General Relativity: An Introduction for Physicists · Problem 4.11

Tensor calculus on manifolds — Hobson, Efstathiou & Lasenby

Physics · killing-vectors-isometry

Problem

Any coordinate transformation

x^a \to x'^a

under which the metric is form invariant, i.e. such that

g'_{ab}(x) = g_{ab}(x)

is called an isometry (note that the argument is the same on both sides of the above equation). Show that the infinitesimal coordinate transformation in Exercise 4.6 is an isometry, to first order in

\epsilon

, provided that

v^a

satisfies

g_{ac}\partial_b v^c + g_{cb}\partial_a v^c + v^c \partial_c g_{ab} = 0.

Show further that this expression can be written as

\nabla_a v_b + \nabla_b v_a = 0.

This is Killing's equation and any vector satisfying it is known as a Killing vector of the metric

g_{ab}

. Show that if

v^a

and

w^a

are both Killing vectors then so too is any linear combination

\lambda v^a + \mu w^a

, where

\lambda

and

\mu

are constants.

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