The Rotating Dyonic Black Holes Of Kaluza-Klein Theory
Dean Rasheed
TL;DR
This work derives the most general rotating dyonic black hole solutions in five-dimensional Kaluza–Klein theory (b=√3), showing how dimensional reduction yields four-dimensional gravity coupled to electromagnetism and a dilaton. By using restricted SO(1,2) transformations on the Kerr solution within a three-dimensional SL(3,ℝ)/SO(3) sigma-model framework, the authors construct explicit metrics with four charges (M,P,Q,J) and a dilaton charge Σ, and they analyze their horizons, ergoregions, and extremal limits. The extreme solutions separate into two distinct surfaces S and W with qualitatively different properties, including zero horizon angular velocity on W despite nonzero ADM angular momentum, and the surprising possibility of increasing angular momentum on certain extremal states without losing extremality. The paper also derives the gyromagnetic and gyroelectric ratios (which can be unbounded), provides generalized Smarr and first-law relations, and discusses stability and thermodynamics, placing KK dyons in a broader context of black hole physics and string-inspired theories.
Abstract
The most general electrically and magnetically charged rotating black hole solutions of 5 dimensional \KK\ theory are given in an explicit form. Various classical quantities associated with the black holes are derived. In particular, one finds the very surprising result that the gyromagnetic and gyroelectric ratios can become {\tenit arbitrarily large}. The thermodynamic quantities of the black holes are calculated and a Smarr-type formula is obtained leading to a generalized first law of black hole thermodynamics. The properties of the extreme solutions are investigated and it is shown how they naturally separate into two classes. The extreme solutions in one class are found to have two unusual properties: (i). Their event horizons have zero angular velocity and yet they have non-zero ADM angular momentum. (ii). In certain circumstances it is possible to add angular momentum to these extreme solutions without changing the mass or charges and yet still maintain an extreme solution. Regarding the extreme black holes as elementary particles, their stability is discussed and it is found that they are stable provided they have sufficient angular momentum.