# Gravitational time dilation

jeremyfiennes
What is the mathematical formula for the time dilation (clock-slowing factor) for a clock in a gravitational field g, equivalent to the Lorentz factor γ for a clock traveling at a relative speed v?

Staff Emeritus
No correspondence, because gravitational time dilation is a function of potential difference not g. However if you assume Rindler observers, and your reference is a clock with acceleration of g, then the time rate for one ‘higher’ by h is faster by a factor of 1 + gh. This is also true to first order for the surface of planet.

[edit: in units with c=1. In common units, 1 + gh/c2 ]

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jeremyfiennes
Not quite with you. I have a clock A in outer space where there is no gravity. And one, B, stationary in a gravitational field g. By what factor does B run slower than A?

Staff Emeritus
Gold Member
Not quite with you. I have a clock A in outer space where there is no gravity. And one, B, stationary in a gravitational field g. By what factor does B run slower than A?

For example, if you compare a clock sitting on the surface of the Earth to a clock sitting on the surface of a world with twice the radius and 4 times the mass, they will run at different rates (with the on on the larger world running slower) even though both clocks are at 1g.

jeremyfiennes
2022 Award
Not quite with you. I have a clock A in outer space where there is no gravity. And one, B, stationary in a gravitational field g. By what factor does B run slower than A?
It depends on the gravitational potential (usually denoted ##\phi##), not the gravitational acceleration (usually denoted ##g##). So your question has no answer as asked.

The rate at which a clock at Schwarzschild coordinate ##r## (assuming that it's outside the mass, therefore) ticks compared to a clock at infinity is ##\sqrt{1-2GM/c^2r}=\sqrt{1-2\phi/c^2}##. The approximations @PAllen gave derive from this under various circumstances.

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Staff Emeritus
Homework Helper
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The approximations @PAllen gave derive from this under various circumstances.
You actually do not need the general expression to derive the approximations. Just using the equivalence principle will work perfectly fine.

Ibix
2022 Award
You actually do not need the general expression to derive the approximations. Just using the equivalence principle will work perfectly fine.
Indeed. Start with a light pulse of frequency f at one height and send it upwards, convert it to a mass, drop the mass, and convert it back into energy. The light needs to have lost the same amount of energy on the upwards leg as the mass gained on the downwards leg, or else we have an energy-creating device here. Thus gravitational redshift, which is the same as gravitational time dilation.