general relativity - How can Schwarzschild coordinates be measured?

In special relativity, we make a big fuss about setting up inertial frames of reference, and then constructing coordinate systems using networks of clocks and rulers. This gives an unambiguous physical definition of the spacetime point $(x, t)$. In general relativity, you can take whatever coordinate system you want, but then I don't know what the coordinates mean.

For example, the event horizon of a Schwarzschild black hole is $r = 2GM$. Naively interpreting $r$ as a radial coordinate, this suggests that the event horizon is "$2GM$ away from the center of the black hole", but that statement doesn't make sense either mathematically (the distance $\int ds$ is not $2GM$ at all) or physically (you can't extend your ruler network inside the black hole). But the books I've seen seem to treat $r$ just like the radial coordinate, and talk about "the radius of a stable circular orbit" or stuff like that.

In the rest of physics, we relentlessly focus on how mathematical quantities can be measured, but I don't know how that works here, for the Schwarzschild coordinate $r$.

• Can the statement "the event horizon is at $r = 2GM$" be phrased in a coordinate-independent way?


• How can the coordinate $r$ be measured?

Answer

Coordinates can be measured in GR, though all too often this fact is overlooked or even contradicted by people getting caught up in coordinate invariance.

As you well note, in Schwarzschild $r$ isn't really a radius in the "integrate at constant angle from the center and recover this value" sense. It is, however, radial in the sense of being orthogonal to the angular coordinates, Moreover, it matches Euclidean intuition with regards to circumferences and areas at fixed $r$.

How can the coordinate $r$ be measured?

One measurement procedure you can adopt is this: Sit in your rocket with a fixed amount of thrust pushing directly away from the black hole, so that you are hovering at constant $r$. Get all your friends to do the same around the black hole, everyone experiencing the same acceleration. Everyone can then lay down rulers in a circle passing through all the rockets, and the sum of the readings (assuming you've adjusted positions so as to maximize this value) is in fact $2\pi r$.

Can the statement "the event horizon is at $r =2GM$" be phrased in a coordinate-independent way?

Sort of, though perhaps not in as direct a way as you would want. Certainly the event horizon is simply the surface delineating what events can influence future null infinity -- no coordinates involved.

Using the discussion above, though, we could say that for any $r > 2GM$ that the surface of constant $r$ is the locus of points such that rockets with a prescribed radial acceleration hold stationary there, with the event horizon being the limit of such surfaces.

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In general, what I'm pushing is the idea that coordinates can be measured as long as you can come up with some experiment where they appear in the formula. This is slightly broader than the notion of measurement of "integrate $\sqrt{g_{\mu\mu}}$ along a line where all coordinates except $x^\mu$ are constant" that suffices for simple spaces.