Bond convexity

Gravitational time dilation is an actual difference of elapsed time between two events as measured by observers differently situated from gravitational masses, in regions of different gravitational potential. The lower the gravitational potential (the closer the clock is to the source of gravitation), the more slowly 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 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. 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.

It can also be manifested by any other kind of accelerating 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 a consequence of centripetal acceleration.

This is supported by the general theory of relativity due to the equivalence principle that states that all accelerated reference frames are physically equivalent to a gravitational field of the same strength. For example, a person standing on the surface of the Earth experiences exactly the same effect as a person standing in a space ship accelerating at 9.8 m/s² (that is, generating a force of 9.8 N/kg, equal to the gravitational field strength of Earth at its surface). 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 spacecraft. Any kind of g-load contributes to gravitational time dilation.

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

$T_{d}(h)=e^{\left(\int _{0}^{h}g(\iota )d\iota \right)/c^{2}}$

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
• $e^{\cdots }$ is the natural exponentiation; see e

For example, for a Rindler'r 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.

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

$T_{d}={\sqrt {1-r^{2}\omega ^{2}/c^{2}}}$

where

• $r$ is the distance from the center of the disk (which is the location of the base observer), and
• $\omega$ 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 ${\sqrt {1-v^{2}/c^{2}}}$ ).

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 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 $E=mc^{2}$). 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

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

References

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