A calculation for a black hole of Schwarzschild radius km reads: 'Two shells surround…

PHYSICS · Multi-part · Fermi University Physics · Relativity & Gravitation · Schwarzschild spacetime and the tests of general relativity · Schwarzschild spacetime and the tests of general relativity

The Geometry of Gravity, Ch The Schwarzschild geometry and Birkhoff's theorem

Problem

A calculation for a black hole of Schwarzschild radius rs=4.00r_s=4.00 km reads: 'Two shells surround the hole at areal radius r1=6.00r_1=6.00 km and r2=10.00r_2=10.00 km. The proper radial distance a static observer measures between them, moving radially outward with a tape measure, is just the difference in their radial coordinates: Δℓ=r2−r1=4.00\Delta\ell = r_2-r_1 = 4.00 km.' ((a)) Identify the error in this reasoning. ((b)) Calculate the correct proper radial distance between the two shells, using ∫r1r2dr1−rs/r=[r(r−rs)+rsln⁡(r+r−rs)]r1r2.\int_{r_1}^{r_2}\frac{dr}{\sqrt{1-r_s/r}} = \Big[\sqrt{r(r-r_s)}+r_s\ln\big(\sqrt r+\sqrt{r-r_s}\big)\Big]_{r_1}^{r_2}.

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