Addendum to Supersymmetric Dyonic Black Holes in Kaluza-Klein Theory

TL;DR

This paper completes the classification of supersymmetric black holes in 4D Kaluza-Klein theory, showing charge constraints and how all solutions relate via SO(n) rotations to simpler seed solutions.

Contribution

It demonstrates that all supersymmetric black holes in this setting can be generated from basic solutions using SO(n) rotations, revealing charge constraints and solution structure.

Findings

Electric and magnetic charges satisfy a orthogonality constraint.

All solutions are related by SO(n) rotations to basic seed solutions.

The metric and internal volume remain unaffected by these rotations.

Abstract

We complete the study of 4-dimensional (4-d), static, spherically symmetric, supersymmetric black holes (BH's) in Abelian $(4+n)$-d Kaluza-Klein theory, by showing that for such solutions $n$ electric charges $\vec{\cal Q} \equiv (Q_1,...,Q_n)$ and $n$ magnetic charges $\vec{\cal P} \equiv (P_1,...,P_n)$ are subject to the constraint $\vec{\cal P}\cdot \vec{\cal Q}=0$. All such solutions can be obtained by performing the SO(n) rotations, which do not affect the 4-d space-time metric and the volume of the internal space, on the supersymmetric $U(1)_M\times U(1)_E$ BH's, {\it i.e.}, supersymmetric BH's with a diagonal internal metric.