Counting the Microstates of a Kerr Black Hole

TL;DR

The paper demonstrates that the entropy of an extremal Kerr black hole, $S = 2\pi|J|$, can be reproduced by counting microstates once Kerr is lifted to M-theory and mapped to a Kaluza-Klein black hole whose D0-D6 microstates are counted. This is accomplished via a three-step chain of dualities and a discrete transformation that exchange angular momentum for charge, ultimately relating the Kerr state to a D0-D6 system with large charges where the microstate count yields $S = \pi N_0 N_6 = 2\pi|J'|$. The work also shows that horizon topology is not invariant under these dualities in M-theory, with dual descriptions yielding different topologies (e.g., $S^2$ vs $S^3$). Finally, the authors discuss limitations and open questions, including extension to other $J$, higher dimensions, non-extremal cases, as well as different compactifications.

Abstract

We show that an extremal Kerr black hole, appropriately lifted to M-theory, can be transformed to a Kaluza-Klein black hole in M-theory, or a D0-D6 charged black hole in string theory. Since all the microstates of the latter have recently been identified, one can exactly reproduce the entropy of an extremal Kerr black hole. We also show that the topology of the event horizon is not well defined in M-theory.

Figures (2)

• Figure 1: The branes wrap a rational direction $k/l$ of the torus (in the figure, $k=3$, $l=1$), so there are $2kl$ intersection points on each $T^2$.
• Figure 2: By taking $R$ large, the geometry becomes a Myers-Perry black hole at the tip of Taub-NUT. A simple reflection now changes configuration (a) with $N_0 = 0$ and $J\ne 0$ into (b) with $N_0 \ne 0$ and $J=0$. Although it appears that the black hole has been rotated by $90^o$, this is just an artifact of the projection down to two dimensions.