My question pertains to entire General Relativity, but to be specific, I'll restrict to black holes.

The black hole solution of the Einstein equations might be represented in a number of different metrics. The most commonly used is the metric written in the Schwarzschild, a.k.a "curvature", coordinates. As well known, these coordinates do not cover the entire manifold, and to treat the full spacetime, one then moves to Kruskal-Szekeres coordinates which do not have spurious singularity at horizon. There are also Eddington-Finkelstein coordinates, which are regular at horizon as well. Another form is the isotropic coordinates, in which the spatial part is conformally flat.

As far as theory is concerned, I think I don't have trouble understanding these forms of the metric and which features they highlight. After all, the major principle of GR is general covariance, so that all coordinate systems are equally "good". What obscures me is how to apply this knowledge in practice. To be specific, let's consider observation of the supermassive black hole in the center of the Milky Way, as currently being undertaken by the Event Horizon Telescope. Which form of the metric should be used to describe what we should see in the direction of that black hole from the Earth (e.g., the expected black hole "shadow")? My intuition is that Finkelstein or Kruskal coordinates are not the best choice here. But what about curvature and isotropic coordinates? Which of them is suitable for this purpose? In fact, a priori, I find the latter more "physical" or logical than traditional curvature coordinates, since in isotropic coordinates both $r$ and $dr$ are equally affected by gravity.

  • $\begingroup$ Remember it's not just what what you are describing but what you want to compute. Numerical schemes in GR for example have strong implications on what coordinates to choose. $\endgroup$ – JamalS Nov 17 '17 at 15:18

There is no recipe for this, just try them all (and usually there are not so many) and pick the one that makes the calculations simpler.

If your system is symmetric, you may want to use that symmetry to reduce the complexity. For instance if the system is spherically symmetric use spherical coordinates, so that if the motion is radial and the problem has been reduce to 2d.


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