In the previous post, we took a tour through the literature and found that when general relativity is translated into the elegant mathematical language of the geometric algebra, the natural coordinates for a black hole turn out to be Painleve or Gullstrand-Painleve coordinates instead of the more common Schwarzschild coordinates. Our next post will derive the equations of motion for orbits in this coordinate system, but before we get into the difficult mathematics, we should take a quick look at the Painleve coordinates.
First of all, most of my readers will know that in general relativity, the choice of coordinates is quite arbitrary. Both Schwarzschild and Painleve coordinates describe the same object, the gravity field of a gravitating object which is spherically symmetric (and therefore non rotating), i.e. they are all descriptions of the black hole. In this sense they correspond to the same solution to Einstein’s field equations, which is sometimes called “Schwarzschild’s Solution”, or the “Schwarzschild Metric”. This is a little confusing, “Schwarzschild” was the person who found the Schwarzschild metric and he found it using the Schwarzschild coordinates, so his name is used twice here.
I guess I should put a pretty picture from the gravity simulation that resulted from all this here so it will show above the fold. This is a set of “knife edge” orbits, that is, orbits that quite nearly fall into the black hole but do not. Due to the time spent near the black hole, whose event horizon is marked in gray, the test masses get huge precession:
We will be discussing the less pretty, but more mathematical subject of Painleve and Schwarzschild coordinates in this post.