Posted: Mar 01, 2011 8:03 am
OK I'll have a go at translating the maths to English:

You probably know a lot of this already, but I need to mention it for completeness.

Much of special relativity can be encapsulated in the statement that the "Minkowski metric"

ds2 = c2dt2 - dx2- dy2- dz2 is "invariant under Lorentz transformations".

ds2 is the spacetime distance between two points (t, x, y, z) and (t+dt, x+dx, y+dy, z+dz).

"Invariant under Lorentz transformations" means that if I transform to a new frame with coordinates t', x', y', z' using a Lorentz transformation, I get the same number for ds2

In general relativity, you have a similar concept except this time, the metric components making up ds2 are functions of the coordinates, and are not just +1, -1, -1, -1 like in the Minkowski case. Also, you have to consider general coordinate transformations, not just Lorentz transformations.

The metric appropriate for describing the gravitational field from a spherically symmetric non rotating mass is called the Schwarzschild solution. This describes the metric outside the object. The field inside the object is given by the Schwarzschild Interior Solution which is that complicated expression for ds2 I gave in the above post. That expression applies when the density of the object is constant. The metric in that post was given in spherical polar form for the spatial coordinates.

Now if you choose points where the two objects whose times we want to compare are at rest, you can set dr=0, dθ=0, dφ=0, so you're left with

ds2 = f(r)c2dt2

Now we can also write ds2 = c2d𝜏2 where 𝜏 is the proper time. Note t is the "coordinate time" - t=const is a spacelike slice of all of spacetime, i.e. it's a picture of space at a given time. So I can write

c2d𝜏12 = f(r1)c2dt2 and

c2d𝜏22 = f(r2)c2dt2

where 𝜏1 is the proper time measured by a clock at r1, and 𝜏2 is the proper time measured by a clock at r2. By dividing these equations the result follows.