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 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.
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 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 is closely related to gravitational redshift: 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 the frequency of the light it emits, as measured by a fixed observer.
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 8,848m), would be about 39 hours ahead of a clock set at sea level. 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.
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.
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, a skyscraper, a shaft on a planet). Let formula_1 be the dependence of g-force on "height", a coordinate along the aforementioned line. The equation with respect to a base observer at formula_2 is
where formula_4 is the "total" time dilation at a distant position formula_5, formula_1 is the dependence of g-force on "height" formula_5, formula_8 is the speed of light, and formula_9 denotes exponentiation by e.
For simplicity, in a Rindler's family of observers in a flat spacetime, the dependence would be
with constant formula_11, which yields
On the other hand, when formula_13 is nearly constant and formula_14 is much smaller than formula_15, the linear "weak field" approximation formula_16 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 formula_29 (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:
Both dilations are shown in the figure below.
Experimental confirmation.
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.
Additionally, time dilations due to height differences of less than one metre have been experimentally verified in the laboratory.
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.