Black hole uniqueness theorems and new thermodynamic identities in eleven dimensional supergravity

Abstract

We consider stationary, non-extremal black holes in 11-dimensional supergravity having isometry group $\mathbb{R} \times U(1)^8$. We prove that such a black hole is uniquely specified by its angular momenta, its electric charges associated with the various 7-cycles in the manifold, together with certain moduli and vector valued winding numbers characterizing the topological nature of the spacetime and group action. We furthermore establish interesting, non-trivial, relations between the thermodynamic quantities associated with the black hole. These relations are shown to be a consequence of the hidden $E_{8(+8)}$ symmetry in this sector of the solution space, and are distinct from the usual "Smarr-type" formulas that can be derived from the first law of black hole mechanics. We also derive the "physical process" version of this first law applicable to a general stationary black hole spacetime without any symmetry assumptions other than stationarity, allowing in particular arbitrary horizon topologies. The work terms in the first law exhibit the topology of the horizon via the intersection numbers between cycles of various dimensions.