Past exam of the mathematics course of the University of Cambridge / 2012 / iii / Paper 56 / 4 / a

A Killing vector field $\xi$ generates a local one-parameter family of isometries of the metric tensor. Equivalently, its tensor Lie derivative of the metric vanishes:The latter is the Killing equation; the equivalence uses metric compatibility and the torsion-free connection.

Let $u$ be the tangent vector to a geodesic with affine parameter $\lambda$. Its geodesic equation is ${\nabla }_{u}u=0$. For a Killing vector field $\xi$, metric compatibility givesThusThis geodesic conserved quantity from a Killing vector is independent of whether the geodesic is timelike, spacelike or null. For a timelike geodesic one can use proper time; for a null geodesic choose an affine parameter. An arbitrary nonaffine tangent rescales the displayed quantity and need not make it constant.

The metric tensor of the planar warped spacetime depends only on $z$. The three translations therefore preserve it, givingThey are linearly independent Killing vector fields everywhere. The associated geodesic conserved quantities from Killing vectors are $-A^2\dot{t}$, $A^2\dot{x}$ and $A^2\dot{y}$, for an affine parameter; the conventional positive energy is $E=A^2\dot{t}$. No restriction on the positive function $A$ is needed for these three symmetries.