A timelike Killing vector of a static metric gives a conserved quantity along any…

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

A timelike Killing vector

\xi^\mu=(\partial_t)^\mu

of a static metric gives a conserved quantity

E=-g_{\mu\nu}\xi^\mu k^\nu

along any geodesic with tangent

k^\mu

, timelike or null. The weak-field line element has

g_{tt}\approx-(1-2GM/\tilde r c^2)c^2

, with

\tilde r

the isotropic radial coordinate, and it was shown from the equivalence principle alone that a photon climbing from a static source at radius

r_1

to a static receiver at radius

r_2>r_1

in a weak field is redshifted by

z\approx GM(1/r_1-1/r_2)/c^2

. ((a)) For a metric of the static spherically symmetric ansatz,

g_{tt}=-e^{2\mathcal{A}(r)}

, use the conservation of

E

along a radial null geodesic, together with the fact that a static metric's components do not depend on

t

, to write the ratio

\nu_2/\nu_1

of received to emitted frequency in terms of

e^{\mathcal{A}}

r_1

and

r_2

. ((b)) Match

e^{2\mathcal{A}}

at large

r

to the weak-field form to identify the integration constant of the vacuum solution as

2GM/c^2

, and show that the ratio found in part (a) reduces, to first order in

GM/rc^2

, to the equivalence-principle redshift formula above. ((c)) A photon leaves the surface of the Earth (

R_\oplus=6371

km,

GM_\oplus/c^2=4.44

mm) and is received by an observer at rest very far away (

r_2\to\infty

). Evaluate the first-order redshift

z

numerically.

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