Kerr-AdS type higher dimensional black holes with non-spherical cross-sections of horizons

TL;DR

This work shows that in all even spacetime dimensions a family of Kerr–AdS–like black holes with negatively curved, non-spherical horizons can be constructed without curvature singularities when all rotation parameters are nonzero. The construction uses an analytic continuation of the higher-dimensional Kerr–AdS metrics and a Kerr–Schild extension to achieve well-behaved horizons and a timelike conformal boundary, with horizon cross-sections that are negatively curved and non-compact. The paper provides detailed analyses of causality, horizons, and asymptotics, including stable causality criteria and projection diagrams that visualize the global structure; it also proves that compact horizon cross-sections are incompatible with nonzero rotation, underlining the novel horizon topology. Limitations include the absence of a general geodesic completeness result and the restriction to even dimensions with negative cosmological constant; the odd-dimensional and positive-$\\Lambda$ extensions are not developed, highlighting directions for future work.

Abstract

We construct, in even spacetime dimensions, a family of singularity-free Kerr-Anti-de Sitter-like black holes with negatively curved cross-sections of conformal infinity and non-spherical cross-sections of horizons.

Figures (5)

• Figure 3.1: Examples of the function $F$ with three (top) or four (bottom) zeros. In all plots, $\ell = 1$.
• Figure 4.1: The projection diagram for an extended spacetime with $\bm>0$, in which there are no closed causal curves, with $\bar{V} - 2 \bm \bar{r}$ having two zeros $-\ell <\bar{r}_- <\ell < \bar{r}_+$. Both horizons are of bifurcate-type, with non-zero surface gravity. The light blue regions are where $\nabla\bar{r}$ is timelike and the pink ones where $\nabla\bar{t}$ is timelike. The solid-blue curves represent the level sets of constant $\bar{r}$ while the dashed-red those of constant $\bar{t}$. The Kerr--Schild extension is indicated by the regions marked I-III. The diagram can be continued indefinitely in the vertical direction so that a shift of the figure by two diamonds generates a $\mathbbm{Z}$-group of isometries. Distinct maximal analytic extensions can be obtained by quotienting the spacetime by distinct subgroups of $\mathbbm{Z}$, leading to closed timelike curves through every point. Further distinct maximal analytic extensions can be obtained by removing, e.g., one or more bifurcation surfaces and passing to covering spaces.
• Figure 4.2: The projection diagram for an extended spacetime with $\bm>0$ in which there are no closed causal curves and $\bar{V} - 2 \bm \bar{r}$ has three zeros $-\ell <\bar{r}_1 < \bar{r}_2 <\ell < \bar{r}_3$, with the zero at $\bar{r}_2$ of order two. The coloring of diamonds and the dotting of curves are as in Figure \ref{['F17IX24.1']}. The extension can be continued in all directions. The horizons at $\bar{r}_2$ are degenerate, in the sense that they have vanishing surface gravity $\kappa$, the other ones have non-zero $\kappa$.
• Figure 4.3: The projection diagram for an extended spacetime with $\bm>0$ in which there are no closed causal curves and $\bar{V} - 2 \bm \bar{r}$ has four zeros $-\ell< \bar{r}_1 < \bar{r}_2< \bar{r}_3 <\ell < \bar{r}_4$. The extension can be continued to the whole Minkowskian plane in all directions. All horizons are of bifurcate-type, with non-zero surface gravity. The nature of the diamonds and of the curves are as in Figure \ref{['F17IX24.1']}. The Kerr--Schild extension is the union of the regions marked I to V. The extension is symmetric with respect to reflections across vertical and horizontal lines passing through the bifurcation points of the Killing horizons. Shifts of the plane by two diamonds along the vertical axis and by two diamonds along the horizontal axis generate a $\mathbbm{Z}^2$ group of isometries of the depicted spacetime when extended to the whole of $\mathbb{R}^2$; taking quotients by distinct subgroups of $\mathbbm{Z}^2$ leads to distinct, non-isometric, analytic extensions of the original metric, possibly but not necessarily introducing causality violations. Further distinct extensions can be obtained by passing to distinct coverings of the projection plane.
• Figure 4.4: Projection diagrams for the Kerr--Newman anti-de Sitter metrics with two distinct zeros of the Boyer-Lindquist function $\Delta_r$ (left diagram) and one double zero (right diagram), from COS, generated using SzybkaOelz. The arrows represent the orientation of the orbits of the Killing vector $\partial_t$. The shaded area is the projection of the Carter time-machine region, including the singular ring, and should be removed from the diagram for a faithful representation of causality. An identical diagram is obtained for our metrics in the two-zeros case of Figure \ref{['F17IX24.1']} when causality violations occur with $\bm>0$.