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 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.
This has been demonstrated by noting that atomic clocks at differing altitudes 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, and later refined by Gravity Probe A and other experiments.

Definition

s that are far from massive bodies run more quickly, and clocks close to massive bodies run more slowly. For example, considered over the total time-span of Earth, 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, 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 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. 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 space-time, 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 space-time.

Outside a non-rotating sphere

A common equation used to determine gravitational time dilation is derived from the Schwarzschild metric, which describes space-time 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 formula for a clock at rest is given above; the formula below gives the gravitational time dilation over one orbit for a clock in a circular orbit:
Both dilations are shown in the figure below.

Important features of gravitational time dilation

Gravitational time dilation has been experimentally measured using atomic clocks on airplanes. 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 has also been confirmed by the Pound–Rebka experiment, observations of the spectra of the white dwarf Sirius B, and experiments with time signals sent to and from Viking 1 Mars lander.