Consider the static spherically symmetric ansatz

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

Consider the static spherically symmetric ansatz ds2=−e2A(r)c2dt2+e2B(r)dr2+r2dΩ2ds^2 = -e^{2\mathcal{A}(r)}c^2dt^2 + e^{2\mathcal{B}(r)}dr^2 + r^2d\Omega^2, with rr defined so that a sphere of coordinate rr has area exactly 4πr24\pi r^2. ((a)) State, for each of the following features of the ansatz, which Killing symmetry of a static, spherically symmetric spacetime is responsible for it: (i) the absence of a dt drdt\,dr cross term; (ii) A\mathcal{A} and B\mathcal{B} depending on rr only, not tt; (iii) A\mathcal{A} and B\mathcal{B} depending on rr only, not θ,ϕ\theta,\phi, and the angular part being exactly r2dΩ2r^2d\Omega^2. ((b)) For this ansatz, after rescaling tt so that A=−B\mathcal{A}=-\mathcal{B}, the vacuum field equation Rθθ=0R_{\theta\theta}=0 reduces to ddr(re−2B)=1\dfrac{d}{dr}\big(re^{-2\mathcal{B}}\big) = 1. Solve this equation for e−2B(r)e^{-2\mathcal{B}}(r), and fix the constant of integration by matching gttg_{tt} to the weak-field metric of a mass MM at large rr. ((c)) The Kretschmann scalar of this solution is K=48G2M2/(c4r6)\mathcal{K} = 48G^2M^2/(c^4r^6). Show that K\mathcal{K} is finite at r=rs=2GM/c2r=r_s=2GM/c^2 (give its value in terms of GG, MM, cc) but diverges as r→0r\to0, and explain why this identifies r=0r=0, not r=rsr=r_s, as a genuine curvature singularity. ((d)) In ingoing Eddington-Finkelstein coordinates (vEF,r,θ,ϕ)(v_{\rm EF},r,\theta,\phi), with vEF=t+r∗/cv_{\rm EF}=t+r_*/c, the metric is ds2=−(1−rsr)c2dvEF2+2c dvEF dr+r2dΩ2.ds^2 = -\left(1-\frac{r_s}{r}\right)c^2dv_{\rm EF}^2 + 2c\,dv_{\rm EF}\,dr + r^2d\Omega^2. Show that every metric component is finite at r=rsr=r_s and that the metric determinant g=−c2r4sin⁡2θg=-c^2r^4\sin^2\theta is non-zero there, and explain why this, rather than the finite Kretschmann scalar of part (c), is what shows the misbehaviour of the original (t,r)(t,r) coordinates at rsr_s is a coordinate artefact.

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