Gravitational time dilation

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Gravitational time dilation is the effect of time passing at different rates in regions of different gravitational potential; the higher the local distortion of spacetime due to gravity, 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 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.

1 Definition
2 Outside a non-rotating sphere
3 Inside a non-rotating sphere
4 Circular orbits
5 Important things to stress
6 Experimental confirmation
7 See also:
8 References

[edit] Definition

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Background knowledge the reader may need to learn: What is a gravitational field? What is time dilation? What is spacetime? The reader may also research gravitational redshift or ordinary redshift. In addition, the much more fundamental, but crucial question: What really is time itself?

Gravitational time dilation is manifested in accelerated frames of reference or, by virtue of the equivalence principle, in the gravitational field of massive objects. In more simple terms, clocks which 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.

It can also be manifested by any other kind of accelerated reference frame such as an accelerating dragster or space shuttle. Spinning objects such as merry-go-rounds and ferris wheels are subjected to gravitational time dilation as an effect of their angular spin.

This is supported by the general theory of relativity due to the equivalence principle that states that all accelerated reference frames possess a gravitational field. According to general relativity, inertial mass and gravitational mass are the same. Not all gravitational fields are "curved" or "spherical"; some are flat as in the case of an accelerating dragster or space shuttle. Any kind of g-load contributes to gravitational time dilation.

In an accelerated box, the equation with respect to an arbitrary base observer is T_d = 1 + gh / c², where
- $T d$ is the total time dilation at a distant position,
- $g$ is the acceleration of the box as measured by the base observer, and
- $h$ is the "vertical" distance between the observers.
On a rotating disk when the base observer is located at the center of the disk and co-rotating with it (which makes their view of spacetime non-inertial), the equation is , where
- $r$ is the distance from the center of the disk (which is the location of the base observer), and
- $ω$ is the angular velocity of the disk.

(It is no accident that in an inertial frame of reference this becomes the familiar velocity time dilation $\sqrt{1 - v^2/c^2}$ ).

[edit] 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 proper time between events A and B for a fast-ticking observer distant from the massive object (and therefore outside of the gravitational field),
$G$ 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
$r 0 = 2 G M / c 2$ is the called the Schwarzschild Radius of M. If a mass collapses so that its surface lies at less than this radial coordinate (or in other words covers an area of less than $16π G 2 M 2 / c 4$ ), then the object exists within a black hole.

[edit] Inside a non-rotating sphere

The equation above is only valid outside the non-rotating massive spherically-symmetric object. Inside the sphere the equation is:

$t_0 = t_f \sqrt{1 - \frac{2G(\frac{r_i}{R})^3M}{r_i c^2}} = t_f \sqrt{1 - r_i^2 \frac{r_0}{R^3}}$ , where

$r i$ is the distance of a point on the inside of the original sphere to the center of that sphere,
$R$ is the radius of the original sphere, and
$M$ is the mass of the original sphere with radius $R$ .

If one is inside the sphere, the sphere can be split in two parts: a hollow sphere above and a solid sphere below. One is weightless anywhere in the interior of a uniform hollow sphere. With respect to one's gravitational potential, it is as if the hollow sphere is not there^[1]^[2]. What is left is the solid sphere below, and its mass is:

$M_i = V_i \rho = \frac{4}{3}\pi r_i^3\rho = \frac{4}{3}\pi r_i^3\frac{M}{V} = \frac{4}{3}\pi r_i^3\frac{M}{\frac{4}{3}\pi R^3} = \frac{r_i^3}{R^3}M$ , where

$r i$ , $R$ and $M$ are the same as described above,
$V$ is the volume of the original sphere with radius $R$ ,
$M i$ is the mass of a sphere with radius $r i$ ,
$V i$ is the volume of a sphere with radius $r i$ , and
$ρ$ is the (uniform) density of any part of the sphere.

The implication is that the gravitational time dilation reaches its maximum at the surface of the non-rotating massive spherically-symmetric object, and that the gravitational time dilation reaches its minimum at the center of the sphere.

[edit] Circular orbits

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

$t_0 = t_f \sqrt{1 - \frac{r_0}{r}}$

[edit] Important things to stress

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 $E = m c 2$ ). 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 more or less a 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 intercept 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.

[edit] Experimental confirmation

Gravitational time dilation has been experimentally measured using atomic clocks on airplanes. The clocks that traveled aboard the airplanes upon return were slightly fast with respect to clocks on the ground. The effect is significant enough that the Global Positioning System needs to correct for its effect on clocks aboard artificial satellites, providing a further experimental confirmation of the effect.

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.

[edit] See also:

Gravitational redshift

[edit] References

Einstein, Albert. "Relativity : the Special and General Theory by Albert Einstein." Project Gutenberg. <http://www.gutenberg.org/etext/5001.>
Einstein, Albert. "The effect of gravity on light" (1911), translated and reprinted in The Principle of Relativity <http://einstein.relativitybook.com/Einstein_gravity.html>
Nave, C.R. "Gravity and the Photon." Hyperphysics. <http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/blahol.html#c2.>

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