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

The distance the space ship 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.