Time dilation

According to the theory of relativity, time dilation is a difference in the elapsed time measured by two observers, either due to a velocity difference relative to each other, or by being differently situated relative to a gravitational field. As a result of the nature of spacetime,[2] a clock that is moving relative to an observer will be measured to tick slower than a clock that is at rest in the observer's own frame of reference. A clock that is under the influence of a stronger gravitational field than an observer's will also be measured to tick slower than the observer's own clock.

Such time dilation has been repeatedly demonstrated, for instance by small disparities in a pair of atomic clocks after one of them is sent on a space trip, or by clocks on the Space Shuttle running slightly slower than reference clocks on Earth, or clocks on GPS and Galileo satellites running slightly faster.[1][2][3] Time dilation has also been the subject of science fiction works, as it technically provides the means for forward time travel.[4]

Soyuz TMA-1 at the ISS
Time dilation explains why two working clocks will report different times after different accelerations. For example, at the ISS time goes slower, lagging 0.007 seconds behind for every six months. For GPS satellites to work, they must adjust for similar bending of spacetime to coordinate with systems on Earth.[1]


Time dilation by the Lorentz factor was predicted by several authors at the turn of the 20th century.[5][6] Joseph Larmor (1897), at least for electrons orbiting a nucleus, wrote "... individual electrons describe corresponding parts of their orbits in times shorter for the [rest] system in the ratio :".[7] Emil Cohn (1904) specifically related this formula to the rate of clocks.[8] 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.[9] Subsequently, Hermann Minkowski (1907) introduced the concept of proper time which further clarified the meaning of time dilation.[10]

Velocity time dilation

Nonsymmetric velocity time dilation
From the local frame of reference of the blue clock, the red clock, being in motion, is perceived as ticking slower[11] (Exaggerated)

Special relativity indicates that, for an observer in an inertial frame of reference, a clock that is moving relative to him will be measured to tick slower than a clock that is at rest in his 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 the rate of time reaching zero as one approaches the speed of light (299,792,458 m/s). This causes massless particles that travel at the speed of light to be unaffected by the passage of time.

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.[2] 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.[12].

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) (which orbits Earth at a speed of about 7,700 m/s[3]) an astronaut would have aged about 0.005 seconds less than those on Earth. The cosmonauts Sergei Krikalev and Sergei Avdeyev both experienced time dilation of about 20 milliseconds compared to time that passed on Earth.[13][14]

Simple inference of velocity time dilation

Left: Observer at rest measures time 2L/c between co-local events of light signal generation at A and arrival at A.
Right: Events according to an observer moving to the left of the setup: bottom mirror A when signal is generated at time t'=0, top mirror B when signal gets reflected at time t'=D/c, bottom mirror A when signal returns at time t'=2D/c

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.[15][16][17][18]

This constancy of the speed of light means that, counter to intuition, 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 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 either of the mirrors.

In the frame in which the clock is at rest (diagram on the left), the light pulse traces out a path of length 2L and the period of the clock is 2L 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 (diagram at right), 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, in a frame moving relative to the local clock, this clock will appear to 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.


Time UV of a clock in S is shorter compared to Ux′ in S′, and time UW of a clock in S′ is shorter compared to Ux in S.
Time dilation02
Transversal time dilation. The blue dots represent a pulse of light. Each pair of dots with light "bouncing" between them is a clock. For each group of clocks, the other group appears to be ticking more slowly, because the moving clock's light pulse has to travel a larger distance than the stationary clock's light pulse. That is so, even though the clocks are identical and their relative motion is perfectly reciprocal.

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's 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 moving 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.[19]

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 a 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 his brother on Earth. The dilemma posed by the paradox, however, can be explained by the fact that the traveling twin must markedly accelerate in at least three phases of the trip (beginning, direction change, and end), while the other will only experience negligible acceleration, due to rotation and revolution of Earth. During the acceleration phases of the space travel, time dilation is not symmetric.

Experimental testing

Doppler effect

  • 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[20]
as deduced by Einstein (1905) from the Lorentz transformation, when the source is running slow by the Lorentz factor.
  • Hasselkamp, Mondry, and Scharmann[21] (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).[22] For ϕ = 90° (cos ϕ = 0) this reduces to fdetected = frestγ. 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 meters per second using optical atomic clocks connected by 75 meters of optical fiber.[23]

Moving particles

  • 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.[24]
  • The lifetime of particles produced in particle accelerators appears 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.[25]

Proper time and Minkowski diagram

Clock C in relative motion between two synchronized clocks A and B. C meets A at d, and B at f.
Twin paradox. One twin has to change frames, leading to different proper times in the twin's world lines.

In the Minkowski diagram from the second 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.[26] 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.[26]

Derivation and formulation

Time dilation
Lorentz factor as a function of speed (in natural units where c = 1). Notice that for small speeds (less than 0.1), γ is approximately 1

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.[27] 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 Δt 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 his clock), known as the proper time, Δt′ 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 vc, 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. It is only when an object approaches speeds on the order of 30,000 km/s (1/10 the speed of light) that time dilation becomes important.[28]

Hyperbolic motion

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 v0 and defining the following abbreviation

the following formulas hold:[29]



Proper time as function of coordinate time:

In the case where v(0) = v0 = 0 and τ(0) = τ0 = 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:[30]



Coordinate time as function of proper time:

Clock hypothesis

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 the light of experimental verification up to very high accelerations in particle accelerators.[31][32]

Gravitational time dilation

The Earth seen from Apollo 17
Time passes more quickly further from a center of gravity, as is witnessed with massive objects (like the Earth)

Gravitational time dilation is experienced by an observer that, being under the influence of a gravitational field, will measure his own clock to slow down, compared to another that is under a weaker gravitational field.

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 at 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.[33] "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." [34] Travel to regions of space where extreme gravitational time dilation is taking place, such as near 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.

Experimental testing

  • In 1959 Robert Pound and Glen A. Rebka measured the very slight gravitational red shift 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.[35] (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.[23]

Combined effect of velocity and gravitational time dilation

Daily satellite time dilation
Daily time dilation (gain or loss if negative) in microseconds as a function of (circular) orbit radius r = rs/re, where rs is satellite orbit radius and re is the equatorial Earth radius, calculated using the Schwarzschild metric. At r ≈ 1.497 [Note 1] there is no time dilation. Here the effects of motion and reduced gravity cancel. ISS astronauts fly below, whereas GPS and Geostationary satellites fly above.[1]
Time Dilation vs Orbital Height
Daily time dilation over circular orbit height split into its components.

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[37][38]


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


is the radial velocity,
is the escape velocity,
, and are velocities as a percentage of speed of light c,
is the Newtonian potential, equivalent to half of the escape velocity squared.

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. .

Experimental testing

  • 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).[39] In 2005, the National Physical Laboratory in the United Kingdom reported their limited replication of this experiment.[40] 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.[41]

See Also


  1. ^ Average time dilation has a weak dependence on the orbital inclination angle (Ashby 2003, p.32). The r ≈ 1.497 result corresponds to [36] the orbital inclination of modern GPS satellites, which is 55 degrees.


  1. ^ a b c Ashby, Neil (2003). "Relativity in the Global Positioning System" (PDF). Living Reviews in Relativity. 6 (1): 16. Bibcode:2003LRR.....6....1A. doi:10.12942/lrr-2003-1. PMC 5253894. PMID 28163638.
  2. ^ a b c Toothman, Jessika (2010-09-28). "How Do Humans age in space?". HowStuffWorks. Retrieved 2018-04-08.
  3. ^ a b Lu, Ed. "Expedition 7: Relativity". Ed's Musing from Space. NASA. Retrieved 2018-04-08.
  4. ^ "Is time travel possible?". NASA Space Place. Retrieved 2018-08-03.
  5. ^ Miller, Arthur I. (1981). Albert Einstein's Special Theory of Relativity: Emergence (1905) and Early Interpretation (1905–1911). Reading, Massachusetts: Addison–Wesley. ISBN 978-0-201-04679-3..
  6. ^ Darrigol, Olivier (2005). The Genesis of the Theory of Relativity (PDF). Séminaire Poincaré. 1. pp. 1–22. doi:10.1007/3-7643-7436-5_1. ISBN 978-3-7643-7435-8.
  7. ^ Larmor, Joseph (1897). "On a Dynamical Theory of the Electric and Luminiferous Medium, Part 3, Relations with Material Media" . Philosophical Transactions of the Royal Society. 190: 205–300. Bibcode:1897RSPTA.190..205L. doi:10.1098/rsta.1897.0020.
  8. ^ Cohn, Emil (1904), "Zur Elektrodynamik bewegter Systeme II" [On the Electrodynamics of Moving Systems II], Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, 1904/2 (43): 1404–1416
  9. ^ Einstein, Albert (1905). "Zur Elektrodynamik bewegter Körper". Annalen der Physik. 322 (10): 891–921. Bibcode:1905AnP...322..891E. doi:10.1002/andp.19053221004.. See also: English translation.
  10. ^ Minkowski, Hermann (1908) [1907], "Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern"  [The Fundamental Equations for Electromagnetic Processes in Moving Bodies], Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 53–111
  11. ^ Hraskó, Péter (2011). Basic Relativity: An Introductory Essay (illustrated ed.). Springer Science & Business Media. p. 60. ISBN 978-3-642-17810-8. Extract of page 60
  12. ^ Calder, Nigel (2006). Magic Universe: A grand tour of modern science. Oxford University Press. p. 378. ISBN 978-0-19-280669-7.
  13. ^ Overbye, Dennis (2005-06-28). "A Trip Forward in Time. Your Travel Agent: Einstein". The New York Times. Retrieved 2015-12-08.
  14. ^ Gott, J., Richard (2002). Time Travel in Einstein's Universe. p. 75.
  15. ^ Cassidy, David C.; Holton, Gerald James; Rutherford, Floyd James (2002). Understanding Physics. Springer-Verlag. p. 422. ISBN 978-0-387-98756-9.
  16. ^ Cutner, Mark Leslie (2003). Astronomy, A Physical Perspective. Cambridge University Press. p. 128. ISBN 978-0-521-82196-4.
  17. ^ Lerner, Lawrence S. (1996). Physics for Scientists and Engineers, Volume 2. Jones and Bartlett. pp. 1051–1052. ISBN 978-0-7637-0460-5.
  18. ^ 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.
  19. ^ Adams, Steve (1997). Relativity: An introduction to space-time physics. CRC Press. p. 54. ISBN 978-0-7484-0621-0.
  20. ^ Blaszczak, Z. (2007). Laser 2006. Springer. p. 59. ISBN 978-3540711131.
  21. ^ Hasselkamp, D.; Mondry, E.; Scharmann, A. (1979). "Direct observation of the transversal Doppler-shift". Zeitschrift für Physik A. 289 (2): 151–155. Bibcode:1979ZPhyA.289..151H. doi:10.1007/BF01435932.
  22. ^ Einstein, A. (1905). "On the electrodynamics of moving bodies". Fourmilab.
  23. ^ a b Chou, C. W.; Hume, D. B.; Rosenband, T.; Wineland, D. J. (2010). "Optical Clocks and Relativity". Science. 329 (5999): 1630–1633. Bibcode:2010Sci...329.1630C. doi:10.1126/science.1192720. PMID 20929843.
  24. ^ Stewart, J. V. (2001). Intermediate electromagnetic theory. World Scientific. p. 705. ISBN 978-981-02-4470-5.
  25. ^ Bailey, J. et al. Nature 268, 301 (1977)
  26. ^ a b Edwin F. Taylor, John Archibald Wheeler (1992). Spacetime Physics: Introduction to Special Relativity. New York: W. H. Freeman. ISBN 978-0-7167-2327-1.
  27. ^ Born, Max (1964), Einstein's Theory of Relativity, Dover Publications, ISBN 978-0-486-60769-6
  28. ^ Petkov, Vesselin (2009). Relativity and the Nature of Spacetime (2nd, illustrated ed.). Springer Science & Business Media. p. 87. ISBN 978-3-642-01962-3. Extract of page 87
  29. ^ See equations 3, 4, 6 and 9 of Iorio, Lorenzo (2005). "An analytical treatment of the Clock Paradox in the framework of the Special and General Theories of Relativity". Foundations of Physics Letters. 18 (1): 1–19. arXiv:physics/0405038. Bibcode:2005FoPhL..18....1I. doi:10.1007/s10702-005-2466-8.
  30. ^ Rindler, W. (1977). Essential Relativity. Springer. pp. 49–50. ISBN 978-3540079705.
  31. ^ Bailey, H.; Borer, K.; Combley F.; Drumm H.; Krienen F.; Lange F.; Picasso E.; Ruden W. von; Farley F. J. M.; Field J. H.; Flegel W. & Hattersley P. M. (1977). "Measurements of relativistic time dilatation for positive and negative muons in a circular orbit". Nature. 268 (5618): 301–305. Bibcode:1977Natur.268..301B. doi:10.1038/268301a0.
  32. ^ Roos, C. E.; Marraffino, J.; Reucroft, S.; Waters, J.; Webster, M. S.; Williams, E. G. H. (1980). "σ+/- lifetimes and longitudinal acceleration". Nature. 286 (5770): 244–245. Bibcode:1980Natur.286..244R. doi:10.1038/286244a0.
  33. ^ "New calculations show Earth's core is much younger than thought". Phys.org. 26 May 2016.
  34. ^ Burns, M. Shane; Leveille, Michael D.; Dominguez, Armand R.; Gebhard, Brian B.; Huestis, Samuel E.; Steele, Jeffrey; Patterson, Brian; Sell, Jerry F.; Serna, Mario; Gearba, M. Alina; Olesen, Robert; O'Shea, Patrick; Schiller, Jonathan (18 September 2017). "Measurement of gravitational time dilation: An undergraduate research project". American Journal of Physics. 85 (10): 757–762. arXiv:1707.00171. doi:10.1119/1.5000802.
  35. ^ Pound, R. V.; Snider J. L. (November 2, 1964). "Effect of Gravity on Nuclear Resonance". Physical Review Letters. 13 (18): 539–540. Bibcode:1964PhRvL..13..539P. doi:10.1103/PhysRevLett.13.539.
  36. ^ Ashby, Neil (2002). "Relativity in the Global Positioning System". Physics Today. 55 (5): 45. Bibcode:2002PhT....55e..41A. doi:10.1063/1.1485583.
  37. ^ See equations 2 & 3 (combined here and divided throughout by c2) at pp. 35–36 in Moyer, T. D. (1981). "Transformation from proper time on Earth to coordinate time in solar system barycentric space-time frame of reference". Celestial Mechanics. 23 (1): 33–56. Bibcode:1981CeMec..23...33M. doi:10.1007/BF01228543. hdl:2060/19770007221.
  38. ^ A version of the same relationship can also be seen at equation 2 in Ashbey, Neil (2002). "Relativity and the Global Positioning System" (PDF). Physics Today. 55 (5): 45. Bibcode:2002PhT....55e..41A. doi:10.1063/1.1485583.
  39. ^ Nave, C. R. (22 August 2005). "Hafele and Keating Experiment". HyperPhysics. Retrieved 2013-08-05.
  40. ^ "Einstein" (PDF). Metromnia. National Physical Laboratory. 2005. pp. 1–4.
  41. ^ Kaplan, Elliott; Hegarty, Christopher (2005). Understanding GPS: Principles and Applications. Artech House. p. 306. ISBN 978-1-58053-895-4. Extract of page 306

Further reading

External links

Barycentric Coordinate Time

Barycentric Coordinate Time (TCB, from the French Temps-coordonnée barycentrique) is a coordinate time standard intended to be used as the independent variable of time for all calculations pertaining to orbits of planets, asteroids, comets, and interplanetary spacecraft in the Solar system. It is equivalent to the proper time experienced by a clock at rest in a coordinate frame co-moving with the barycenter of the Solar system: that is, a clock that performs exactly the same movements as the Solar system but is outside the system's gravity well. It is therefore not influenced by the gravitational time dilation caused by the Sun and the rest of the system.

TCB was defined in 1991 by the International Astronomical Union, in Recommendation III of the XXIst General Assembly. It was intended as one of the replacements for the problematic 1976 definition of Barycentric Dynamical Time (TDB). Unlike former astronomical time scales, TCB is defined in the context of the general theory of relativity. The relationships between TCB and other relativistic time scales are defined with fully general relativistic metrics.

Because the reference frame for TCB is not influenced by the gravitational potential caused by the Solar system, TCB ticks faster than clocks on the surface of the Earth by 1.550505 × 10−8 (about 490 milliseconds per year). Consequently, the values of physical constants to be used with calculations using TCB differ from the traditional values of physical constants (The traditional values were in a sense wrong, incorporating corrections for the difference in time scales). Adapting the large body of existing software to change from TDB to TCB is an ongoing task, and as of 2002 many calculations continue to use TDB in some form.

Time coordinates on the TCB scale are conventionally specified using traditional means of specifying days, carried over from non-uniform time standards based on the rotation of the Earth. Specifically, both Julian Dates and the Gregorian calendar are used. For continuity with its predecessor Ephemeris Time, TCB was set to match ET at around Julian Date 2443144.5 (1977-01-01T00Z). More precisely, it was defined that TCB instant 1977-01-01T00:00:32.184 exactly corresponds to the International Atomic Time (TAI) instant 1977-01-01T00:00:00.000 exactly, at the geocenter. This is also the instant at which TAI introduced corrections for gravitational time dilation.

Coordinate time

In the theory of relativity, it is convenient to express results in terms of a spacetime coordinate system relative to an implied observer. In many (but not all) coordinate systems, an event is specified by one time coordinate and three spatial coordinates. The time specified by the time coordinate is referred to as coordinate time to distinguish it from proper time.

In the special case of an inertial observer in special relativity, by convention the coordinate time at an event is the same as the proper time measured by a clock that is at the same location as the event, that is stationary relative to the observer and that has been synchronised to the observer's clock using the Einstein synchronisation convention.

Experimental testing of time dilation

Time dilation as predicted by special relativity is often verified by means of particle lifetime experiments. According to special relativity, the rate of a clock C traveling between two synchronized laboratory clocks A and B, as seen by a laboratory observer, is slowed down relative to the laboratory clock rates. Since any periodic process can be considered a clock, the lifetimes of unstable particles such as muons must also be affected, so that moving muons should have a longer lifetime than resting ones. A variety of experiments confirming this effect have been performed both in the atmosphere and in particle accelerators. Another type of time dilation experiments is the group of Ives–Stilwell experiments measuring the relativistic Doppler effect.

Geocentric Coordinate Time

Geocentric Coordinate Time (TCG - Temps-coordonnée géocentrique) is a coordinate time standard intended to be used as the independent variable of time for all calculations pertaining to precession, nutation, the Moon, and artificial satellites of the Earth. It is equivalent to the proper time experienced by a clock at rest in a coordinate frame co-moving with the center of the Earth: that is, a clock that performs exactly the same movements as the Earth but is outside the Earth's gravity well. It is therefore not influenced by the gravitational time dilation caused by the Earth.

TCG was defined in 1991 by the International Astronomical Union, in Recommendation III of the XXIst General Assembly. It was intended as one of the replacements for the ill-defined Barycentric Dynamical Time (TDB). Unlike former astronomical time scales, TCG is defined in the context of the general theory of relativity. The relationships between TCG and other relativistic time scales are defined with fully general relativistic metrics.

Because the reference frame for TCG is not rotating with the surface of the Earth and not in the gravitational potential of the Earth, TCG ticks faster than clocks on the surface of the Earth by a factor of about 7.0 × 10−10 (about 22 milliseconds per year). Consequently, the values of physical constants to be used with calculations using TCG differ from the traditional values of physical constants. (The traditional values were in a sense wrong, incorporating corrections for the difference in time scales.) Adapting the large body of existing software to change from TDB to TCG is a formidable task, and as of 2002 many calculations continue to use TDB in some form.

Time coordinates on the TCG scale are conventionally specified using traditional means of specifying days, carried over from non-uniform time standards based on the rotation of the Earth. Specifically, both Julian Dates and the Gregorian calendar are used. For continuity with its predecessor Ephemeris Time, TCG was set to match ET at around Julian Date 2443144.5 (1977-01-01T00Z). More precisely, it was defined that TCG instant 1977-01-01T00:00:32.184 exactly corresponds to TAI instant 1977-01-01T00:00:00.000 exactly. This is also the instant at which TAI introduced corrections for gravitational time dilation.

TCG is a Platonic time scale: a theoretical ideal, not dependent on a particular realisation. For practical purposes, TCG must be realised by actual clocks in the Earth system. Because of the linear relationship between Terrestrial Time (TT) and TCG, the same clocks that realise TT also serve for TCG. See the article on TT for details of the relationship and how TT is realised.

Barycentric Coordinate Time (TCB) is the analog of TCG, used for calculations relating to the solar system beyond Earth orbit. TCG is defined by a different reference frame from TCB, such that they are not linearly related. Over the long term, TCG ticks more slowly than TCB by about 1.6 × 10−8 (about 0.5 seconds per year). In addition there are periodic variations, as Earth moves within the Solar system. When the Earth is at perihelion in January, TCG ticks even more slowly than it does on average, due to gravitational time dilation from being deeper in the Sun's gravity well and also velocity time dilation from moving faster relative to the Sun. At aphelion in July the opposite holds, with TCG ticking faster than it does on average.

Gravitational time dilation

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 higher the gravitational potential (the farther the clock is from the source of gravitation), the faster time passes. Albert Einstein originally predicted this effect in his theory of relativity and it has since been confirmed by tests of general relativity.This has been demonstrated by noting that atomic clocks at differing altitudes (and thus different gravitational potential) will eventually show different times. The effects detected in such Earth-bound experiments are extremely small, with differences being measured in nanoseconds. Relative to Earth's age in billions of years, Earth's core is effectively 2.5 years younger than its surface. Demonstrating larger effects would require greater distances from the Earth or a larger gravitational source.

Gravitational time dilation was first described by Albert Einstein in 1907 as a consequence of special relativity in accelerated frames of reference. In general relativity, it is considered to be a difference in the passage of proper time at different positions as described by a metric tensor of space-time. The existence of gravitational time dilation was first confirmed directly by the Pound–Rebka experiment in 1959.

Hafele–Keating experiment

The Hafele–Keating experiment was a test of the theory of relativity. In October 1971, Joseph C. Hafele, a physicist, and Richard E. Keating, an astronomer, took four cesium-beam atomic clocks aboard commercial airliners. They flew twice around the world, first eastward, then westward, and compared the clocks against others that remained at the United States Naval Observatory. When reunited, the three sets of clocks were found to disagree with one another, and their differences were consistent with the predictions of special and general relativity.

Impulse drive

In the fictional Star Trek universe, the impulse drive is the method of propulsion that starships and other spacecraft use when they are travelling below the speed of light. Typically powered by deuterium fusion reactors, impulse engines let ships travel interplanetary distances readily. For example, Starfleet Academy cadets use impulse engines when flying from Earth to Saturn and back. Unlike the warp engines, impulse engines work on principles used in today's rocketry, throwing mass out the back as fast as possible to drive you forward.

Intergalactic travel

Intergalactic travel is the term used for hypothetical manned or unmanned travel between galaxies. Due to the enormous distances between our own galaxy the Milky Way and even its closest neighbors—hundreds of thousands to millions of light-years—any such venture would be far more technologically demanding than even interstellar travel. Intergalactic distances are roughly a hundred-thousandfold (five orders of magnitude) greater than their interstellar counterparts.The technology required to travel between galaxies is far beyond humanity's present capabilities, and currently only the subject of speculation, hypothesis, and science fiction.

However, theoretically speaking, there is nothing to conclusively indicate that intergalactic travel is impossible. There are several hypothesized methods of carrying out such a journey, and to date several academics have studied intergalactic travel in a serious manner.

Ives–Stilwell experiment

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.

Both time dilation and the relativistic Doppler effect were predicted by Albert Einstein in his seminal 1905 paper.

Einstein subsequently (1907) suggested an experiment based on the measurement of the relative frequencies of light perceived as arriving from a light source in motion with respect to the observer, and he calculated the additional Doppler shift due to time dilation. This effect was later called "transverse Doppler effect" (TDE), since such experiments were initially imagined to be conducted at right angles with respect to the moving source, in order to avoid the influence of the longitudinal Doppler shift. Eventually, Herbert E. Ives and G. R. Stilwell (referring to time dilation as following from the theory of Lorentz and Larmor) gave up the idea of measuring this effect at right angles. They used rays in longitudinal direction and found a way to separate the much smaller TDE from the much bigger longitudinal Doppler effect. The experiment was performed in 1938 and it was reprised several times (see, e.g.). Similar experiments were conducted several times with increased precision, for example by Otting (1939), Mandelberg et al. (1962),

Hasselkamp et al. (1979), and Botermann et al.

Kennedy–Thorndike experiment

The Kennedy–Thorndike experiment, first conducted in 1932 by Roy J. Kennedy and Edward M. Thorndike, is a modified form of the Michelson–Morley experimental procedure, testing special relativity.

The modification is to make one arm of the classical Michelson–Morley (MM) apparatus shorter than the other one. While the Michelson–Morley experiment showed that the speed of light is independent of the orientation of the apparatus, the Kennedy–Thorndike experiment showed that it is also independent of the velocity of the apparatus in different inertial frames. It also served as a test to indirectly verify time dilation – while the negative result of the Michelson–Morley experiment can be explained by length contraction alone, the negative result of the Kennedy–Thorndike experiment requires time dilation in addition to length contraction to explain why no phase shifts will be detected while the Earth moves around the Sun. The first direct confirmation of time dilation was achieved by the Ives–Stilwell experiment. Combining the results of those three experiments, the complete Lorentz transformation can be derived.Improved variants of the Kennedy–Thorndike experiment have been conducted using optical cavities or Lunar Laser Ranging. For a general overview of tests of Lorentz invariance, see Tests of special relativity.

Length contraction

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. This contraction (also known as Lorentz contraction or Lorentz–FitzGerald contraction after Hendrik Lorentz and George Francis FitzGerald) 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.

Quantum clock

A quantum clock is a type of atomic clock with laser cooled single ions confined together in an electromagnetic ion trap. Developed in 2010 by National Institute of Standards and Technology physicists, the clock was 37 times more precise than the then-existing international standard. The quantum logic clock is based on an aluminium spectroscopy ion with a logic atom.

Both the aluminium-based quantum clock and the mercury-based optical atomic clock track time by the ion vibration at an optical frequency using a UV laser, that is 100,000 times higher than the microwave frequencies used in NIST-F1 and other similar time standards around the world. Quantum clocks like this are able to be far more precise than microwave standards.

Relativistic Doppler effect

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

The relativistic Doppler effect is different from the non-relativistic Doppler effect as the equations include the time dilation effect of special relativity and do not involve the medium of propagation as a reference point. They describe the total difference in observed frequencies and possess the required Lorentz symmetry.

Astronomers know of three sources of redshift/blueshift: Doppler shifts; gravitational redshifts (due to light exiting a gravitational field); and cosmological expansion (where space itself stretches). This article concerns itself only with Doppler shifts.

Schwarzschild radius

The Schwarzschild radius (sometimes historically referred to as the gravitational radius) is a physical parameter that shows up in the Schwarzschild solution to Einstein's field equations, corresponding to the radius defining the event horizon of a Schwarzschild black hole. It is a characteristic radius associated with every quantity of mass. The Schwarzschild radius was named after the German astronomer Karl Schwarzschild, who calculated this exact solution for the theory of general relativity in 1916.

The Schwarzschild radius is given as

where G is the gravitational constant, M is the object mass, and c is the speed of light.


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 can be used to visualize relativistic effects such as why different observers perceive where and when events occur differently.

Until the turn of the 20th century, the assumption had been that the three-dimensional geometry of the universe (its spatial expression in terms of coordinates, distances, and directions) was independent of one-dimensional time. However, in 1905, Albert Einstein based his seminal work on special relativity on two postulates: (1) The laws of physics are invariant (i.e., identical) in all inertial systems (i.e., non-accelerating frames of reference); (2) The speed of light in a vacuum is the same for all observers, regardless of the motion of the light source.

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

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

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

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

Technology in Stargate

In the fictional universe of Stargate, a number of technologically advanced races and societies have produced a variety of highly advanced weapons, tools, and spacecraft. By liaising with these races and learning from them, Earth too has begun to create its own futuristic technology. All such technology is SCI-classified top secret, and is used mainly by the SGC, its SG teams, or in Atlantis.

Theory of relativity

The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity. Special relativity applies to elementary particles and their interactions, describing all their physical phenomena except gravity. General relativity explains the law of gravitation and its relation to other forces of nature. It applies to the cosmological and astrophysical realm, including astronomy.The theory transformed theoretical physics and astronomy during the 20th century, superseding a 200-year-old theory of mechanics created primarily by Isaac Newton. It introduced concepts including spacetime as a unified entity of space and time, relativity of simultaneity, kinematic and gravitational time dilation, and length contraction. In the field of physics, relativity improved the science of elementary particles and their fundamental interactions, along with ushering in the nuclear age. With relativity, cosmology and astrophysics predicted extraordinary astronomical phenomena such as neutron stars, black holes, and gravitational waves.

Time travel

Time travel is the concept of movement between certain points in time, analogous to movement between different points in space by an object or a person, typically using a hypothetical device known as a time machine. Time travel is a widely-recognized concept in philosophy and fiction. The idea of a time machine was popularized by H. G. Wells' 1895 novel The Time Machine.

It is uncertain if time travel to the past is physically possible. Forward time travel, outside the usual sense of the perception of time, is an extensively-observed phenomenon and well-understood within the framework of special relativity and general relativity. However, making one body advance or delay more than a few milliseconds compared to another body is not feasible with current technology. As for backwards time travel, it is possible to find solutions in general relativity that allow for it, but the solutions require conditions that may not be physically possible. Traveling to an arbitrary point in spacetime has a very limited support in theoretical physics, and usually only connected with quantum mechanics or wormholes, also known as Einstein-Rosen bridges.

Twin paradox

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, according to 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, and so there is no symmetry between the spacetime paths of the twins. Therefore, the twin paradox is not a paradox in the sense of a logical contradiction.

Starting with Paul Langevin in 1911, there have been various explanations of this paradox. These explanations "can be grouped into those that focus on the effect of different standards of simultaneity in different frames, and those that designate the acceleration [experienced by the travelling twin] as the main reason". Max von Laue argued in 1913 that since the traveling twin must be in two separate inertial frames, one on the way out and another on the way back, this frame switch is the reason for the aging difference, not the acceleration per se. Explanations put forth by Albert Einstein and Max Born invoked gravitational time dilation to explain the aging as a direct effect of acceleration. General relativity is not necessary to explain the twin paradox; special relativity alone can explain the phenomenon.Time dilation has been verified experimentally by precise measurements of atomic clocks flown in aircraft and satellites. For example, gravitational time dilation and special relativity together have been used to explain the Hafele–Keating experiment. It was also confirmed in particle accelerators by measuring the time dilation of circulating particle beams.

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