Balancing Static Vacuum Black Holes with Signed Masses in 4 and 5 Dimensions

TL;DR

This paper constructs new static vacuum solutions in 4 and 5 dimensions representing superpositions of positive and negative mass black holes, analyzing their properties and the conditions for balancing these configurations.

Contribution

It introduces a novel class of solutions with signed masses, including negative mass black rings, and provides detailed analysis of singularities and balancing conditions in higher dimensions.

Findings

Negative mass singularities are manageable in the solutions.

Balancing of black holes is impossible in 4D but possible in 5D.

New negative mass black ring solutions are constructed.

Abstract

We construct a new set of asymptotically flat, static vacuum solutions to the Einstein equations in dimensions 4 and 5, which may be interpreted as a superposition of positive and negative mass black holes. The resulting spacetimes are axisymmetric in 4-dimensions and bi-axisymmetric in 5-dimensions, and are regular away from the negative mass singularities, for instance conical singularities are absent along the axes. In 5-dimensions, the topologies of signed mass black holes used in the construction may be either spheres $S^3$ or rings $S^1 \times S^2$; in particular, the negative mass static black ring solution is introduced. A primary observation that facilitates the superposition is the fact that, in Weyl-Papapetrou coordinates, negative mass singularities arise as overlapping singular support for a particular type of Green's function. Furthermore, a careful analysis of conical singularities along axes is performed, and formulas are obtained for their propagation across horizons, negative mass singularities, and corners. The methods are robust, and may be used to construct a multitude of further examples. Lastly, we show that balancing does not occur between any two signed mass black holes of the type studied here in 4 dimensions, while in 5 dimensions two-body balancing is possible.