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

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.

💡 Research Summary

The authors present a new family of vacuum solutions to the Einstein equations with a negative cosmological constant (Λ < 0) in even spacetime dimensions (d = n + 1 ≥ 4). These solutions are generalizations of the well‑known Kerr‑Anti‑de Sitter (Kerr‑AdS) black holes, but unlike the standard higher‑dimensional Kerr‑AdS metrics whose event‑horizon cross‑sections are topologically spherical (S^{n‑1}), the new metrics possess non‑compact, negatively curved (hyperbolic) horizon sections. The construction proceeds by an analytic continuation of the standard Kerr‑AdS line element: the angular coordinates μ_i are replaced by i · \bar{μ}_i, the time coordinate by i · \bar{t}, the radial coordinate by –i · \bar{r}, and each rotation parameter a_j by –i · \bar{a}j, together with an appropriate redefinition of the mass parameter. This continuation preserves the Lorentzian signature and yields a metric \bar{g}{μν} (eqs. 3.5, 3.10) that still solves the vacuum Einstein equations with Λ = –1/(2n(n–1))ℓ^{–2}.

When all rotation parameters vanish, the metric reduces to the Birmingham‑Kottler (Schwarzschild‑AdS) solution with a hyperbolic (ε = –1) Einstein base, confirming that the construction interpolates smoothly between known spherical and hyperbolic cases. For non‑zero rotation parameters, the function \bar{U}^{–1} remains bounded, and the norm of the timelike Killing vector ∂_{\bar{t}} does not diverge at \bar{r}=0. Consequently the curvature singularity present in the non‑rotating case disappears: the spacetime is C^{2}‑extendible across \bar{r}=0, and the only genuine singularities are associated with zeros of the function \bar{V}–2\bar{m}\bar{r}. These zeros define Killing horizons. By rewriting the metric in a Kerr–Schild form (eqs. 3.22‑3.27) the authors demonstrate that the spacetime can be analytically extended across each horizon, yielding a maximal analytic extension with multiple horizon branches.

A detailed causal analysis is carried out in Section 3.8. The authors derive a sufficient condition (eq. 1.1) and a necessary‑and‑sufficient condition (Proposition 3.4) that guarantee the absence of closed timelike curves and ensure stable causality. Roughly, if the mass parameter satisfies 2| \bar{m} | < N ∏_{i=1}^{N}|\bar{a}_i|^{2/(N–1)} (with N = (n–1)/2), the spacetime is free of causality violations. This is in stark contrast to the four‑dimensional Kerr solution, where large angular momentum typically creates naked singularities and causality problems.

The surface gravity κ of each Killing horizon is computed. The outermost horizon (the largest root of \bar{V}–2\bar{m}\bar{r}=0) has non‑zero κ, corresponding to a finite Hawking temperature, while inner horizons can have κ = 0, representing extremal or “cold” horizons. The signature analysis (Section 3.7) confirms that the metric retains the (–,+,…,+) signature throughout the domain of definition.

Topologically, the horizon cross‑sections are shown to be hyperbolic Einstein manifolds. For small rotation parameters the authors prove that these sections cannot be realized on a compact manifold, implying that the black holes are intrinsically non‑compact (“hyperbolic black holes”). Whether compactifications become possible for larger rotations remains an open question.

Section 4 presents projection (Penrose‑type) diagrams that illustrate the causal structure and the arrangement of multiple horizons. The diagrams depend on the number of real roots of \bar{V}–2\bar{m}\bar{r}=0, which is analyzed in Appendix A. The maximal extensions constructed are not globally hyperbolic, but the authors discuss conditions under which the domains of outer communication might be.

Technical appendices (A–D) provide rigorous proofs of the number of roots of the horizon equation, regularity of auxiliary functions \bar{H}_±, determinant calculations, and the detailed analytic continuation arguments that guarantee the Einstein equations remain satisfied after the complex coordinate transformation.

In the concluding Section 5 the authors speculate on extensions to odd dimensions, to positive cosmological constant (Λ > 0), and to Ricci‑flat horizon geometries. They note that while the present construction is limited to even dimensions with Λ < 0, the methodology could be adapted to broader settings. Open problems include a full analysis of geodesic completeness, thermodynamic properties (entropy, mass, angular momenta) in the absence of compact boundaries, and potential holographic interpretations within the AdS/CFT correspondence.

Overall, the paper establishes the existence of a rich new class of higher‑dimensional Kerr‑AdS black holes with non‑spherical, negatively curved horizons, free of curvature singularities when all rotation parameters are non‑zero, and possessing well‑behaved causal structure under explicit parameter constraints. This advances our understanding of possible black‑hole geometries in higher dimensions and opens new avenues for exploring their physical and holographic implications.