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 closer the clock is to the source of gravitation), the slower time passes, speeding up as the gravitational potential increases (the clock getting away from the source of gravitation). Albert Einstein originally predicted this effect in his theory of relativity and it has since been confirmed by tests of general relativity.[1]

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.[2] 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[3] 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 spacetime. The existence of gravitational time dilation was first confirmed directly by the Pound–Rebka experiment in 1959, and later refined by Gravity Probe A and other experiments.

Gravitational time dilation can also equivalently be interpreted as gravitational redshift:[4][5][6] if two oscillators (attached to transmitters producing electromagnetic radiation) are operating at different gravitational potentials, the oscillator at the higher gravitational potential (farther from the attracting body) will seem to ‘tick’ faster; that is, when observed from the same location, it will have a higher measured frequency than the oscillator at the lower gravitational potential (closer to the attracting body).

Definition

Clocks that 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 of Mount Everest (prominence 8,848 m), would be about 39 hours ahead of a clock set at sea level.[7][8] This is because gravitational time dilation is manifested in accelerated frames of reference or, by virtue of the equivalence principle, in the gravitational field of massive objects.[9]

According to general relativity, inertial mass and gravitational mass are the same, and all accelerated reference frames (such as a uniformly rotating reference frame with its proper time dilation) are physically equivalent to a gravitational field of the same strength.[10]

Consider a family of observers along a straight "vertical" line, each of whom experiences a distinct constant g-force directed along this line (e.g., a long accelerating spacecraft,[11][12] a skyscraper, a shaft on a planet). Let be the dependence of g-force on "height", a coordinate along the aforementioned line. The equation with respect to a base observer at is

where is the total time dilation at a distant position , is the dependence of g-force on "height" , is the speed of light, and denotes exponentiation by e.

For simplicity, in a Rindler's family of observers in a flat spacetime, the dependence would be

with constant , which yields

.

On the other hand, when is nearly constant and is much smaller than , the linear "weak field" approximation can also be used.

See Ehrenfest paradox for application of the same formula to a rotating reference frame in flat spacetime.

Outside a non-rotating sphere

A common equation used to determine gravitational time dilation is derived from the Schwarzschild metric, which describes spacetime in the vicinity of a non-rotating massive spherically symmetric object. The equation is

where

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

In the Schwarzschild metric, free-falling objects can be in circular orbits if the orbital radius is larger than (the radius of the photon 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:[13][14]

Both dilations are shown in the figure below.

Important features of gravitational time dilation

Experimental confirmation

See also: Gravitational redshift § Experimental confirmation, and Tests of general relativity

Satellite clocks are slowed by their orbital speed, but accelerated by their distance out of Earth's gravitational well.
Satellite clocks are slowed by their orbital speed, but accelerated by their distance out of Earth's gravitational well.

Gravitational time dilation has been experimentally measured using atomic clocks on airplanes, such as the Hafele–Keating experiment. The clocks aboard the airplanes were slightly faster than clocks on the ground. The effect is significant enough that the Global Positioning System's artificial satellites need to have their clocks corrected.[15]

Additionally, time dilations due to height differences of less than one metre have been experimentally verified in the laboratory.[16]

Gravitational time dilation in the form of gravitational redshift has also been confirmed by the Pound–Rebka experiment and observations of the spectra of the white dwarf Sirius B.

Gravitational time dilation has been measured in experiments with time signals sent to and from the Viking 1 Mars lander.[17][18]

See also

References

  1. ^ Einstein, A. (February 2004). Relativity : the Special and General Theory by Albert Einstein. Project Gutenberg.
  2. ^ 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.
  3. ^ 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, see parts I, II and III.
  4. ^ Eddington, A. S. (1926). "Einstein Shift and Doppler Shift". Nature. 117 (2933): 86. Bibcode:1926Natur.117...86E. doi:10.1038/117086a0. ISSN 1476-4687. S2CID 4092843.
  5. ^ Florides, Petros S. "Einstein's Equivalence Principle and the Gravitational Red Shift" (PDF). School of Mathematics, Trinity College, Ireland.
  6. ^ Cheng, T.P. (2010). Relativity, Gravitation and Cosmology: A Basic Introduction. Oxford Master Series in Physics. OUP Oxford. p. 73. ISBN 978-0-19-957363-9. Retrieved 2022-11-07.
  7. ^ 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
  8. ^ 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
  9. ^ John A. Auping, Proceedings of the International Conference on Two Cosmological Models, Plaza y Valdes, ISBN 9786074025309
  10. ^ Johan F Prins, On Einstein's Non-Simultaneity, Length-Contraction and Time-Dilation
  11. ^ Kogut, John B. (2012). Introduction to Relativity: For Physicists and Astronomers (illustrated ed.). Academic Press. p. 112. ISBN 978-0-08-092408-3.
  12. ^ 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
  13. ^ 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
  14. ^ Taylor, Edwin F.; Wheeler, John Archibald (2000). Exploring Black Holes. Addison Wesley Longman. p. 8-22. ISBN 978-0-201-38423-9.
  15. ^ Richard Wolfson (2003). Simply Einstein. W W Norton & Co. p. 216. ISBN 978-0-393-05154-4.
  16. ^ C. W. Chou, D. B. Hume, T. Rosenband, D. J. Wineland (24 September 2010), "Optical clocks and relativity", Science, 329(5999): 1630–1633; [1]
  17. ^ Shapiro, I. I.; Reasenberg, R. D. (30 September 1977). "The Viking Relativity Experiment". Journal of Geophysical Research. AGU. 82 (28): 4329-4334. Bibcode:1977JGR....82.4329S. doi:10.1029/JS082i028p04329. Retrieved 6 February 2021.
  18. ^ 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.

Further reading