Towards a uniqueness theorem for static black holes in Kaluza-Klein theory with small circle size

Kaluza-Klein theory, by which we mean vacuum gravity in 5-dimensions, with asymptotics that are a product of a circle with Minkowski spacetime, has a variety of different static black hole solutions; localized black holes and the homogeneous and inhomogeneous black strings. There is currently no uniqueness theorem for the solutions, and for fixed circle size multiple solutions with the same mass co-exist. Intuitively for small circle sizes we might expect the theory truncates to become 4-dimensional, and correspondingly the only black holes are the homogeneous black strings. Thus we conjecture that for fixed mass and sufficiently small circle size, the only black holes are homogeneous ones. Here we give evidence that this is indeed the case. Firstly we introduce a toy scalar field model with a potential that allows tachyonic behaviour. Putting this theory on a product of Minkowski with a circle gives an analogous set of static homogeneous and inhomogeneous solutions to that of the black holes. We prove that solutions must be homogeneous for small circle sizes, the analog of our conjecture for this toy model. A weaker statement that is straightforward to derive is a bound on how inhomogeneous a solution can be - putting this scalar theory in a large but finite cavity, a norm of the wavefunction of the Kaluza-Klein modes can be shown to vanish in the small circle limit. Turning to the full gravitational theory, we employ a metric ansatz that imposes static axisymmetry, encompasses the homogeneous and inhomogeneous black strings (but not the localized solutions) and allows us to measure inhomogeneity of a solution. Employing a finite cavity and imposing boundary conditions that are compatible with homogeneity we show a similar result; the norm of certain Kaluza-Klein modes is bounded by the circle size, providing evidence that our conjecture is true.