Gravitational time dilation derivation

[kairama15]

If a beam of light is shot from a fast space ship, it travels a distance c*t1 according to their reference frame.
The same beam of light seen from an outside observer goes at an angle and travels a distance c*t2.
The distance the spaceship travels is equal to v*t2.
Using the triangle made from these lengths, the Pythagorean theorem shows that
V2=v1/sqrt(1-v^2/c^2)

This is the well known time dilation from special relativity.

The equation for gravitational time dilation in the presence of a large mass is very similar (v is replaced with escape velocity it turns out.)

Does anyone know a geometrical proof for gravitational time dilation similar to the one for time dilation due to relative speed? I'm looking for an elegant derivation like special relativity's. Something simple and beautiful like the above example.

 

[PeterDonis]

If a beam of light is shot from a fast space ship, it travels a distance c*t1 according to their reference frame.

A distance from where to where?

The same beam of light seen from an outside observer goes at an angle and travels a distance c*t2.

From where to where?

The distance the spaceship travels is equal to v*t2.

From where to where?

Using the triangle made from these lengths

How can you draw a triangle from these lengths when they are lengths in different frames? That doesn't make sense.

If you are getting this geometric derivation of SR time dilation from some reference, please give the reference.

Does anyone know a geometrical proof for gravitational time dilation similar to the one for time dilation due to relative speed?

The general answer will be "no", since gravitational time dilation is not the same thing as SR time dilation due to relative speed and shouldn't be expected to have the same geometric properties.

 

[pervect]

The usual way to motivate gravitational time dilation is to invoke the equivalence principle to equate the behavior of time on Einstein's accelerating elevator to time in a gravitational field.

It's not too hard to show that a beam of light emitted from the rear of the spaceship will take some amount of time to reach the front, during which the spaceship (including the front of the ship) accelerates. This causes the beam of light to be redshifted when it arrives at the front of the ship. The argument goes then that this redshift is explained by "gravitational time dilation" in the accelerated frame.

A less simple but more convincing calculation uses the ability of tensors to handle arbitrary coordinate systems, and it involves finding the necessary transform between the coordinates of the accelerating ship and the coordinates in some particular inertial frame. Then the Rindler metric gives the gravitational time dilation directly.

This second approach defines the mathematical structure to actually describe all the metric effects of acceleration, and to show that the only effect is on the metric coefficient $g_{00}$. The motivational treatment is a bit haphazard, it doesn't guarantee that there aren't any other relativistic effects, for instance.

For the end result, see for instance https://en.wikipedia.org/w/index.php?title=Rindler_coordinates&oldid=846543331, and note that the metric is the same except for $g_{00}$.

I doubt this is as simple as what you want, but it's what I've got to offer at the moment.

 

[m4r35n357]

Does anyone know a geometrical proof for gravitational time dilation similar to the one for time dilation due to relative speed? I'm looking for an elegant derivation like special relativity's. Something simple and beautiful like the above example.

Gravitation is covered by General Relativity, not Special Relativity. Gravitational time dilation emerges from (for example) the Schwartzschild solution of Einstein's equations. I would consider this beautiful, but would never claim it to be simple. It can also be derived from an accelerated reference frame, but that is not strictly gravitation (equivalence principle). Again, beautiful but not simple.