Kepler’s third law in classical mechanics states that the square of the orbital period, , of a planet or particle is proportional to the cube of the semi-major axis of its orbit. The classical equation describing this is
where is the semi-major axis, the Newtonian time, and the mass of the test particle. We are interested in test particles, so .
An interesting question is how this generalizes to test particles moving around a very massive body in general relativity, such as a black hole. The metric in Schwarzschild coordinates that describes this geometry is
where is the mass of the black hole. The question is whether Kepler’s third law applies to the Schwarzschild time, , or the proper time, , of the test particle? To answer this question let us assume that and that the motion is circular, so .
Using our assumptions this simplifies to
From this we see that
and that it is the Schwarzschild time coordinate that is analogous to the Newtonian time in Kepler’s third law.