Spacetime and Geometry · Problem 7.5

Cosmology — Sean Carroll

Physics — General Relativity · exact-gravitational-plane-wave

Problem

In the exercises for Chapter 3, we introduced the metric

ds^2 = -(\mathrm{d}u\,\mathrm{d}v + \mathrm{d}v\,\mathrm{d}u) + a^2(u)\mathrm{d}x^2 + b^2(u)\mathrm{d}y^2, \tag{7.206}

where

a

and

b

are unspecified functions of

u

. For appropriate functions

a

and

b

, this represents an exact gravitational plane wave. (a) Calculate the Christoffel symbols and Riemann tensor for this metric. (b) Use Einstein's equation in vacuum to derive equations obeyed by

a(u)

and

b(u)

. (c) Show that an exact solution can be found, in which both

a

and

b

are determined in terms of an arbitrary function

f(u)

Related theory

Ligo And Gravitational Wave Detection

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