Extremality conditions for isolated and dynamical horizons

A maximally rotating Kerr black hole is said to be extremal. In this paper we introduce the corresponding restrictions for isolated and dynamical horizons. These reduce to the standard notions for Kerr but in general do not require the horizon to be either stationary or rotationally symmetric. We consider physical implications and applications of these results. In particular we introduce a parameter $e$ which characterizes how close a horizon is to extremality and should be calculable in numerical simulations.

Figure 1

Plot of the extremality parameter, $e$, for a Kerr-Newman horizon as a function of the scaled angular momentum $J/R_H^2$ and electric charge $Q/R_H$. For nonspinning black holes, $J=0$ and $e=Q^2/R_H^2$. The extremal Kerr-Newman solutions which satisfy $(Q/R_H{)}^4+4(J/R_H^2{)}^2=1$ all have an extremality parameter $e=1$.

Figure 2

A schematic of a horizon “jump.” Matter falls into an isolated horizon causing it to expand as a dynamical horizon. However at a certain point the density of the matter is such that a new horizon forms outside the old resulting in a jump which geometrically corresponds to a timelike membrane (a timelike three-surface with ${\theta }_{(\ell )}=0$, ${\theta }_{(n)}<0$) connecting two dynamical horizons. In this figure 45° lines are null, time increases in the vertical direction, and surface area increases to the right.

Figure 3

$2J/R_H^2$ versus $\Lambda$ for the extremal Kerr-(A)dS family of black holes. For $\Lambda =0$, $2J/R_H^2=1$, as expected for the extremal, asymptotically flat Kerr solution. For $\Lambda >0$, $2J/R_H^2<1$ while for $\Lambda <0$, $J/R_H^2>1/2$ and it diverges as $\Lambda$ approaches its minimum allowed value. Therefore, for extremal Kerr anti-de Sitter black holes, the local reformulation of the Kerr bound (19) is violated.

Figure 4

Several $\rho$ that appear in the text. ${\rho }_{△}$ is the maximizing curve, $\rho$ is a typical “saw-toothed” curve, and $\rho +\delta \rho$ (which appears as a dotted line where it does not coincide with $\rho$) is a variation of that curve which increases the value of $\gamma$.