The emphasis of this article is on classical properties of exact higher-dimensional black-hole solutions. We devote most space to a rather pedagogical discussion of vacuum solutions. Since this includes black rings, there is some overlap with our earlier review . The present review discusses material that has appeared since , in particular the “doubly spinning” black-ring solution of . However, we shall not discuss several aspects of black-ring physics that were dealt with at length in , for example, black-ring microphysics. On the other hand, we present some new material: Figures 3, 6, 7, 13, and 14 describe the physical parameter ranges (phase space) of higher-dimensional black holes, and Figure 4 for the area of 5D Myers–Perry solutions, have not been presented earlier. Some of our discussion of the properties of the solutions is also new.
Our discussion of nonvacuum black holes is less pedagogical than that of the vacuum solutions. It is essentially a survey of the literature. In going beyond vacuum solutions, we had to decide what kinds of matter fields to consider. Since much of the motivation for the study of extra dimensions comes from string theory, we have restricted ourselves to considering black-hole solutions of supergravity theories known to arise as consistent truncations of supergravity. We consider both asymptotically flat and asymptotically anti-de Sitter black holes.
In the asymptotically flat case, we consider only solutions of maximal supergravity theories arising from the toroidal reduction of supergravity to five or more dimensions. In particular, this implies that in five dimensions we demand the presence of a Chern–Simons term for the gauge field, with a precise coefficient. A review of charged rotating black holes with other values for the Chern–Simons coupling can be found in .
In the asymptotically AdS case, we consider solutions of gauged supergravity theories arising from the dimensional reduction of supergravity on spheres, in particular the maximal-gauged supergravity theories in . Obviously does not fall within our “higher-dimensional” remit but asymptotically AdS black holes are not as familiar as their asymptotically-flat cousins so it seems worthwhile reviewing them here. In the AdS case, several different asymptotic boundary conditions are of physical interest. We consider only black holes obeying standard “normalizable” boundary conditions . Note that all known black-hole solutions satisfying these restrictions involve only Abelian gauge fields.
Important related subjects that we do not discuss include: black holes in brane-world scenarios (reviewed in ); black holes in spacetimes with Kaluza–Klein asymptotics (reviewed in ), and in general black holes with different asymptotics than flat or AdS; black holes in higher-derivative theories [199, 33]; black-hole formation at the LHC or in cosmic rays, and the spectrum of their radiation (reviewed in [30, 155]).
Sections 3 to 6 are devoted to asymptotically-flat vacuum solutions: Section 3 introduces basic notions and solutions, in particular the Schwarzschild–Tangherlini black hole. Section 4 presents the Myers–Perry solutions, first with a single angular momentum, then with arbitrary rotation. Section 5 reviews the great recent progress in five-dimensional vacuum black holes: first we discuss black rings, with one and two angular momenta; then we introduce the general analysis of solutions with two rotational isometries (or , in general). In Section 6 we briefly describe a first attempt at understanding vacuum black holes beyond the MP solutions.
Section 7 reviews asymptotically-flat black holes with gauge fields (within the restricted class mentioned above). Section 8 concludes our overview of asymptotically-flat solutions (vacuum and charged) with a discussion of general results and some open problems. Finally, Section 9 reviews asymptotically AdS black-hole solutions of gauged supergravity theories.
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