Dynamical black holes: Approach to the final state

Since black holes can be formed through widely varying processes, the horizon structure is highly complicated in the dynamical phase. Nonetheless, as numerical simulations show, the final state appears to be universal, well described by the Kerr geometry. How are all these large and widely varying deviations from the Kerr horizon washed out? To investigate this issue, we introduce a well-suited notion of horizon multipole moments and equations governing their dynamics, thereby providing a coordinate- and slicing-independent framework to investigate the approach to equilibrium. In particular, our flux formulas for multipoles can be used as analytical checks on numerical simulations and, in turn, the simulations could be used to fathom possible universalities in the way black holes approach their final equilibrium.

Figure 1

A quasilocal horizon $M$. The past portion of $M$ consists of a dynamical horizon $H$: This portion is spacelike and foliated by marginally trapped surfaces $S$. ${\widehat{\tau }}^a$ is the unit timelike normal to $H$ and ${\widehat{r}}^a$ the unit spacelike normal within $H$ to the foliation. Although $H$ is spacelike, motions along ${\widehat{r}}^a$ can be regarded as time evolution with respect to observers at infinity. $H$ joins on to an isolated horizon $\Delta$ in the future, representing the equilibrium state of the black hole. $\Delta$ is null, endowed with a preferred null normal ${\overline{\ell }}^a$. The transition from $H$ to $\Delta$ occurs at $S_o$.