How do you calculate time dilation in a gravitational field

[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? I am 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.

 

[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.

 

[mijoon]

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

 

[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!

 

[pervect]

The formula for gravitational time dilation is given at
this link

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.

 

[Gamish]

OK, can someone please give me an example using this equation sqrt(1-(2GM/Rc^2)), and tell me how to fill in all the variables corectly, and what metric units to use. I am 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, I am 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?

 

[mijoon]

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!

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.

 

[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.

 

[Creator]

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.

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

 

[Creator]

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!

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

 

[Gamish]

GR time dilation

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

So, does this mean that using the Special Relativity time dilation equation (t=t*sqr(1-v^2/c^2), I can calculate time dilation in a gravitational field by replacing the "v" variable with the escape velocity of the gravitational field which I am in?

So, if I was on the Earth's surface for an hour, then it only took me 0.999999999305134 hours, from the observer in a vacuum. Therefore, I traveled 2501.5193916289 nanoseconds in the future. I calculated all this with my time dilation software which I programed in VB

 

[Creator]

So, does this mean that using the Special Relativity time dilation equation (t=t*sqr(1-v^2/c^2), I can calculate time dilation in a gravitational field by replacing the "v" variable with the escape velocity of the gravitational field which I am in?
:

That's a big 10 - 4 little buddy. (That's trucker talk in case any big rigs have their ears on).
Very perceptive of you, Gamish.

 

[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 that's a negative number for the time variable. Does this allow time travel to the past. I am 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, I am looking forward to my mornaing coffe and I nice day learing ALG.

 

[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

 

[Creator]

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 -

Probably true.

- how does it fit with experiments that measure time dilation in a powerful centrifuge? -

What experiment are you referring to?

Creator

 

[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.

 

[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?

 

[ohwilleke]

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?

No. This is not correct.

 

[Gamish]

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

 

[hellfire]

My own off-the-top-of-my-head answer is that the t-t component of the metric tensor plays a key role--...

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".