(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=(0,0,0,\epsilon x^0)

and

\epsilon=3.0\times10^{-3}

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) Mars has mass

M=6.42\times10^{23}

kg and radius

R=3.39\times10^6

m. Compute

GM/(Rc^2)

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

g_{00}

is adequate there. (c) Mars's spin angular momentum is

J=1.9\times10^{32}

kg m

^2

^{-1}

. Outside a slowly rotating sphere the gravitomagnetic part of the metric is

h_{0i}=-\dfrac{2G}{c^3r^3}(\vec J\times\vec r)_i

; on the equatorial plane at distance

r

from the centre its magnitude is

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

. Evaluate

|h_{0i}|

at Mars's surface, on the equator (

r=R