# Equivalence of Time Dilation

1. Sep 20, 2008

### Chrisc

Is the time dilation between two frames at different distances from a mass equivalent by some constant of proportionality to the time dilation between two frames in relative motion?
For example: where r0 is the Schwarzschild radius and r is a radial coordinate greater than r0, this would be analogous to the time dilation at r0 wrt r is equivalent to a relative velocity of r0 and r.

2. Sep 20, 2008

### Jonathan Scott

The time dilation at Schwarzschild radial coordinate r from central mass M in the Schwarzschild solution can be read off from the standard metric as sqrt(1-2GM/rc^2), which is approximately equal to the expression (1-GM/rc^2). When this is multiplied by the mass of a test body, the "1" part corresponds to the original rest mass and the "-GM/rc^2" part corresponds to the Newtonian potential energy relative to infinity, which matches the effective decrease in rest mass due to time dilation.

For a test body which has fallen from rest at infinity down to the same radius, the Special Relativity time dilation due to its velocity at that point is the same factor. In Newtonian terms, this is because the potential energy has been converted to kinetic energy, so the fraction of the rest mass which was lost because of potential energy has been added on again as kinetic energy.

The time dilation due to the gravitational potential or due to the corresponding velocity for falling from infinity can be compared at any two points at radial distances r_1 and r_2, and the fractional difference will be approximately (-GM/c^2) (1/r_1-1/r_2), matching the difference in the Newtonian potential. However, the velocity is approximately related to the square root of the energy, not the energy itself, so the difference in velocity between two points is not directly related to the relative time dilation between those points.

3. Sep 21, 2008

### Chrisc

Thanks Jonathan.
Would it then be correct to say the gravitaitonal time dilation between r_1 and r_2 would be approximately equal to a SR time dilation if one considers the gravitational acceleration of r_1 and r_2 resolved to a relative velocity between r_1 and r_2
In other words the fractional difference between the gravitational time dilation at r_1 and r_2, (-GM/c^2)(1/r_1-1/r_2), is approximately equal to the fractional difference in the time dilation of a test body at the instantaneous velocity of gravitational acceleration (from rest at infinity) at r_1 and r_2 wrt M?

4. Sep 21, 2008

### Mentz114

Chrisc,

I've worked out the exact formula for time dilation between two observers with radial separation and radial relative velocities in Schwarzschild space-time here

In the case of the Schwarzschild solution the gravity part and the velocity part can be separated. The formula shows that the velocity part is much bigger than the gravity part. They are not caused by the same thing so comparisons are not hugely meaningful.

( My notation is a bit difficult so if you need to work out a particulat case, let me know)

M

5. Sep 21, 2008

### Jonathan Scott

That's right provided that you realize that the relative time dilation due to velocity ONLY matches the change in gravitational time dilation in the special case where the velocity is assumed to be zero at infinity. In general, if you let a test object fall between two points, the change in velocity depends on the initial velocity at the first point, but the change in kinetic energy only depends on the difference in potential between the points.

If instead you think of the SR relative time dilation as being due to kinetic energy (which for non-relativistic speeds is proportional to v^2/2) instead of velocity then the change in kinetic energy (and hence the fractional change in SR time dilation factor) due to the gravitational field between any two points is equal to the change in potential energy (and hence the gravitational time dilation factor), and that applies regardless of the initial velocity. That's a simplified relativistic way of thinking about Newtonian gravity.

6. Sep 21, 2008

### Jonathan Scott

My previous reply was of course to Chrisc's previous one; I must remember to make that clear.

I don't think that matches Chrisc's case, where we are talking about a test object falling past two observation points at different radii. In that case, for non-relativistic velocities and weak fields, the effect is as for Newtonian physics, in that the change in kinetic energy matches the change in potential energy.

7. Sep 22, 2008

### Chrisc

As Jonathan said, this is not exactly what I had in mind. But in that it is an exact formula for calculating the difference between the two time dilations, I may take you up on your offer shortly when I clarify my idea to be correct or not.

8. Sep 22, 2008

### Chrisc

Forgive me for paraphrasing your answers, I find it a reliable method of confirming my understanding of them.

So if I understand you, instead of thinking of the infinite extent of the field in order to directly relate the relative velocity of a test body's time dilation at two different radial coordinates (points), it is simpler to think of the change in kinetic energy of a test body at two different points as directly proportional to the change in gravitational potential at the same two points and therefore the potential energy of the test body. This allows a direct correlation of (non-relativistic) SR time dilation due to change in kinetic energy (regardless initial velocity) and the change of time dilation due to gravitational potential at the same two points?

9. Sep 22, 2008

### Jonathan Scott

If I understand you, yes!