General theorems about higher dimensional black holes

Higher dimensional black holes seem to behave qualitatively different from 4-dimensional ones, in the sense that there are many qualitatively new solutions with properties that simply have no counterpart in 4 dimensions. It is not easy to get an intuitive understanding of why this is so. But very broadly speaking, the reason seems to be that black holes in higher dimensions have more possibilities to rotate in different directions, and this somehow gives rise to more complicated configurations in which rotational forces balance those of gravitational attraction. A very crude way of explaining why this phenomenon can occur is as follows. In (n-1)-dimensional  Euclidean space, there are N=\lceil (n-2)/2 \rceil independent planes of rotation (or equivalently, the Cartan subgroup of SO(n-1) is U(1)^N). Thus, we have the possibility of having up to N commuting rotational symmetries, with associated angular momenta J_i. The centrifugal barrier for the rotation in a given plane is always of the order of

(1)   \begin{equation*} \sim \frac{J_i^2}{M^2 r^2} \ , \end{equation*}

where the radial dependence does not depend on the dimension because rotations take place in a 2-plane. By contrast, the gravitational potential exerted by a mass M is

(2)   \begin{equation*} \sim \frac{GM}{r^{n-3}} \ , \end{equation*}

which does depend on the dimension. Thus, we see that the competition between the gravitational attraction and the centrifugal force depends on the dimension. For example, in n=5 dimensions both have the same radial dependence, and this can be seen crudely as the reason why black rings (horizon topology S^1 \times S^2) can exist in n=5 dimensions, but not in n=4. It also accounts for the fact that in higher dimensions, one can have black objects with arbitrarily high spin (in 4 dimensions, J is bounded by \sim M), leading e.g. to very flat “pancake-like” horizons, and in the limit as some J_i \to \infty to new solutions describing extended black objects such as “black branes”, which are also of considerable interest.

A more precise way of arguing that a black ring cannot exist in 4 dimensions is provided by Hawking’s famous topology theorem. The upshot is that the topology simply has to be S^2 in fairly general theories of gravity, whereas the topology of a ring would be T^2 = S^1 \times S^1. In Einstein-Maxwell theory, one furthermore has the famous and much more precise black hole uniqueness theorems, which establish that the solution must in fact be given by the known charged Kerr-Newman family of black holes. Black hole uniqueness theorems are known for certain types of higher dimensional black hole solutions, but they are much less powerful than the 4-dimensional cousin. For this reason, and also from a more general perspective, it is of great interest to know general restrictions of the type of the topology theorem also in higher dimensions. Another such general result in 4 dimensions, which is in fact an important stepping stone of the uniqueness theorems, is the so-called “rigidity theorem”, which states that any stationary black holes must either be static, or rotating with a rotational symmetry. Static black holes are usually much easier to classify. For example in vacuum general relativity, the only such solution is shown to be the famous Schwarzschild metric

(3)   \begin{equation*} g = -(1-2M/r) dt^2 + (1-2M/r)^{-1} dr^2 + r^2 d\sigma^2_{S^2} \end{equation*}

and also in many other gravity theories, static solutions have been classified. If the spacetime is not static, we have learned from the rigidity theorem that the spacetime symmetry group must include rotations in a single plane, i.e. must at least be \mathbb{R} \times U(1) (this is in fact precisely the symmetry group of the Kerr-metric).

To what extent do the topology and rigidity theorems hold in dimensions n>4? Concerning the topology theorem, it is easy to see that if we replace the 2-sphere in the Schwarzschild metric by another compact n-2 dimensional Einstein space (B,\gamma) [i.e. Ric_\gamma = c \gamma], then we get another solution g to Einstein’s equation, with c related to M. This solution would not be asymptotically flat, but Hawking’s argument is quasi-local and does not care about asymptotic conditions. If the constant c is negative, then the mass of the solution would be negative, too, so we should take c>0. Thus, it is clear that Hawking’s argument in higher dimensions cannot yield more restrictions on B than that B can be equipped with an Einstein-metric with positive constant c. By going through the argument of Hawking in higher dimensions, Galloway and Schoen were in fact able to show that the restrictions on B afforded by the use of the spacetime Einstein equations (and other reasonable rather general conditions) restrict the topology of B to be of “positive Yamabe type”, meaning, in essence, that B should be able to carry a metric of everywhere positive scalar curvature. In n=4, the dimension of B is 2. If \gamma is a metric on B with positive scalar curvature, then the Gauss-Bonnet theorem tells us that

(4)   \begin{equation*} 2 \pi \ [2 - 2 \ {\rm genus}(B)] = \int_B Scal_\gamma > 0 \ , \end{equation*}

whence the genus is positive and B must be a 2-sphere. This  is Hawking’s topology theorem. In higher dimensions, the positive Yamabe condition is increasingly less restrictive and e.g. satisfied by any product of spheres B \cong S^{q_1} \times ... \times S^{q_i}. There is now convincing evidence that regular, asymptotically flat black holes with such horizons indeed exist. In n=5 dimensions, the condition of positive Yamabe type is shown to imply that B  must be a quotient of S^3 by a discrete subgroup \Gamma \subset SO(4), by S^1 \times S^2, or connected sums thereof.

A version of the  rigidity theorem can also be shown in higher dimensions. It essentially states that, if the black hole is stationary, then it must either be static (in which case classifications are known), or it must have at least one rotational symmetry, i.e. the symmetry group must be at least \mathbb{R} \times U(1). The proof shows in interesting connection to ergodic theory on the horizon manifold. It is also known that one cannot improve this result, in the sense that there exist stationary black holes in higher dimensions which just have that symmetry. Although the rigidity theorem is in this sense less powerful than in 4 dimensions, it is nevertheless useful for many arguments about higher dimensional black holes. It can be combined also with the topology theorem to give a somewhat more refined restriction on the possible black hole topologies, and this has been worked out e.g. in the case n=5:

Theorem:The topology of the horizon B can be one of the following in 5 dimensions:

  1. If the U(1) symmetry has a fixed point on B, then the topology must be B \cong \# l\cdot (S^1 \times S^2) \# L(p_1,q_1) ... \# L(p_k,q_k).
  2. If the U(1) symmetry does not have a fixed point, then B \cong S^3/\Gamma, where \Gamma can be certain finite subgroups of SO(4), or B \cong S^1 \times S^2. This class of manifolds includes the Lens-spaces L(p,q), but also Prism manifolds, the Poincare homology sphere, and various other quotients. All manifolds in this class are Seifert fibred spaces over S^2 with positive orbifold Euler characteristic.

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