Gravitational time dilation

Gravitational time dilationis a form oftime dilation,an actual difference of elapsed time between twoevents,as measured byobserverssituated at varying distances from a gravitatingmass.The lower thegravitational potential(the closer the clock is to the source of gravitation), the slower time passes, speeding up as the gravitational potential increases (the clock moving away from the source of gravitation).Albert Einsteinoriginally predicted this in histheory of relativity,and it has since been confirmed bytests of general relativity.^[1]

This effect has been demonstrated by noting thatatomic clocksat differingaltitudes(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 innanoseconds.Relative to Earth's age in billions of years, Earth's core is in effect 2.5 years younger than its surface.^[2]Demonstrating larger effects would require measurements at greater distances from the Earth, or a larger gravitational source.

Gravitational time dilation was first described by Albert Einstein in 1907^[3]as a consequence ofspecial relativityin accelerated frames of reference. Ingeneral relativity,it is considered to be a difference in the passage ofproper timeat different positions as described by ametric tensorof spacetime. The existence of gravitational time dilation was first confirmed directly by thePound–Rebka experimentin 1959, and later refined byGravity Probe Aand other experiments.

Gravitational time dilation is closely related togravitational redshift,^[4]in which the closer a body emitting light of constant frequency is to a gravitating body, the more its time is slowed by gravitational time dilation, and the lower (more "redshifted" ) would seem to be the frequency of the emitted light, as measured by a fixed observer.

Definition[edit]

Clocksthat are far from massive bodies (or at higher gravitational potentials) run more quickly, and clocks close to massive bodies (or at lower gravitational potentials) run more slowly. For example, considered over the total time-span of Earth (4.6 billion years), a clock set in a geostationary position at an altitude of 9,000 meters above sea level, such as perhaps at the top ofMount Everest(prominence8,848m), would be about 39 hours ahead of a clock set at sea level.^[5]^[6]This is because gravitational time dilation is manifested in acceleratedframes of referenceor, by virtue of theequivalence principle,in the gravitational field of massive objects.^[7]

According to general relativity,inertial massandgravitational massare the same, and all accelerated reference frames (such as auniformly rotating reference framewith its proper time dilation) are physically equivalent to a gravitational field of the same strength.^[8]

Consider a family of observers along a straight "vertical" line, each of whom experiences a distinct constantg-forcedirected along this line (e.g., a long accelerating spacecraft,^[9]^[10]a skyscraper, a shaft on a planet). Let $g(h)$ be the dependence of g-force on "height", a coordinate along the aforementioned line. The equation with respect to a base observer at $h=0$ is

T_{d}(h)=\exp \left[{\frac {1}{c^{2}}}\int _{0}^{h}g(h')dh'\right]

where $T_{d}(h)$ is thetotaltime dilation at a distant position $h$ , $g(h)$ is the dependence of g-force on "height" $h$ , $c$ is thespeed of light,and $\exp$ denotesexponentiationbye.

For simplicity, in aRindler's family of observersin aflat spacetime,the dependence would be

g(h)=c^{2}/(H+h)

with constant $H$ ,which yields

T_{d}(h)=e^{\ln(H+h)-\ln H}={\tfrac {H+h}{H}}

.

On the other hand, when $g$ is nearly constant and $gh$ is much smaller than $c^{2}$ ,the linear "weak field" approximation $T_{d}=1+gh/c^{2}$ can also be used.

SeeEhrenfest paradoxfor application of the same formula to a rotating reference frame in flat spacetime.

Outside a non-rotating sphere[edit]

A common equation used to determine gravitational time dilation is derived from theSchwarzschild metric,which describes spacetime in the vicinity of a non-rotating massivespherically symmetricobject. The equation is

t_{0}=t_{f}{\sqrt {1-{\frac {2GM}{rc^{2}}}}}=t_{f}{\sqrt {1-{\frac {r_{\rm {s}}}{r}}}}=t_{f}{\sqrt {1-{\frac {v_{e}^{2}}{c^{2}}}}}=t_{f}{\sqrt {1-\beta _{e}^{2}}}<t_{f}

where

$t_{0}$ is the proper time between two events for an observer close to the massive sphere, i.e. deep within the gravitational field
$t_{f}$ is the coordinate time between the events for an observer at an arbitrarily large distance from the massive object (this assumes the far-away observer is usingSchwarzschild coordinates,a coordinate system where a clock at infinite distance from the massive sphere would tick at one second per second of coordinate time, while closer clocks would tick at less than that rate),
$G$ is thegravitational constant,
$M$ is themassof the object creating the gravitational field,
$r$ is the radial coordinate of the observer within the gravitational field (this coordinate is analogous to the classical distance from the center of the object, but is actually a Schwarzschild coordinate; the equation in this form has real solutions for $r>r_{\rm {s}}$ ),
$c$ is thespeed of light,
$r_{\rm {s}}=2GM/c^{2}$ is theSchwarzschild radiusof $M$ ,
$v_{e}={\sqrt {\frac {2GM}{r}}}$ is the escape velocity, and
$\beta _{e}=v_{e}/c$ is the escape velocity, expressed as a fraction of the speed of light c.

To illustrate then, without accounting for the effects of rotation, proximity to Earth's gravitational well will cause a clock on the planet's surface to accumulate around 0.0219 fewer seconds over a period of one year than would a distant observer's clock. In comparison, a clock on the surface of the Sun will accumulate around 66.4 fewer seconds in one year.

Circular orbits[edit]

In the Schwarzschild metric, free-falling objects can be in circular orbits if the orbital radius is larger than ${\tfrac {3}{2}}r_{s}$ (the radius of thephoton sphere). The formula for a clock at rest is given above; the formula below gives the general relativistic time dilation for a clock in a circular orbit:^[11]^[12]

t_{0}=t_{f}{\sqrt {1-{\frac {3}{2}}\!\cdot \!{\frac {r_{\rm {s}}}{r}}}}\,.

Both dilations are shown in the figure below.

Important features of gravitational time dilation[edit]

According tothe general theory of relativity,gravitational time dilation is copresent with the existence of anaccelerated reference frame.Additionally, all physical phenomena in similar circumstances undergo time dilation equally according to theequivalence principleused inthe general theory of relativity.
The speed of light in a locale is always equal tocaccording to the observer who is there. That is, every infinitesimal region of spacetime may be assigned its own proper time, and the speed of light according to the proper time at that region is alwaysc.This is the case whether or not a given region is occupied by an observer. Atime delaycan be measured for photons which are emitted from Earth, bend near the Sun, travel to Venus, and then return to Earth along a similar path. There is no violation of the constancy of the speed of light here, as any observer observing the speed of photons in their region will find the speed of those photons to bec,while the speed at which we observe light travel finite distances in the vicinity of the Sun will differ fromc.
If an observer is able to track the light in a remote, distant locale which intercepts a remote, time dilated observer nearer to a more massive body, that first observer tracks that both the remote light and that remote time dilated observer have a slower time clock than other light which is coming to the first observer atc,like all other light the first observerreallycan observe (at their own location). If the other, remote light eventually intercepts the first observer, it too will be measured atcby the first observer.
Gravitational time dilation $T$ in a gravitational well is equal to thevelocity time dilationfor a speed that is needed to escape that gravitational well (given that the metric is of the form $g=(dt/T(x))^{2}-g_{space}$ ,i. e. it is time invariant and there are no "movement" terms $dxdt$ ). To show that, one can applyNoether's theoremto a body that freely falls into the well from infinity. Then the time invariance of the metric implies conservation of the quantity $g(v,dt)=v^{0}/T^{2}$ ,where $v^{0}$ is the time component of the4-velocity $v$ of the body. At the infinity $g(v,dt)=1$ ,so $v^{0}=T^{2}$ ,or, in coordinates adjusted to the local time dilation, $v_{loc}^{0}=T$ ;that is, time dilation due to acquired velocity (as measured at the falling body's position) equals to the gravitational time dilation in the well the body fell into. Applying this argument more generally one gets that (under the same assumptions on the metric) the relative gravitational time dilation between two points equals to the time dilation due to velocity needed to climb from the lower point to the higher.

Experimental confirmation[edit]

Gravitational time dilation has been experimentally measured using atomic clocks on airplanes, such as theHafele–Keating experiment.The clocks aboard the airplanes were slightly faster than clocks on the ground. The effect is significant enough that theGlobal Positioning System's artificial satellitesneed to have their clocks corrected.^[13]

Additionally, time dilations due to height differences of less than one metre have been experimentally verified in the laboratory.^[14]

Gravitational time dilation in the form ofgravitational redshifthas also been confirmed by thePound–Rebka experimentand observations of the spectra of thewhite dwarf Sirius B.

Gravitational time dilation has been measured in experiments with time signals sent to and from theViking 1Mars lander.^[15]^[16]

References[edit]

^Einstein, A. (February 2004).Relativity: the Special and General Theory by Albert Einstein.Project Gutenberg.
^Uggerhøj, U I; Mikkelsen, R E; Faye, J (2016). "The young centre of the Earth".European Journal of Physics.37(3): 035602.arXiv:1604.05507.Bibcode:2016EJPh...37c5602U.doi:10.1088/0143-0807/37/3/035602.S2CID 118454696.
^A. Einstein, "Über das Relativitätsprinzip und die aus demselben gezogenen Folgerungen", Jahrbuch der Radioaktivität und Elektronik 4, 411–462 (1907); English translation, in "On the relativity principle and the conclusions drawn from it", in "The Collected Papers", v.2, 433–484 (1989); also in H M Schwartz, "Einstein's comprehensive 1907 essay on relativity, part I", American Journal of Physics vol.45, no.6 (1977) pp.512–517; Part II in American Journal of Physics vol.45 no.9 (1977), pp.811–817; Part III in American Journal of Physics vol.45 no.10 (1977), pp.899–902, seeparts I, II and III Archived2020-11-28 at theWayback Machine.
^Cheng, T.P. (2010).Relativity, Gravitation and Cosmology: A Basic Introduction.Oxford Master Series in Physics. OUP Oxford. p. 72.ISBN 978-0-19-957363-9.Retrieved2022-11-07.
^Hassani, Sadri (2011).From Atoms to Galaxies: A Conceptual Physics Approach to Scientific Awareness.CRC Press. p. 433.ISBN 978-1-4398-0850-4.Extract of page 433
^Topper, David (2012).How Einstein Created Relativity out of Physics and Astronomy(illustrated ed.). Springer Science & Business Media. p. 118.ISBN 978-1-4614-4781-8.Extract of page 118
^John A. Auping,Proceedings of the International Conference on Two Cosmological Models,Plaza y Valdes,ISBN 9786074025309
^Johan F Prins,On Einstein's Non-Simultaneity, Length-Contraction and Time-Dilation
^Kogut, John B. (2012).Introduction to Relativity: For Physicists and Astronomers(illustrated ed.). Academic Press. p. 112.ISBN 978-0-08-092408-3.
^Bennett, Jeffrey (2014).What Is Relativity?: An Intuitive Introduction to Einstein's Ideas, and Why They Matter(illustrated ed.). Columbia University Press. p. 120.ISBN 978-0-231-53703-2.Extract of page 120
^Keeton, Keeton (2014).Principles of Astrophysics: Using Gravity and Stellar Physics to Explore the Cosmos(illustrated ed.). Springer. p. 208.ISBN 978-1-4614-9236-8.Extract of page 208
^Taylor, Edwin F.;Wheeler, John Archibald(2000).Exploring Black Holes.Addison Wesley Longman. p.8-22.ISBN 978-0-201-38423-9.
^Richard Wolfson (2003).Simply Einstein.W W Norton & Co. p. 216.ISBN 978-0-393-05154-4.
^C. W. Chou, D. B. Hume, T. Rosenband, D. J. Wineland (24 September 2010), "Optical clocks and relativity",Science,329(5999): 1630–1633;[1]
^Shapiro, I. I.; Reasenberg, R. D. (30 September 1977)."The Viking Relativity Experiment".Journal of Geophysical Research.82(28). AGU: 4329–4334.Bibcode:1977JGR....82.4329S.doi:10.1029/JS082i028p04329.Retrieved6 February2021.
^Thornton, Stephen T.; Rex, Andrew (2006).Modern Physics for Scientists and Engineers(3rd, illustrated ed.). Thomson, Brooks/Cole. p. 552.ISBN 978-0-534-41781-9.

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