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For a blackhole of mass $M$, radius $R$ and Schwarzschild radius $R^*$ (where $R<R^*$), its density $\rho$ is defined as $$\rho=\frac{M}{(R^*)^3}.$$

One reason I read for using the Schwarzschild radius $R^*$ instead of the actual radius $R$ is that no measurements can be made inside of $R^*$ (i.e. for $r<R^*$).

I know that the Schwarzschild metric is $$g_{\mu\nu}=\text{diag}\left[\left(-1+{R^*\over r}\right),\left(1-{R^*\over r}\right)^{-1},r^2,r^2\sin^2\theta\right],$$ so there is a singular point at $r=R^*$ where $g_{rr}$ is infinite and hence the invariant interval $ds^2$ is infinite.

However, for points inside $R^*$ (i.e. $r<R^*$), the metric and $ds^2$ is finite. Why then is it not possible to make a measurement inside for points inside $R^*$? Is it somehow required to first pass the singular point at $r=R^*$ before a measurement can be made inside $R^*$?

StephenG - Help Ukraine
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TaeNyFan
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  • Do you mean “to compute density inside" rather than "to make a measurement inside"? – dan Nov 15 '20 at 09:20
  • Why do you think a black hole has an “actual radius” $R$ smaller than its Schwarzschild radius? Do you see this anywhere in the metric? – G. Smith Nov 15 '20 at 18:18
  • https://physics.stackexchange.com/questions/26515/what-is-exactly-the-density-of-a-black-hole-and-how-can-it-be-calculated – safesphere Nov 16 '20 at 06:23

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Yes, at the Schwarzschild radius, the metric is so messed up but the fault is in the coordinate system. The metric in Schwarzschild coordinate is faking a singularity. You can call it "A Coordinate Singularity". The Real singularity lies at zero radius (At least in the Schwarzschild coordinate). You can choose a better coordinate system to avoid it (Like the Kruskal-Szekeres Coordinate, And It's easier to understand the cause and effect in this coordinate because the null geodesics are straight 45 degree lines). Being not a true singularity does not mean that the insides of it could be observed, This is because there is a horizon that breaks the causal connection to the insides. And Yes, It is absolutely right that you should make a pass through to make a causal connection with the insides.

Vikash Kotteeswaran
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