Angular momentum conservation for dynamical black holes

Angular momentum can be defined by rearranging the Komar surface integral in terms of a twist form, encoding the twisting around of space-time due to a rotating mass, and an axial vector. If the axial vector is a coordinate vector and has vanishing transverse divergence, it can be uniquely specified under certain generic conditions. Along a trapping horizon, a conservation law expresses the rate of change of angular momentum of a general black hole in terms of angular momentum densities of matter and gravitational radiation. This identifies the transverse-normal block of an effective gravitational-radiation energy tensor, whose normal-normal block was recently identified in a corresponding energy conservation law. Angular momentum and energy are dual, respectively, to the axial vector and a previously identified vector, the conservation equations taking the same form. Including charge conservation, the three conserved quantities yield definitions of an effective energy, electric potential, angular velocity and surface gravity, satisfying a dynamical version of the so-called first law of black-hole mechanics. A corresponding zeroth law holds for null trapping horizons, resolving an ambiguity in taking the null limit.

Figure 1

A non-null hypersurface $H$ foliated by spatial surfaces $S$, with generating vector $\xi$ and its normal dual $\tau ={\xi }^{*}$. If $H$ becomes null, $\xi$ and $\tau$ coincide.

Figure 2

A transverse vector $\psi$.

Figure 3

Curves $\gamma$ of constant expansion ${\theta }_{\xi }$.