Physics Asked by user12262 on June 7, 2021
This accepted answer to the question “What is the physical meaning of the Eddington-Finkelstein coordinates?” (PSE/q/91724) prescribes to
1. Enclos[e] the origin of our Schwarzschild spacetime in a series of concentric spheres […]
In a spacetime region of Schwarzschild geometry (including “its origin“, if necessary), what is the explicit physical meaning* of a set of sufficiently many distinct participants (presumably at least four) being geometrically related to each other as (a subset of) a unique “sphere“;
and what is the explicit physical meaning of two distinct such “spheres” being “concentric” to each other?
(*: In the given context of general-relativity the notion of “physical meaning” is surely understood in the sense spelt out by Einstein, that “All our well-substantiated space-time propositions amount to the determination of space-time coincidences {such as} encounters between two or more material points.“.)
One of the lessons of relativity is that we are free to choose any coordinates to describe the geometry of spacetime, though obviously some coordinates make more physical sense than others. In the case of a Schwarzschild black hole we choose a set of polar coordinates $(t, r, theta, phi)$ called the Schwarzschild coordinates. The singularity at the centre of the black hole is at $r = 0$.
When we say a sphere we simply mean the spacial surface with constant $r$ in the Schwarzschild coordinates. All such spheres are automatically concentric because they are constructed around the same centre i.e. the singularity at $r = 0$.
The Eddington-Finkelstein coordinates use the same $r$ coordinates as the Schwarzschild coordinates, so the meaning of sphere is the same in both.
Answered by John Rennie on June 7, 2021
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