuracy timekeeping, low-Earth-orbit satellite tracking, and pulsar timing are applications that require the consideration of the combined effects of mass and motion in producing time dilation. Practical examples include the International Atomic Time standard and its relationship with the Barycentric Coordinate Time standard used for interplanetary objects.

Relativistic time dilation effects for the solar system and the Earth can be modeled very precisely by the Schwarzschild solution to the Einstein field equations. In the Schwarzschild metric, the interval is given by:^{ [36] }^{ [37] }

where:

- is a small increment of proper time (an interval that could be recorded on an atomic clock),
- is a small increment in the coordinate (coordinate time),
- are small increments in the three coordinates of the clock's position,
- represents the sum of the Newtonian gravitational potentials due to the masses in the neighborhood, based on their distances from the clock. This sum includes any tidal potentials.

The coordinate velocity of the clock is given by:

The coordinate time is the time that would be read on a hypothetical "coordinate clock" situated infinitely far from all gravitational masses (), and stationary in the system of coordinates (). The exact relation between the rate of proper time and the rate of coordinate time for a clock with a radial component of velocity is:

where:

- is the radial velocity,
- is the escape speed,
- , and are velocities as a percentage of speed of light
*c*, - is the Newtonian potential; hence equals half the square of the escape speed.

The above equation is exact under the assumptions of the Schwarzschild solution. It reduces to velocity time dilation equation in the presence of motion and absence of gravity, i.e. . It reduces to gravitational time dilation equation in the absence of motion and presence of gravity, i.e. .

- Hafele and Keating, in 1971, flew caesium atomic clocks east and west around the Earth in commercial airliners, to compare the elapsed time against that of a clock that remained at the U.S. Naval Observatory. Two opposite effects came into play. The clocks were expected to age more quickly (show a larger elapsed time) than the reference clock since they were in a higher (weaker) gravitational potential for most of the trip (c.f. Pound–Rebka experiment). But also, contrastingly, the moving clocks were expected to age more slowly because of the speed of their travel. From the actual flight paths of each trip, the theory predicted that the flying clocks, compared with reference clocks at the U.S. Naval Observatory, should have lost 40±23 nanoseconds during the eastward trip and should have gained 275±21 nanoseconds during the westward trip. Relative to the atomic time scale of the U.S. Naval Observatory, the flying clocks lost 59±10 nanoseconds during the eastward trip and gained 273±7 nanoseconds during the westward trip (where the error bars represent standard deviation).
^{ [38] }In 2005, the National Physical Laboratory in the United Kingdom reported their limited replication of this experiment.^{ [39] }The NPL experiment differed from the original in that the caesium clocks were sent on a shorter trip (London–Washington, D.C. return), but the clocks were more accurate. The reported results are within 4% of the predictions of relativity, within the uncertainty of the measurements. - The Global Positioning System can be considered a continuously operating experiment in both special and general relativity. The in-orbit clocks are corrected for both special and general relativistic time dilation effects as described above, so that (as observed from the Earth's surface) they run at the same rate as clocks on the surface of the Earth.
^{ [40] }

Velocity and gravitational time dilation have been the subject of science fiction works in a variety of media. Some examples in film are the movies * Interstellar * and * Planet of the Apes *.^{ [41] } In *Interstellar*, a key plot point involves a planet, which is close to a rotating black hole and on the surface of which one hour is equivalent to seven years on Earth due to time dilation.^{ [42] } Physicist Kip Thorne collaborated in making the film and explained its scientific concepts in the book * The Science of Interstellar *.^{ [43] }^{ [44] }

Time dilation was used in the * Doctor Who * episodes "World Enough and Time" and "The Doctor Falls", which take place on a spaceship in the vicinity of a black hole. Due to the immense gravitational pull of the black hole and the ship's length (400 miles), time moves faster at one end than the other. When The Doctor's companion, Bill, gets taken away to the other end of the ship, she waits years for him to rescue her; in his time, only minutes pass.^{ [45] } Furthermore, the dilation allows the Cybermen to evolve at a "faster" rate than previously seen in the show.

* Tau Zero *, a novel by Poul Anderson, is an early example of the concept in science fiction literature. In the novel, a spacecraft uses a Bussard ramjet to accelerate to high enough speeds that the crew spends five years on board, but thirty-three years pass on the Earth before they arrive at their destination. The velocity time dilation is explained by Anderson in terms of the tau factor which decreases closer and closer to zero as the ship approaches the speed of light—hence the title of the novel.^{ [46] } Due to an accident, the crew is unable to stop accelerating the spacecraft, causing such extreme time dilation that the crew experiences the Big Crunch at the end of the universe.^{ [47] } Other examples in literature, such as * Rocannon's World * and * The Forever War *, similarly make use of relativistic time dilation as a scientifically plausible literary device to have certain characters age slower than the rest of the universe.^{ [48] }^{ [49] }

- ↑ Average time dilation has a weak dependence on the orbital inclination angle (Ashby 2003, p.32). The
*r*≈ 1.497 result corresponds to^{ [35] }the orbital inclination of modern GPS satellites, which is 55 degrees.

In physics and general relativity, **gravitational redshift** is the phenomenon that electromagnetic waves or photons travelling out of a gravitational well lose energy. This loss of energy corresponds to a decrease in the wave frequency and increase in the wavelength, known more generally as a *redshift*. The opposite effect, in which photons gain energy when travelling into a gravitational well, is known as a **gravitational blueshift**. The effect was first described by Einstein in 1907, eight years before his publication of the full theory of relativity.

In physics, the **special theory of relativity**, or **special relativity** for short, is a scientific theory of the relationship between space and time. In Albert Einstein's 1905 treatment, the theory is presented as being based on just two postulates:

- The laws of physics are invariant (identical) in all inertial frames of reference.
- The speed of light in vacuum is the same for all observers, regardless of the motion of light source or observer.

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

In physics, the **twin paradox** is a thought experiment in special relativity involving identical twins, one of whom makes a journey into space in a high-speed rocket and returns 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, as a consequence of an incorrect and naive application of time dilation and the principle of relativity, each should paradoxically find the other to have aged less. However, this scenario can be resolved within the standard framework of special relativity: the travelling twin's trajectory involves two different inertial frames, one for the outbound journey and one for the inbound journey. Another way of looking at it is to realize the travelling twin is undergoing acceleration, which makes him a non-inertial observer. In both views there is no symmetry between the spacetime paths of the twins. Therefore, the twin paradox is not actually a paradox in the sense of a logical contradiction. There is still debate as to the resolution of the twin paradox.

In physics, in particular in special relativity and general relativity, a **four-velocity** is a four-vector in four-dimensional spacetime that represents the relativistic counterpart of velocity, which is a three-dimensional vector in space.

**Length contraction** is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is the length as measured in the object's own rest frame. It is also known as **Lorentz contraction** or **Lorentz–FitzGerald contraction** and is usually only noticeable at a substantial fraction of the speed of light. Length contraction is only in the direction in which the body is travelling. For standard objects, this effect is negligible at everyday speeds, and can be ignored for all regular purposes, only becoming significant as the object approaches the speed of light relative to the observer.

The **relativistic Doppler effect** is the change in frequency, wavelength and amplitude of light, caused by the relative motion of the source and the observer, when taking into account effects described by the special theory of relativity.

In relativity, **proper time** along a timelike world line is defined as the time as measured by a clock following that line. The **proper time interval** between two events on a world line is the change in proper time, which is independent of coordinates, and is a Lorentz scalar. The interval is the quantity of interest, since proper time itself is fixed only up to an arbitrary additive constant, namely the setting of the clock at some event along the world line.

The **Lorentz factor** or **Lorentz term** is a quantity that expresses how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativity, and it arises in derivations of the Lorentz transformations. The name originates from its earlier appearance in Lorentzian electrodynamics – named after the Dutch physicist Hendrik Lorentz.

**Gravitational time dilation** is a form of time dilation, an actual difference of elapsed time between two events as measured by observers situated at varying distances from a gravitating mass. The lower the gravitational potential, the slower time passes, speeding up as the gravitational potential increases. Albert Einstein originally predicted this effect in his theory of relativity and it has since been confirmed by tests of general relativity.

The **Shapiro time delay** effect, or **gravitational time delay** effect, is one of the four classic Solar System tests of general relativity. Radar signals passing near a massive object take slightly longer to travel to a target and longer to return than they would if the mass of the object were not present. The time delay is caused by spacetime dilation, which increases the time it takes light to travel a given distance from the perspective of an outside observer. In a 1964 article entitled *Fourth Test of General Relativity*, Shapiro wrote:

Because, according to the general theory, the speed of a light wave depends on the strength of the gravitational potential along its path, these time delays should thereby be increased by almost 2×10

^{−4}sec when the radar pulses pass near the sun. Such a change, equivalent to 60 km in distance, could now be measured over the required path length to within about 5 to 10% with presently obtainable equipment.

Rindler coordinates are a coordinate system used in the context of special relativity to describe the hyperbolic acceleration of a uniformly accelerating reference frame in flat spacetime. In relativistic physics the coordinates of a *hyperbolically accelerated reference frame* constitute an important and useful coordinate chart representing part of flat Minkowski spacetime. In special relativity, a uniformly accelerating particle undergoes hyperbolic motion, for which a uniformly accelerating frame of reference in which it is at rest can be chosen as its proper reference frame. The phenomena in this hyperbolically accelerated frame can be compared to effects arising in a homogeneous gravitational field. For general overview of accelerations in flat spacetime, see Acceleration and Proper reference frame.

The **Ehrenfest paradox** concerns the rotation of a "rigid" disc in the theory of relativity.

The **Ives–Stilwell experiment** tested the contribution of relativistic time dilation to the Doppler shift of light. The result was in agreement with the formula for the transverse Doppler effect and was the first direct, quantitative confirmation of the time dilation factor. Since then many Ives–Stilwell type experiments have been performed with increased precision. Together with the Michelson–Morley and Kennedy–Thorndike experiments it forms one of the fundamental tests of special relativity theory. Other tests confirming the relativistic Doppler effect are the Mössbauer rotor experiment and modern Ives–Stilwell experiments.

The **geodetic effect** represents the effect of the curvature of spacetime, predicted by general relativity, on a vector carried along with an orbiting body. For example, the vector could be the angular momentum of a gyroscope orbiting the Earth, as carried out by the Gravity Probe B experiment. The geodetic effect was first predicted by Willem de Sitter in 1916, who provided relativistic corrections to the Earth–Moon system's motion. De Sitter's work was extended in 1918 by Jan Schouten and in 1920 by Adriaan Fokker. It can also be applied to a particular secular precession of astronomical orbits, equivalent to the rotation of the Laplace–Runge–Lenz vector.

**Lemaître coordinates** are a particular set of coordinates for the Schwarzschild metric—a spherically symmetric solution to the Einstein field equations in vacuum—introduced by Georges Lemaître in 1932. Changing from Schwarzschild to Lemaître coordinates removes the coordinate singularity at the Schwarzschild radius.

In relativity, **proper velocity****w** of an object relative to an observer is the ratio between observer-measured displacement vector and proper time τ elapsed on the clocks of the traveling object:

**Test theories of special relativity** give a mathematical framework for analyzing results of experiments to verify special relativity.

In physics, time is defined by its measurement: time is what a clock reads. In classical, non-relativistic physics, it is a scalar quantity and, like length, mass, and charge, is usually described as a fundamental quantity. Time can be combined mathematically with other physical quantities to derive other concepts such as motion, kinetic energy and time-dependent fields. *Timekeeping* is a complex of technological and scientific issues, and part of the foundation of *recordkeeping*.

**Accelerations in special relativity** (SR) follow, as in Newtonian Mechanics, by differentiation of velocity with respect to time. Because of the Lorentz transformation and time dilation, the concepts of time and distance become more complex, which also leads to more complex definitions of "acceleration". SR as the theory of flat Minkowski spacetime remains valid in the presence of accelerations, because general relativity (GR) is only required when there is curvature of spacetime caused by the energy–momentum tensor. However, since the amount of spacetime curvature is not particularly high on Earth or its vicinity, SR remains valid for most practical purposes, such as experiments in particle accelerators.

- 1 2 3 Ashby, Neil (2003). "Relativity in the Global Positioning System".
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*Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905–1911)*. Reading, Massachusetts: Addison–Wesley. ISBN 978-0-201-04679-3.. - ↑ Darrigol, Olivier (2005). "The Genesis of the Theory of Relativity".
*Einstein, 1905–2005*(PDF). Vol. 1. pp. 1–22. doi:10.1007/3-7643-7436-5_1. ISBN 978-3-7643-7435-8.`{{cite book}}`

:`|work=`

ignored (help) - ↑ Larmor, Joseph (1897).
*Philosophical Transactions of the Royal Society*.**190**: 205–300. Bibcode:1897RSPTA.190..205L. doi: 10.1098/rsta.1897.0020 . . - ↑ Cohn, Emil (1904), "Zur Elektrodynamik bewegter Systeme II" [ On the Electrodynamics of Moving Systems II ],
*Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften*, vol. 1904/2, no. 43, pp. 1404–1416 - ↑ Einstein, Albert (1905). "Zur Elektrodynamik bewegter Körper".
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*The New York Times*. Retrieved 2015-12-08. - ↑ Gott, Richard J. (2002).
*Time Travel in Einstein's Universe*. p. 75. - ↑ Cassidy, David C.; Holton, Gerald James; Rutherford, Floyd James (2002).
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*Physics for Scientists and Engineers, Volume 2*. Jones and Bartlett. pp. 1051–1052. ISBN 978-0-7637-0460-5. - ↑ Ellis, George F. R.; Williams, Ruth M. (2000).
*Flat and Curved Space-times*(2n ed.). Oxford University Press. pp. 28–29. ISBN 978-0-19-850657-7. - ↑ Galli, J. Ronald; Amiri, Farhang (Apr 2012). "The Square Light Clock and Special Relativity" (PDF).
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