(a) Flat spacetime is described in coordinates in which everywhere. A small…

PHYSICS · Subjective · Fermi University Physics · Relativity & Gravitation · Einstein's field equations · Einstein's field equations

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

Problem

(a) Flat spacetime is described in coordinates in which

h_{\mu\nu}=0

everywhere. A small coordinate change

x^\mu\to x^\mu+\zeta^\mu

is made, with

\zeta^\mu=(\epsilon x^1,0,0,0)

and

\epsilon=5.0\times10^{-4}

a dimensionless constant. Using

h_{\mu\nu}\to h_{\mu\nu}-\partial_\mu\zeta_\nu-\partial_\nu\zeta_\mu

, find every non-zero component of

h_{\mu\nu}

in the new coordinates. (b) A white dwarf has mass

M=1.0\,M_\odot

and radius

R=6.0\times10^3

km. Using

GM_\odot/c^2=1.48

km, compute

GM/(Rc^2)

at its surface and decide whether the weak-field linear approximation to

g_{00}

is adequate there. (c) The white dwarf spins with angular momentum

J=2.0\times10^{42}

kg m

^2

^{-1}

. Outside a slowly rotating sphere, on the equatorial plane at distance

r

from the centre,

|h_{0i}|=2GJ/(c^3r^2)

. Evaluate

|h_{0i}|

at its surface, on the equator (

r=R