- #31
PeterDonis
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grav-universe said:
I should point out that the equation in post #16 is *not* the correct equation for the "potential" inside a uniformly dense non-rotating sphere. The "potential" *outside* the sphere, in the exterior vacuum region, is given by the Schwarzschild metric (which is where the formula in post #16 comes from); but *inside* the sphere, in the interior non-vacuum region, the potential is given by a different formula, which is *not* the sum of contributions from individual pieces of the object, as I said in my previous post. The correct formula for the "potential" at radius r inside a non-rotating spherical object with uniform density is this:
\frac{d\tau}{dt} = \frac{3}{2} \sqrt{1 - \frac{2M}{R}} - \frac{1}{2} \sqrt{\frac{1 - 2M r^{2}}{R^{3}}}
where M is the total mass of the object and R is the radius at its outer surface. You will note that at r = R, i.e., at the object's surface, this becomes
\frac{d\tau}{dt} = \sqrt{1 - \frac{2M}{R}}
which is the same as the "potential" from the exterior Schwarzschild metric at r = R; this shows that the "potential" changes smoothly as the surface of the object is crossed.
