Black Holes in Higher-Dimensional Gravity

TL;DR

The work surveys the phase structure of black holes in higher-dimensional vacuum gravity, focusing on Kaluza-Klein black holes and stationary asymptotically flat solutions. It combines analytical constructions (uniform/non-uniform strings, localized black holes, multi-black-hole configurations, thin black rings) with perturbative methods (matched asymptotic expansions) and numerical insights to map $(oldsymbol{mu},n)$-type phase diagrams and torus generalizations. Key contributions include the explicit construction of multi-black-hole configurations on the cylinder, Newtonian thermodynamics for these systems, and a leading-order thin black-ring solution in any $D\, ext{(} \,D\ge 5)$, plus a unifying phase-diagram picture that links KK and rotating black holes via a membrane/torus analogy. The findings illuminate rich, dimension-dependent phase structures, topology-changing transitions, and potential implications for stability analyses and gauge/gravity duality in higher dimensions.

Abstract

These lectures review some of the recent progress in uncovering the phase structure of black hole solutions in higher-dimensional vacuum Einstein gravity. The two classes on which we focus are Kaluza-Klein black holes, i.e. static solutions with an event horizon in asymptotically flat spaces with compact directions, and stationary solutions with an event horizon in asymptotically flat space. Highlights include the recently constructed multi-black hole configurations on the cylinder and thin rotating black rings in dimensions higher than five. The phase diagram that is emerging for each of the two classes will be discussed, including an intriguing connection that relates the phase structure of Kaluza-Klein black holes with that of asymptotically flat rotating black holes.

Figures (5)

• Figure 1: Black hole and string phases for $d=5$, drawn in the $(\mu,n)$ phase diagram. The horizontal (dotted) line is the uniform string branch. The rightmost solid branch emanating from this at the Gregory-Laflamme point is the non-uniform string branch and the rightmost dashed branch starting in the origin is the localized black hole branch. The solid and dashed branches to the left are the $k=2$ copies of the non-uniform and localized branch. The results strongly suggest that the black hole and non-uniform black string branches meet.
• Figure 2: $a_H (\ell)$ phase diagram in seven dimensions ($\mathcal{M}^5 \times \mathbb{T}^2$) for Kaluza-Klein black hole phases with one uniform direction. Shown are the uniform black membrane phase (dotted), the non-uniform black membrane phase (solid) and the localized black string phase (dashed). For the latter two phases, we have also shown their $k=2$ copy. The non-uniform black membrane phase emanates from the uniform black membrane phase at the GL point $\ell_{\rm GL} = 0.811$, while the $k=2$ copy starts at the 2-copied GL point $\ell_{\rm GL}^{(2)} = \sqrt{2} \ell_{\rm GL} =1.15$. This figure is representative for the phase diagram of phases on $\mathcal{M}^{D-2} \times \mathbb{T}^2$ for all $6 \leq D \leq 14$. Reprinted from Ref. obeEmparan:2007wm.
• Figure 3: Area vs spin for fixed mass, $a_H(j)$, in seven dimensions. For large $j$, the thin curve is the result for thin black rings and is extrapolated here down to $j\sim O(1)$. The thick curve is the exact result for the MP black hole. The gray line corresponds to the conjectured phase of pinched black holes (see Sec. \ref{['obesec:phas']}), which branch off tangentially from the MP curve at a value $j_{\rm GL}> j_{\rm mem}$. At any given dimension, the phases should not necessarily display the swallowtail as shown in this diagram, but could also connect more smoothly via a pinched black hole phase that starts tangentially in $j_{\rm GL}$ and has increasing $j$. Reprinted from Ref. obeEmparan:2007wm.
• Figure 4: Correspondence between phases of black membranes wrapped on a $\mathbb{T}^2$ of side $L$ (left) and fastly-rotating MP black holes with rotation parameter $a\sim L\geq r_0$ (right: must be rotated along a vertical axis): (i) Uniform black membrane and MP black hole. (ii) Non-uniform black membrane and pinched black hole. (iii) Pinched-off membrane and black hole. (iv) Localized black string and black ring. Reprinted from Ref. obeEmparan:2007wm.
• Figure 5: Proposal for the phase diagram of thermal equilibrium phases of rotating black holes in $D\geq 6$ with one angular momentum. The solid lines and figures have significant arguments in their favor, while the dashed lines and figures might not exist and admit conceivable, but more complicated, alternatives. Some features have been drawn arbitrarily, $e.g.$ at any given bifurcation and in any dimension one may either have smooth connections or swallowtails with cusps. If thermal equilibrium is not imposed, the whole semi-infinite strip $0<a_H <a_H(j=0)$, $0\leq j<\infty$ is covered, and multi-rings are possible. Reprinted from Ref. obeEmparan:2007wm.