Gravitational time dilation

You must add a |reason= parameter to this Cleanup template – replace it with {{Cleanup|December 2005|reason=<Fill reason here>}}, or remove the Cleanup template.

Gravitational time dilation exists when the rate of passage of proper time at distant positions differs the local rate due to the presence of an accelerated reference frame.

When an observer in a geostationary orbit compares their own clock with clock lower in the gravitational field below it (so that it should not be subject to velocity time dilation), the clock lower in gravitational potential will be found to be ticking slower than the observer's clock. (Similarly, a clock at a higher potential will be found to be ticking faster.)

Time dilation in accelerated reference frames, or Gravitational time dilation, was first described by Albert Einstein in 1907 as a consequence of special relativity. It is a feature of general relativity, and of all other metric theories of gravitation.

Experimental verification

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 = time dilation of an accelerated reference frame

Contrary to the to what the name may suggest, curvature of space time is not necessary for there to be gravitational time dilation, but rather the time dilation is the result of an accelerated reference frame. An accelerated frame of reference may represent the acceleration due to a gravitational field or its equivalent, such as a rocket ship undergoing acceleration (see Equivalence principle).

Accelerated (non-inertial) reference frames underwent length contraction and they experience Gravitational time dilation (a slowing of internal processes). Accelerating reference frames undergo changes of length contraction and experience more and more Gravitational time dilation (more slowing of internal processes).

Gravitational shift and gravitational time dilation

When using special relativity's relativistic Doppler relationships to calculate the change in energy and frequency (assuming no complicating route-dependent effects), then the Gravitational redshift and blueshift frequency ratios are the inverse of each other, suggesting that the "seen" frequency-change corresponds to the actual difference in underlying clockrate. In General Relativity, where route-dependence comes into play, establishing globally-agreed differences in underlying clockrate can be more difficult.

Common mistakes

Quite too often, the following two formulas are used to (wrongly) describe gravitational shift's direct correspondence to the gravitational time dilation.

${\frac {\lambda _{observed}}{\lambda _{initial}}}=1+{\frac {aL}{c^{2}}}$

${\frac {f_{observed}}{f_{initial}}}=1-{\frac {aL}{c^{2}}}$

Where:

$\lambda$ is the wavelength.

$f$ is the frequency.

$a$ can be one of two things, either the constant acceleration of a light source which possesses an accelerated reference frame, or the acceleration due to uniform (false) gravitational field.

$L$ is the change in distance.

The error exists because the relationship between wavelength and frequency should be inverse of each other, where as in these Newtonian approximations, they are not.

The equation(s) for Gravitational Time Dilation

Time dilation caused by any mass $M$ may be described using the following (exact) equation:

$T={\frac {t}{\sqrt {1-\left({\frac {2GM}{rc^{2}}}\right)}}}$

$T$ is time dilation for an object within the vicinity of M's gravitational field.

$t$ is time dilation for an observer not influenced by this field.

${\frac {2GM}{c^{2}}}$ is the Schwarzschild Radius of M.

$r$ is the r-coordinate as used in the Schwarzschild solution to Einstein's field equation.

By substituting ${\frac {2GM}{c^{2}}}$ with $R_{s}$ , we get the following equation:

$T={\frac {t}{\sqrt {1-\left({\frac {R_{s}}{r}}\right)}}}$

$R_{s}$ is the Schwarzschild Radius of that mass.

Yet another way to write this is the following:

$T={\frac {t}{\sqrt {1-\left({\frac {v_{e}^{2}}{c^{2}}}\right)}}}$

$v_{e}^{2}$ is the square of the escape velocity as function of the radius. $v_{e}^{2}\approx {\frac {2GM}{r}}$

The equation for escape velocity in Einstein's General Relativity differs from the Newton's equation for escape velocity. Despite that, for the Gravitational Time Dilation formula derived from the Schwarzschild solution in particular, the substitution of some of the variables with a variable for escape velocity is valid.

This formula has the same form as the velocity time dilation formula, aside from replacing the variable for relative velocity with a variable for escape velocity.

Another version of this formula involves substituting part of the formula with the value for Angle of Deflection caused by null space-time geodesics:

"It may be added that, according to the theory, half of this deflection is produced by the Newtonian field of attraction of the sun, and the other half by the geometrical modification (" curvature ") of space caused by the sun." - Albert Einstein (The General Theory of Relativity: Appendix III)

$angleofdeflection={\frac {4GM}{rc^{2}}}$

$r$ is the photon's closest approach to the black hole expressed as the r-coordinate.

$T={\frac {t}{\sqrt {1-\left({\frac {a}{2}}\right)}}}$

These four equations can apply for any mass M, not just black holes. However, for the second equation, you must remember to use the Schwarzschild radius and not the radius of mass " $M$ " itself.

Note that if you let angleofdeflection = 2 radians, you have the time dilation at the schwarzchild radius where (a/2)=1.

Extreme gravitational time dilation

An extreme example of gravitational time dilation occurs near a black hole. A clock falling towards the event horizon would appear (to observers far away) to slow down to a halt as it approached the horizon. A small and sturdy enough clock could conceivably cross the horizon without suffering adverse effects at the horizon, but to far-away observers it would seem to "freeze" and flattened out as it approached the critical horizon radius.

The following chart details the effects of black hole's time dilation as imposed on the black hole explorers, relative to an outside observer.

${\frac {r}{R_{s}}}-1$	Time experienced by outside observer per explorer second
1/3rd	2 seconds
1/8th	3 seconds
1/224th	15 seconds
1/3599th	1 minute
1/12,959,999th	1 hour
1/7,464,959,999th	1 day
1/995,839,578,165,547th	1 year
1/186,909,130,425,891,581,603,655,782,399,999,999th	Age of the Universe (according to WMAP) = 13.7 billion years

If the outside observer could somehow watch the action near the black hole, she would perceive the object moving and evolving at a staggeringly slow pace (rightfully so). If the crew could watch the life of the outside observer, the outsiders would appear to be passing by at a very fast pace (rightfully so). All of them would experience time passing normally, since the speed of light, which governs the brain, is a local constant, and all of them would experience light going 299,792,458 meters for every one of their "local seconds". The proper time for the crew would be many times as brief than the proper time for people on Earth.

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
Nave, C.R. "Gravity and the Photon." Hyperphysics. <http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/blahol.html#c2.>
Weisstein, Eric. "Gravitational Redshift" Eric Weisstein's World of Physics. <http://scienceworld.wolfram.com/physics/GravitationalRedshift.html.>