Topology, Entropy and Witten Index of Dilaton Black Holes

TL;DR

Gibbons and Kallosh investigate the topology, entropy, and Witten index of dilaton black holes. They show that extreme dilaton black holes require an inner boundary to yield χ=0, while non-extreme holes have χ=2, leading to vanishing entropy for all extreme holes. The work connects these geometric insights to the vanishing of quantum corrections and recasts the extreme-hole path integral as a Witten index, then extends the discussion to multi-hole moduli spaces and supersymmetric quantum mechanics on black-hole moduli spaces, including torsionful, non-HyperKähler geometries. They also explore the potential fission of extreme black holes and the associated energy release, highlighting a topological perspective on black-hole thermodynamics and quantum behavior with possible implications for string theory and soliton physics.

Abstract

We have found that for extreme dilaton black holes an inner boundary must be introduced in addition to the outer boundary to give an integer value to the Euler number. The resulting manifolds have (if one identifies imaginary time) topology $S^1 \times R \times S^2 $ and Euler number $χ= 0$ in contrast to the non-extreme case with $χ=2$. The entropy of extreme $U(1)$ dilaton black holes is already known to be zero. We include a review of some recent ideas due to Hawking on the Reissner-Nordström case. By regarding all extreme black holes as having an inner boundary, we conclude that the entropy of {\sl all} extreme black holes, including $[U(1)]^2$ black holes, vanishes. We discuss the relevance of this to the vanishing of quantum corrections and the idea that the functional integral for extreme holes gives a Witten Index. We have studied also the topology of ``moduli space'' of multi black holes. The quantum mechanics on black hole moduli spaces is expected to be supersymmetric despite the fact that they are not HyperKähler since the corresponding geometry has torsion unlike the BPS monopole case. Finally, we describe the possibility of extreme black hole fission for states with an energy gap. The energy released, as a proportion of the initial rest mass, during the decay of an electro-magnetic black hole is 300 times greater than that released by the fission of an ${}^{235} U$ nucleus.

Figures (4)

• Figure 1: The geometry of the $r-\tau$ space for non-extreme black holes. The circles are lines of constant $r$. The shaded region has an outer boundary but no inner boundary. The vector field ${\partial \over \partial \tau}$ has a fixed point set at the horizon. The space has the topology ${ R}^2$. Thus $\chi=1$.
• Figure 2: The geometry of the $r-\tau$ space for the extreme black holes. The circles are lines of constant $r$. The shaded region has an outer and inner boundary. The vector field ${\partial \over \partial \tau}$ has no fixed points. The surface has an infinitely long spine, and has topology $S^1 \times{ R} \sim R^2 - \{ 0 \}$. Thus $\chi = 0$.
• Figure 3: The entropy $S$ of the dilaton black holes as the function of the electric $Q$ and magnetic $P$ charges. The shaded region presents the entropy of non-extreme and near extreme $[U(1)]^2$ black holes, whose topology is shown on Fig. 1. The bold line shows the vanishing entropy of extreme black holes, whose topology is shown on Fig. 2.
• Figure 4: The relative moduli space for two extreme holes with $0\leq a < {1\over 3}$. What is illustrated is the covering space $\tilde{{\cal M}}^{\rm rel} _2$. Each circle represents a 2-sphere. To obtain ${\cal M} ^ {\rm rel} _2$ one must identify antipodal points on these spheres. The identified space is non-orientable and has $\chi = 1$