The Schwarzschild line element is ds^2 = - (1- r_s r )c^2dt^2 + dr^2 1-r_s/r + r^2d ^2

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

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

Problem

The Schwarzschild line element is

ds^2 = -\left(1-\frac{r_s}{r}\right)c^2dt^2 + \frac{dr^2}{1-r_s/r} + r^2d\Omega^2,

with

r_s=2GM/c^2

. Define the tortoise coordinate

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

and the ingoing Eddington-Finkelstein time coordinate

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

. ((a)) Show that

dr_*/dr = (1-r_s/r)^{-1}

. ((b)) Use this to write

dt

in terms of

dv_{\rm EF}

and

dr

, and substitute into the line element above to show it becomes

ds^2 = -\left(1-\frac{r_s}{r}\right)c^2dv_{\rm EF}^2 + 2c\,dv_{\rm EF}\,dr + r^2d\Omega^2.

((c)) Write the metric components

g_{\mu\nu}

in the coordinates

(v_{\rm EF},r,\theta,\phi)

, evaluate each at

r=r_s

, and compute the determinant

g

there. State what this shows about the misbehaviour at

r=r_s

in the original

(t,r)

coordinates.

Related theory

Black holes

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