Extremal vacuum black holes in higher dimensions

TL;DR

<3-5 sentence high-level summary> This work develops a near-horizon analysis for extremal vacuum black holes in dimensions $D>5$, proving an $SO(2,1)$ symmetry under special rotational enhancements and constructing explicit near-horizon geometries for extremal MP black holes and boosted MP strings. It proposes a deep connection between extremal black rings in odd dimensions and tensionless boosted MP strings, arguing that the near-horizon geometry of a higher-dimensional extremal ring is globally isometric to that of the corresponding string, and uses this to derive phase diagrams and conserved charges. The results illuminate the structure and non-uniqueness landscape of higher-dimensional extremal objects and provide a framework for understanding (and potentially counting) microstates in regimes lacking supersymmetry. The analysis also clarifies which physical quantities can be inferred from near-horizon data alone and highlights the role of asymptotic data in fixing mass and angular velocities.

Abstract

We consider extremal black hole solutions to the vacuum Einstein equations in dimensions greater than five. We prove that the near-horizon geometry of any such black hole must possess an SO(2,1) symmetry in a special case where one has an enhanced rotational symmetry group. We construct examples of vacuum near-horizon geometries using the extremal Myers-Perry black holes and boosted Myers-Perry strings. The latter lead to near-horizon geometries of black ring topology, which in odd spacetime dimensions have the correct number rotational symmetries to describe an asymptotically flat black object. We argue that a subset of these correspond to the near-horizon limit of asymptotically flat extremal black rings. Using this identification we provide a conjecture for the exact ``phase diagram'' of extremal vacuum black rings with a connected horizon in odd spacetime dimensions greater than five.

Figures (8)

• Figure 1: The phase space of extremal six dimensional MP black holes. Left: The extremal locus in the $(j_1, j_2)$ plane. Right: The area as a function of $j_1$; note that the maximum area configuration is the symmetric one, $j_1 = j_2$.
• Figure 2: The phase space of extremal seven dimensional MP black holes. Left: The extremal locus as a function of the reduced angular momenta variables $(j_1, j_2, j_3)$. Here the allowed region is non-compact, with the arms of the fan extending off to infinity. Also note that as one of the $j_i$, say $j_1$ gets large, one the surface projected onto the $(j_2,j_3)$ plane starts to resemble a rescaled version of the five dimensional MP extremal locus, see Fig. \ref{['fig:fivedphasediag']}. Right: The phase plot $a_H(j_1,j_2)$. Note that the maximum occurs at the symmetric point $j_1 = j_2 = j_3$ and the limiting behaviour of the surface at large values of $j_i$ coincides with that of the five dimensional MP curve, see Fig. \ref{['fig:fivedphasediag']}
• Figure 3: The phase space of extremal eight dimensional MP black holes. Left: The extremal locus as a function of the reduced angular momenta variables $(j_1, j_2, j_3)$. Right: The phase plot $a_H(j_1,j_2)$. The maximum occurs at the symmetric point $j_1 = j_2 = j_3$ and the limiting behaviour of the surface at large values of one of the $j_i$ coincides with that of the six dimensional MP curve. We can also take two angular momenta large in which case the solution behaves like a four dimensional Kerr black hole.
• Figure 4: The extremal loci of MP black holes illustrating the membrane limit. Left: The behaviour for 7d MP; the curves correspond to increasing values of $j_3$ moving from right to left. As $j_3$ gets large the extremal locus degenerates into a straight line as for the 5d MP, cf., (\ref{['fdmpextl']}) and Fig. \ref{['fig:fivedphasediag']}. Right: Eight dimensional MP where again we plot the curves are for different $j_3$ values (increasing from right to left) -- compare the limiting behaviour with the 6d MP extremal locus, Fig. \ref{['fig:sixdextl']}.
• Figure 5: Phase diagram for extremal MP black holes and doubly spinning black rings in $D=5$. The gray curve corresponds to the MP black hole and the black one to the black ring. Left: Plot of the $j_\phi$ vs. $j_\psi$ curve, where $j_\phi$ is the ${\bf S}^{2}$ angular momentum. Note that the point of intersection of the black ring and the black hole is excluded by the bounds on the angular momenta for the ring. Right: Plot of the $a_\textrm{H}$ vs. $j_\psi$ curve.