Does proper time of a test particle depend on the density of the body, around which the test particle is moving? Does the proper time of a test particle depend on the density of the body, around which the test particle is moving? Say the proper time on earth is not the same as the proper time on black hole, since black hole is much denser than earth.
 A: In some sense, you can say that it does depend; But actually, it doesn't in general. Truly speaking, proper time is a geometrical concept. So from that point of view, it may seemed to be independent of any physical parameter.
But if you take Einstein's field equation into consideration, then the proper time should depend on the energy-momentum tensor of that particular system.
Now back to your question:
The dependence may not be on the density, may be on the mass, may be on the energy density or may be on something else. So if your targeted energy-momentum tensor will appear to be the density (of the body) or not, that solely depends on your problem.
For example, if you take your example and consider the Earth to be a perfectly spherically symmetric non-rotating static body, then the metric outside the Earth and a Schwarzschild black-hole will be identical.
$d\tau^2 = -\left(1-\dfrac{2M_{body}}{r}\right)dt^2 + \dfrac{dr^2}{\left(1-\dfrac{2M_{body}}{r}\right)} + r^2d\Omega^2$
So clearly, the proper time does depend on the mass of the body, not on the density.
