General Relativity: An Introduction for Physicists · Problem 2.7

Manifolds and coordinates — Hobson, Efstathiou & Lasenby

Physics · conformal-transformations-null-curves

Problem

A conformal transformation is not a change of coordinates but an actual change in the geometry of a manifold such that the metric tensor transforms as g~ab(x)=Ω2(x)gab(x),\tilde{g}_{ab}(x) = \Omega^2(x)g_{ab}(x), where Ω(x)\Omega(x) is some non-vanishing scalar function of position. In a pseudo-Riemannian manifold, show that if xa(λ)x^a(\lambda) is a null curve with respect to gabg_{ab} (i.e. ds2=0ds^2 = 0 along the curve), then it is also a null curve with respect to g~ab\tilde{g}_{ab}. Is this true for timelike curves?

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