On the nonexistence of a vacuum black lens

TL;DR

This paper proves that five-dimensional vacuum black holes with lens space topology and simple rod structure cannot exist without singularities, using analytical and numerical methods to demonstrate unavoidable conical singularities.

Contribution

It provides a proof of nonexistence for certain five-dimensional black lens solutions and shows that all such configurations have unavoidable singularities.

Findings

No smooth black lens solutions with the specified topology exist.

All solutions exhibit conical singularities on the inner axis.

Numerical evidence supports nonexistence in the doubly spinning case.

Abstract

We demonstrate that five-dimensional, asymptotically flat, stationary and biaxisymmetric, vacuum black holes with lens space $L(n, 1)$ topology, possessing the simplest rod structure, do not exist. In particular, we show that the general solution on the axes and horizon, which we recently constructed by exploiting the integrability of this system, must suffer from a conical singularity on the inner axis component. We give a proof of this for two distinct singly spinning configurations and numerical evidence for the generic doubly spinning solution.