Abstract
We demonstrate that five-dimensional, asymptotically flat, stationary and bi-axisymmetric, 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.
Highlights
The axes and horizons and the orbit space for such spacetimes can be identified with a half-plane {(ρ, z) | ρ > 0}
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 this note we will consider the most basic open question in this context: do regular black holes with lens space topology exist? A number of attempts at constructing such black lens solutions have resulted in singular spacetimes, the mildest being a conical singularity on the inner axis [20,21,22]
Summary
We consider the rod structure for the simplest L(n, 1) black lens as in figure 1.1 we have four rods: IL = (−∞, z1), IH = (z1, z2), ID = (z2, z3) and IR = (z3, ∞), with z1 < z2 < z3, where IH is a horizon rod and IL, ID, IR are axis rods. Up to irrelevant discrete choices this is the most general rod structure with one horizon and one finite axis rod, that is compatible with asymptotic flatness and obeys the admissibility condition det(vD, vR) = ±1 (this latter condition ensures the absence of orbifold singularities at z = z3) [8]. For n = ±1 the horizon topology L(1, 1) ∼= S3 is spherical, the rod structure is distinct to that of the Myers-Perry black hole
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