(a) Starting from the field equations and the fact that, in Lorenz gauge , the…

PHYSICS · Subjective · Fermi University Physics · Relativity & Gravitation · Einstein's field equations · Gravitational waves and gravitational-wave astronomy

The Geometry of Gravity, Ch Linearised gravity: weak fields, gravitomagnetism and the wave equation

Problem

(a) Starting from the field equations

G_{\mu\nu}=(8\pi G/c^4)T_{\mu\nu}

and the fact that, in Lorenz gauge

\partial^\mu\bar h_{\mu\nu}=0

, the linearised Einstein tensor collapses to

G_{\mu\nu}^{(1)}=-\tfrac12\Box\bar h_{\mu\nu}

(with

\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12h\eta_{\mu\nu}

), derive

\Box\bar h_{\mu\nu}=-\frac{16\pi G}{c^4}T_{\mu\nu}.

(b) For a static, non-relativistic source of total mass

M

with

T_{00}\approx\rho_{\rm m} c^2

and all other components negligible, solve the static (

\partial_0=0

) version of this equation outside the source by comparison with the Newtonian equation

\nabla^2\Phi=4\pi G\rho_{\rm m}

, where

\Phi=-GM/\tilde r

. Hence find

\bar h_{00}

, and by undoing the trace reversal find

h_{00}

and

h_{ij}

in terms of

\Phi

. (c) Evaluate

h_{00}

and

h_{ij}

at the Earth's surface,

\tilde r\approx R_\oplus=6.371\times10^6

m, using

GM_\oplus/c^2=4.44

mm, and check the result against the gravitational redshift relation

g_{00}=-(1+2\Phi/c^2)

. (d) Now set

T_{\mu\nu}=0

(vacuum). Write the plane-wave solution

\bar h_{\mu\nu}=A_{\mu\nu}e^{ik_\alpha x^\alpha}

of the wave equation from part (a), state the resulting condition on the wave four-vector

k^\mu

, and say how many of the ten independent components of

A_{\mu\nu}

survive once the Lorenz condition is imposed.

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