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 differently situated from gravitational masses. The stronger the gravitational potential (the closer the clock is to the source of gravitation), the slower 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 experiments are extremely small, with differences being measured in nanoseconds.

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.

Definition

Clocks that are far from massive bodies (or at higher gravitational potentials) run faster, and clocks close to massive bodies (or at lower gravitational potentials) run slower. 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.

Let us consider a family of observers along a straight "vertical" line each of whom experiences a constant g-force along this line (e.g., a long accelerating spacecraft, a skyscraper, a shaft on a planet). Let $g(h)$ be the dependence of g-force on "height", a coordinate along 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 the total time dilation at a distant position $h$, $g(h)$ is the dependence of g-force on "height" $h$, $c$ is the speed of light, and $\exp$ denotes exponentiation by e.

For simplicity, in a Rindler's family of observers in a flat space-time 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 $T_d = 1 + gh/c^2$ may also be used.

See Ehrenfest paradox for application of the same formula to a rotating reference frame in the 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 spacetime in the vicinity of a non-rotating massive spherically symmetric object. The equation is:

$t_0 = t_f \sqrt{1 - \frac{2GM}{rc^2}} = t_f \sqrt{1 - \frac{r_0}{r}}$

where

• $t_0$ is the proper time between events A and B for a slow-ticking observer within the gravitational field,
• $t_f$ is the coordinate time between events A and B for a fast-ticking observer at an arbitrarily large distance from the massive object (this assumes the fast-ticking observer is using Schwarzschild 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 the gravitational constant,
• $M$ is the mass of the object creating the gravitational field,
• $r$ is the radial coordinate of the observer (which is analogous to the classical distance from the center of the object, but is actually a Schwarzschild coordinate),
• $c$ is the speed of light, and
• $r_0 = 2GM/c^2$ is the Schwarzschild radius of $M$.

To illustrate then, a clock on the surface of the Earth (assuming it does not rotate) will accumulate around 0.0219 seconds less than a distant observer over a period of one year. In comparison, a clock on the surface of the sun will accumulate around 66.4 seconds less in a year.

Circular orbits

In the Schwarzschild metric, free-falling objects can be in circular orbits if the orbital radius is larger than $\tfrac{3}{2} r_0$. The formula for a clock at rest is given above; for a clock in a circular orbit, the formula is instead.

$t_0 = t_f \sqrt{1 - \frac{3}{2} \! \cdot \! \frac{r_0}{r}}\, .$

Important features of gravitational time dilation

• According to General Relativity, gravitational time dilation is copresent with the existence of an accelerated reference frame.

• The speed of light in a locale is always equal to c according to the observer who is there. The stationary observer's perspective corresponds to the local proper time. Every infinitesimal region of space time may have its own proper time that corresponds to the gravitational time dilation there, where electromagnetic radiation and matter may be equally affected, since they are made of the same essence (as shown in many tests involving the famous equation $E=mc^2$). Such regions are significant whether or not they are occupied by an observer. A time delay is measured for signals that bend near the Sun, headed towards Venus, and bounce back to Earth along a more or less similar path. There is no violation of the speed of light in this sense, as long as an observer is forced to observe only the photons which intercept the observing faculties and not the ones that go passing by in the depths of more (or even less) gravitational time dilation.

• If a distant observer is able to track the light in a remote, distant locale which intercepts a time dilated observer nearer to a more massive body, he sees that both the distant light and that distant time dilated observer have a slower proper time clock than other light which is coming nearby him, which intercepts him, at c, like all other light he really can observe. When the other, distant light intercepts the distant observer, it will come at c from the distant observer's perspective.

Experimental confirmation

Satellite clocks are slowed by their orbital speed but sped up by their distance out of the Earth's gravitational well.

Gravitational time dilation has been experimentally measured using atomic clocks on airplanes. The clocks aboard the airplanes were slightly faster with respect to 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 1 meter 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.

See also

• Gravitational redshift
• Hafele–Keating experiment
• Twin paradox
• Clock hypothesis

External links

• Øyvind Grøn, Arne Næss, Einstein's Theory: A Rigorous Introduction for the Mathematically Untrained, 2002, ISBN 82-92261-07-9