For the static planet metric ds^2=- (1+ 2 (r) c^2 )c^2dt^2+dr^2+r^2d ^2+r^2 ^2 ,d ^2

PHYSICS · Subjective · Fermi University Physics · Relativity & Gravitation · Differential geometry of curved spacetime · Differential geometry of curved spacetime

The Geometry of Gravity, Ch Symmetry: Lie derivatives, Killing vectors and conserved quantities

Problem

For the static planet metric ds2=−(1+2Φ(r)c2)c2dt2+dr2+r2dθ2+r2sin⁡2θ dϕ2,ds^2=-\Big(1+\frac{2\Phi(r)}{c^2}\Big)c^2dt^2+dr^2+r^2d\theta^2+r^2\sin^2\theta\,d\phi^2, someone checks whether ξ=∂θ\xi=\partial_\theta is a Killing vector as follows: "L∂θgθθ=∂θ(r2)=0\mathcal L_{\partial_\theta}g_{\theta\theta}=\partial_\theta(r^2)=0, and gttg_{tt}, grrg_{rr} plainly do not depend on θ\theta either. So every metric component is unchanged by L∂θ\mathcal L_{\partial_\theta}, and therefore ∂θ\partial_\theta is a Killing vector." Find the error in this argument and state the correct conclusion.

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