Re: I am beginning to get suspicious about this...

Pi-R wrote:
There is a spherical mass M of uniform density and of radius R.

There is a clock A at a theoretical infinite distance from mass M, such
that gravitational influence of M on clock A is null or negligible.

There is a clock B at a distance r from the center of mass M.

Clock B runs slower than clock A by a factor f(r) defined as follows:

- for r>R: f(r) = 1/sqrt(1-2GM/rc^2)

Ze question: what is f(r) for r<R ?

The (Newtonian) gravitational potential outside the body is Phi(r) = -GM/r. Inside, it's Phi(r) = a + br^2, where a and b are chosen such that the overall potential and its first derivative are continuous. I think this gives a = -3GM/2R and b = GM/2R^3. As I recall, the observed z factor is given by (Phi(r_o) - Phi(r_e))/c^2, which in this case would be GM/rc^2 outside and (GM/(2Rc^2))(3-(r/R)^2) inside. Add 1 to get the relative clock rates.

That's an approximation. To get the exact answer you'd need to stitch together the exterior and interior Schwarzschild solutions and then calculate some null geodesics. It's not that hard, but it's more time than I want to spend at the moment.

-- Ben
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