Sagittarius A*

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

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

Problem

Sagittarius A*, the black hole at the Galactic Centre, has mass M=4.3×106 M⊙M = 4.3\times10^{6}\,M_\odot. Solving Rμν=0R_{\mu\nu}=0 for the static spherically symmetric ansatz ds2=−e2Ac2dt2+e2Bdr2+r2dΩ2ds^2 = -e^{2\mathcal{A}}c^2dt^2 + e^{2\mathcal{B}}dr^2 + r^2d\Omega^2 and matching to the weak-field metric fixes the integration constant rs=2GM/c2r_s = 2GM/c^2. The resulting spacetime has Kretschmann scalar K=48G2M2/(c4r6)\mathcal{K} = 48G^2M^2/(c^4r^6). ((a)) Using GM⊙/c2=1.48GM_\odot/c^2 = 1.48 km, find rsr_s for Sagittarius A* in metres. ((b)) Because K\mathcal{K} is a fully contracted scalar built from the Riemann tensor, its value at an event does not depend on the coordinates used to compute it. Evaluate K\mathcal{K} at r=rsr = r_s and at r=rs/10r = r_s/10, and explain what the behaviour of K\mathcal{K} as r→0r\to0 shows about that point, in contrast with r=rsr = r_s.

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