General black holes in Kaluza-Klein theory

TL;DR

This work surveys general black holes in five-dimensional Kaluza-Klein theory, emphasizing both homogeneous solutions invariant along the KK circle and inhomogeneous configurations. It shows how KK reduction yields a four-dimensional Einstein–Maxwell–dilaton framework and derives charged and rotating KK black holes, including extremal limits and entropy relations, with key results such as G_4 M, Q, P, J and the slow/fast rotation extremal behavior. It then maps the static solution space beyond translation symmetry, detailing localized black holes and inhomogeneous black strings via perturbative and numerical methods, and highlights a topology-changing merger near M_* that connects branches; stability analyses reveal a rich pattern, including a dimension-dependent transition around D_* ≈ 13.5. Overall, the paper demonstrates nonuniqueness of static KK black holes, intricate horizon topologies, and the intricate interplay between mass, charges, rotation, and extra-dimensional structure in determining black hole solutions and their stability.

Abstract

A brief review is given of black holes in Kaluza-Klein theory. This includes both solutions which are homogeneous around the compact extra dimension and those which are not.

Figures (3)

• Figure 1.: Plot of area $A$ against mass $M$ for fixed Kaluza-Klein circle size $L$ for the homogeneous (dashed) and inhomogeneous (solid thin line) black strings and the localized black holes (solid thick line). The curves for the latter two are the numerical solutions of Headrick:2009pv. The inhomogeneous black strings and localized black holes are compatible with a merger at $G_5 M_{\star} / L^2 \simeq 0.12$, and there is a maximum mass localized solution, with $G_5 M_{max} / L^2 \simeq 0.17$. The gray dotted line is the small localized black hole approximation in equation (\ref{['eq:smallloc4']}) and we see the approximation is excellent for increasing mass up to $M \sim M_{max}$.
• Figure 2.: Plot of inverse temperature $2\pi/\kappa$ against mass $M$ for fixed circle size $L$ for the same solutions as in the previous figure. The labels 'A' to 'I' denote solutions whose horizon embeddings are displayed in the following figure. Again we see consistency with a merger. We also note that there is a minimum surface gravity solution at $M_{\kappa}$ (near the label 'C') which is rather close to, but slightly less than $M_{max}$. The perturbative approximation in equation (\ref{['eq:smallloc4']}) is also plotted and again very good agreement is seen for localized solutions with increasing mass up to around $M \sim M_{max}$.
• Figure 3.: Figure showing the spatial geometry of the horizon for a number of localized black holes and inhomogeneous black strings all with the same asymptotic circle size. The embeddings are labelled 'A' to 'I' and correspond to the solutions annotated in the previous figure. We emphasize that these geometries are those actually found in the numerical solutions of Headrick:2009pv. For the localized black holes the proper length of the axis of symmetry is also depicted.