All the Four Dimensional Static, Spherically Symmetric Solutions of Abelian Kaluza-Klein Theory

TL;DR

This paper derives all static, spherically symmetric solutions in four-dimensional Abelian Kaluza-Klein theory by applying specific symmetry transformations to the Schwarzschild solution, revealing various black hole configurations.

Contribution

It explicitly constructs all such solutions using symmetry transformations, including parameters for mass, charges, and Taub-NUT charge, and classifies their black hole types.

Findings

Non-extreme black holes resemble Reissner-Nordström or Schwarzschild solutions.

Supersymmetric extreme black holes have singularities or naked singularities.

Non-supersymmetric extreme solutions resemble extreme Reissner-Nordström black holes.

Abstract

We present the explicit form for all the four dimensional, static, spherically symmetric solutions in $(4+n)$-d Abelian Kaluza-Klein theory by performing a subset of $SO(2,n)$ transformations corresponding to four $SO(1,1)$ boosts on the Schwarzschild solution, supplemented by $SO(n)/SO(n-2)$ transformations. The solutions are parameterized by the mass $M$, Taub-Nut charge $a$, $n$ electric $\vec{\cal Q}$ and $n$ magnetic $\vec{\cal P}$ charges. Non-extreme black holes (with zero Taub-NUT charge) have either the Reissner-Nordstr\" om or Schwarzschild global space-time. Supersymmetric extreme black holes have a null or naked singularity, while non-supersymmetric extreme ones have a global space-time of extreme Reissner-Nordstr\" om black holes.