This problem follows the static spherically symmetric geometry from its symmetry…

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

This problem follows the static spherically symmetric geometry from its symmetry properties through to the coordinate that removes the trouble at

r=r_s

. Throughout,

\mathcal{A}(r)

and

\mathcal{B}(r)

are the two metric functions of the ansatz

ds^2=-e^{2\mathcal{A}(r)}c^2dt^2+e^{2\mathcal{B}(r)}dr^2+r^2d\Omega^2

r_s=2GM/c^2

, and

f(r)\equiv 1-r_s/r

. ((a)) Name the Killing-vector property responsible for (i) the absence of any

dt\,dr

dt\,d\theta

dt\,d\phi

cross-term in the line element, and (ii) the absence of

t

-dependence in

\mathcal{A}

and

\mathcal{B}

. ((b)) The vacuum field equations

R_{\mu\nu}=0

for this ansatz reduce to

\mathcal{A}+\mathcal{B}=\text{constant}

(set to zero by rescaling

t

) and to

\frac{d}{dr}\left(re^{-2\mathcal{B}}\right)=1.

Solve this equation for

e^{-2\mathcal{B}}(r)

, writing the constant of integration as

-r_s

, and hence give

e^{2\mathcal{A}}(r)

. ((c)) The Kretschmann scalar of this geometry is

\mathcal{K}=R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}=48G^2M^2/(c^4r^6)

. Evaluate

\lim_{r\to0}\mathcal{K}

and

\mathcal{K}(r_s)

(the latter left as an expression in

G,M,c,r_s

), and state what each result shows about

r=0

and about

r=r_s

. ((d)) Define the tortoise coordinate

r_*=r+r_s\ln|r/r_s-1|

and the ingoing Eddington-Finkelstein time

v_{\rm EF}=t+r_*/c

. Starting from

ds^2=-f(r)c^2dt^2+f(r)^{-1}dr^2+r^2d\Omega^2

, change coordinates from

(t,r)

(v_{\rm EF},r)

and show that every component of the resulting metric, and its determinant, is finite at

r=r_s

, with the determinant non-zero there.

Browse the problem library