Past exam of the general relativity II course of the University of Oxford / 2017 / 3 / b

The metric tensor is independent of $t$ and $\varphi$, soare commuting Killing vector fields. For $0<a<M$ they span the continuous Killing vector field algebra; every constant linear combination is also a Killing vector field. They express stationary spacetime and axial symmetry, respectively. At the allowed endpoint $a=0$ the spherical symmetry of Schwarzschild spacetime enlarges the algebra: in addition to $T$, a basis of rotational Killing vector fields isTheir polar-coordinate expressions are singular at the axis, but the rotational fields are smooth there.

Let $U^a=d{x}^a/ds$ be the tangent to any affinely parametrized geodesic. For a Killing vector field $X$,This is the geodesic conserved quantity from a Killing vector. With $p_a=g_{ab}{U}^b$, choose the energy and axial angular momentum conventionsFor a massive particle with proper time parameter these are per unit rest mass; for a null geodesic their common scale depends on the affine parameter normalization. Expanding in the supplied Boyer-Lindquist coordinates givesThe factor of two between $g_{t\varphi }$ and the coefficient of $dtd\varphi$ is essential for these signs and coefficients.