General Relativity: An Introduction for Physicists · Problem 4.6

Tensor calculus on manifolds — Hobson, Efstathiou & Lasenby

Physics · infinitesimal-coordinate-transformation-metric

Problem

Consider the infinitesimal coordinate transformation x′a=xa+ϵva(x),x'^a = x^a + \epsilon v^a(x), where va(x)v^a(x) is a vector field and ϵ\epsilon is a small scalar quantity. Show that, to first order in ϵ\epsilon, gab′(x′)=gab(x)−ϵ(gac∂bvc+gcb∂avc).g'_{ab}(x') = g_{ab}(x) - \epsilon(g_{ac}\partial_b v^c + g_{cb}\partial_a v^c).

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