Properties of Kaluza-Klein black holes

TL;DR

This work numerically constructs static vacuum Kaluza-Klein black holes localized on a circle in six dimensions using elliptic relaxation of the Einstein equations. It develops a boundary-value formulation with a horizon at $\rho_0$ in $(\rho,\chi)$ coordinates and computes thermodynamics from horizon data, while comparing to Wiseman's non-uniform string solutions. The results show horizons become prolate and the symmetry axis decompactifies as mass grows, and the largest black holes found already exceed the mass and horizon volume of the most non-uniform strings, challenging a simple topology-changing merger. The authors discuss the possible existence of an upper mass bound, the role of axis decompactification, and the need for radius-stabilising mechanisms, suggesting extensions to 5D and stabilised models for future work.

Abstract

We detail numerical methods to compute the geometry of static vacuum black holes in 6 dimensional gravity compactified on a circle. We calculate properties of these Kaluza-Klein black holes for varying mass, while keeping the asymptotic compactification radius fixed. For increasing mass the horizon deforms to a prolate ellipsoid, and the geometry near the horizon and axis decompactifies. We are able to find solutions with horizon radii approximately equal to the asymptotic compactification radius. Having chosen 6-dimensions, we may compare these solutions to the non-uniform strings compactified on the same radius of circle found in previous numerical work. We find the black holes achieve larger masses and horizon volumes than the most non-uniform strings. This sheds doubt on whether these solution branches can merge via a topology changing solution. Further work is required to resolve whether there is a maximum mass for the black holes, or whether the mass can become arbitrarily large.

Figures (13)

• Figure 1: Schematic illustration of the boundaries we would intuitively take in $r,z$ coordinates. At least for small black holes, taking the horizon boundary to be circular would ensure the metric functions $A,B,C$ would be small and finite, as the geometry near the horizon would only be a weak distortion of 6-dimensional Schwarzschild, which we build into the metric ansatz via $A_{bg},B_{bg},C_{bg}$.
• Figure 2: Illustration of isosurfaces of $\rho, \chi$ as functions of $r,z$. The schematic boundaries of figure \ref{['fig:rzboundaries']} can be mapped to constant $\rho$ and $\chi$ values, making for convenient numerical implementation. For $\rho<0$, $\chi$ behaves as an angular coordinate, and $\rho$ as a radial one, whereas for large positive $\rho$, we find $\rho, \chi$ behave as $r, z$ respectively. We will take the horizon boundary at constant $\rho = \rho_0$, finding later that specifying $\rho_0$ determines the physical size of the black hole solution.
• Figure 3: Plots of the metric functions $A,B,C$ for a black hole solution with $\rho_0 = -0.28$. This is quite near to the maximal size ($\rho_0 = -0.18$) we were able to construct before we become limited by gradients and lattice resolution near the symmetry axis at $\chi = \pi/2$ for $\rho <0$. Already this black hole solution has equal horizon volume and mass to the most non-uniform strings compactified on the same asymptotic radius. Whilst the lattice is large, being $140*420$ in $\chi,\rho$, since $\rho_0$ is very close to zero the number of points along the symmetry axis is only around $\sim 20$. This is still enough to see good behaviour in the metric functions. Note that $B, C$ are much less than $A$ in magnitude. The maximum $\rho$ for the lattice is $\sim 5$, and not all the domain is shown as the functions simply go smoothly to zero at large $\rho$.
• Figure 4: Embedding of horizons into Euclidean space for a moderate black hole on the left, with $\rho_0 = -0.71$, and the maximal black hole to the right with $\rho = -0.18$. Whilst we interpolate the numerical functions to determine the embedding, we plot here the actual positions of the lattice points for our highest resolution to give an indication of how the resolution varies with position on the horizon. For all the solutions, we may fit a prolate ellipsoid to the horizon, the dashed line, and in all cases find perfect agreement.
• Figure 5: On the left we plot the equatorial (red) and polar (blue) radii of the ellipses fitted to the horizon embeddings, against the parameter $\rho_0$ specifying the size of the black hole. It is unclear what happens in the limit $\rho_0 \rightarrow 0$ where the $\rho$ coordinate changes topology. It would be very interesting to know if $R_{eq}, R_{polar}$ remain finite or not, and thus whether there is a maximum mass black hole or not. Since the geometry near the axis and horizon decompactifies, even though the largest black holes found have comparable radii to the corresponding asymptotic half period distance $L/2 = \pi/2$, the horizons are still quite spherical. We plot the ratio of the radii in the right hand diagram. Again it is unclear what will occur in the limit $\rho_0 \rightarrow 0$.