How do you calculate time dilation in a gravitational field

1. Jan 1, 2005

Gamish

How do you calculate time dilation in a gravitational field. I already know SR time dilation equations, t=t*sqr(1-v^2/c^2), I have even designed a software in VB to calculate it. But I do not know how to calculate time dilation in General Relativity. Can someone please tell me? Im trying to work GR into one of my theories.

According to my software, if you travel at a speed of 55 MPH for 1 hour, you will have traveled 0.0119904086659517 nanoseconds into the future , hehe. And if it is at all off, blame google.

2. Jan 1, 2005

Janitor

A great question for the experts here. My own off-the-top-of-my-head answer is that the t-t component of the metric tensor plays a key role--but don't take that too seriously, coming from a non-expert.

The metric tensor as a whole is determined by the specifics of the gravitating bodies (sizes, masses, shapes, rotation), along with boundary conditions.

3. Jan 2, 2005

mijoon

Multiply the time as measured by a distant observer by sqrt(1-(2GM/Rc^2)).
This is independent of any velocity induced dilation.

4. Jan 2, 2005

yogi

Mijoon - correct formula - but the answer precisely corresponds to the escape velocity at the distance R - or what is the same, the velocity that is acquired at a distance R from the center of a spherically symmetrical mass M if the clock fell from infinity in the gravitational field of the M. Something more than a coincidence, maybe!

5. Jan 2, 2005

pervect

Staff Emeritus
The formula for gravitational time dilation is given at

This is close to the same as mijmoon's formula, except that there may be some possibility of a confusion of multiplication / division in the previous answer.

6. Jan 2, 2005

Gamish

OK, can someone please give me an example using this equasion sqrt(1-(2GM/Rc^2)), and tell me how to fill in all the variables corectly, and what metric units to use. Im assuming the M variable is mass in KG, the R variable is the radius or something, probally in meters, and c is the speed of light in a vaccume. G, im assuming has something to do with the gravitational force, but I am not sure how to use it. Can someone give me an example of the dilation of time if I was 1 KM above the survace of the earth or something simple like that?

7. Jan 2, 2005

mijoon

Very excellent point, Yogi.
The formula I gave is derived from the Schwartzchild solution, but I suspect from your post that you are familiar with the "alternate" derivation of the Schwartzchild metric.
Indeed, something more than a coincidence.

8. Jan 2, 2005

Chronos

The expression used to calculate redshift due surface gravity of stars is z = GM/Rc^2. In binary star systems, masses can be derived from their orbits. The radius of the star can then be calculated by plugging the observed redshift and mass into this formula.

9. Jan 2, 2005

Creator

That's an interesting use of the grav. time dilation formula, Chronos. It's a clever use of the fact that the redshifted radiation originates from a certain gravitational potential at the surface of the star.

Creator

10. Jan 2, 2005

Creator

True, gravitational time dilation is equivalent to velocity time dilation at the escape velocity. To find out the place where they are equivalent simply set

sqrt(1-(2GM/Rc^2)) = sqrt(1-v^2/c^2), and solve for v,
which results in the escape vel.

This appears on the surface to have some underlying physical significance. However, it may be otherwise when we realize that it may simply be a mathematical artifact of the fact that both the escape velocity and gravitational time dilation have the gravitational potential as the basis of their derivation.
Remember we originally derive escape velocity by equating kinetic energy to gravitational potential energy. In other words, it is in itself an expression of the work done against the gravitational potential of the planet.

Creator

11. Jan 2, 2005

Gamish

GR time dilation

12. Jan 2, 2005

Creator

Last edited: Jan 3, 2005
13. Jan 3, 2005

Gamish

OK, this is nice to know. When I made my time dilation software (email me if you want), I only designed it to calculate Special Relativity, but now, I can also calculate General Relativity, fairly easy :rolleseyes:

This also gives me another question, if you were traveling at .9c in a gravitational field that has an escape velocity of .2c, then thats a negative number for the time variable. Does this allow time travel to the past. Im pretty sure this is not posible, for some reason, I think that the speed of light slows down in a gravitational field, preventing FTL travel. Maybe, the stronger the gravitational field, the slower the speed of light is. Maybe I will post another thread for this question sometime, I just threw it out there :)

ZZZzzzzzzzzzz. time to sleep, im looking forward to my mornaing coffe and I nice day learing ALG.

14. Jan 3, 2005

yogi

Good point in Post 10 Creator - the relationship is definitely an energy thing - this may suggests that time dilation in both SR and GR is geared to some hard to describe physical dynamic - it is easier to comeup with a causal bases for time dilation in GR that is potential dependent - but not easy to see how energy relationships can be adapted to explain time dilation in SR.

But upon further consideration of your comment in 10 - how does it fit with experiments that measure time dilation in a powerful centrifuge? - here the clock rate corresponds to the tangential velocity "a la" SR - which is the same as the acceleration field that corresponds (v^2)/r

15. Jan 5, 2005

Creator

Probably true.

What experiment are you referring to?

Creator

16. Jan 6, 2005

makc

didn't anybody noted radius in the formula? as long it changes, T and T0 becomes dT and dT0, so you have to integrate it along the path where r changes.

17. Jan 6, 2005

Gamish

I have a question, if this is somewhat irrelavent here, I will open up a new thread.

I am in a gravitational field with an escape velocity of .3c

I am traveling .9c in the gravitational field

My speed is 1.2c

Is this correct?

18. Jan 6, 2005

ohwilleke

No. This is not correct.

19. Jan 6, 2005

Gamish

Ok, then what is the whole thing of traveling faster than light in a light cone or something? Explaine the logic behind that......

20. Jan 9, 2005

hellfire

This is right for the Schwarzschild spacetime, which was discussed in this thread. The phenomenon is usually called "gravitational time dilation". But there are other scenarios within general relativity (other spacetimes), in which a time dilation does also arise. For example the Robertson-Walker spacetime, in which the time dilation is not determined by the t-t component of the metric but by the spatial components (the scale factor). This is usually called "cosmological time dilation".