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We construct a new set of asymptotically flat, static vacuum solutions to theEinstein equations in dimensions 4 and 5, which may be interpreted as asuperposition of positive and negative mass black holes. The resultingspacetimes are axisymmetric in 4-dimensions and bi-axisymmetric in5-dimensions, and are regular away from the negative mass singularities, forinstance conical singularities are absent along the axes. In 5-dimensions, thetopologies of signed mass black holes used in the construction may be eitherspheres $S^3$ or rings $S^1 \times S^2$; in particular, the negative massstatic black ring solution is introduced. A primary observation thatfacilitates the superposition is the fact that, in Weyl-Papapetrou coordinates,negative mass singularities arise as overlapping singular support for aparticular type of Green's function. Furthermore, a careful analysis of conicalsingularities along axes is performed, and formulas are obtained for theirpropagation across horizons, negative mass singularities, and corners. Themethods are robust, and may be used to construct a multitude of furtherexamples. Lastly, we show that balancing does not occur between any two signedmass black holes of the type studied here in 4 dimensions, while in 5dimensions two-body balancing is possible.

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