Uniqueness theorem for Kaluza-Klein black holes in five-dimensional minimal supergravity

We show a uniqueness theorem for Kaluza-Klein black holes in the bosonic sector of five-dimensional minimal supergravity. More precisely, under the assumptions of the existence of two commuting axial isometries and a nondegenerate connected event horizon of the cross-section topology $S^3$, or lens space, we prove that a stationary charged rotating Kaluza-Klein black hole in five-dimensional minimal supergravity is uniquely characterized by its mass, two independent angular momenta, electric charge, magnetic flux, and nut charge, provided that there exists neither a nut nor a bolt (a bubble) in the domain of outer communication. We also show that under the assumptions of the same symmetry, same asymptotics, and the horizon cross section of $S^1\times S^2$, a black ring within the same theory—if it exists—is uniquely determined by its dipole charge and rod intervals besides the charges and magnetic flux.

Figure 1

The rod structures of spacetimes with Kaluza-Klein asymptotics: (a) the Gross-Perry-Sorkin monopole, (b) the black hole, and (c) the black ring. Here, the solid finite rods correspond to the horizons, the assigned vectors on the spacelike rods denote the rod vectors; i.e., the pairs of numbers $(1,\pm N)$ means that the Killing vectors, $v=(\partial /\partial \varphi )\pm N(\partial /\partial w)$, have fixed points there—more precisely, the metric, $g_{ij}(0,z)$, has an eigenvalue zero for a given $z$. See Ref. 77 about the rod structures of well-known gravitational instantons—for example, Euclidean self-dual Taub-NUT space—with $U(1)\times U(1)$ symmetry and its classification.