general relativity - Behavior of black holes in higher- and lower-dimensional space-times

The behavior of black holes in 3+1 dimensional space-time as our own is rather well known: formation, event-horizon size, mass, spin, radiation etc.

However, my question is what would black holes behave like in universes with different number of space-time dimensions, such as 2+1 dimensional or 4+1 dimensional space-time?

Would they be able to form? Would the event horizon radius be smaller or larger? Would their densities be higher or lower? Would they spin, and how quickly? Would they emit Hawking radiation? How would the space-time behave close to it compared to in 3+1 dimensional space-time?

Answer

First of all, in lower dimensions (2+1 and 1+1) the gravity is much simpler. This is because in 3d curvature tensor is completely defined by Ricci tensor (and metric at a given point) while in 2d curvature tensor is completely defined by scalar curvature. This means that there are no purely gravitational dynamical degrees of freedom, in particular no gravitational waves.

General note: horizon (which is the defining feature of black hole) representing our inability to obtain information about events under it would always imply the entropy of corresponding solution. So, in all of black hole models there is some black hole thermodynamics. For Hawking radiation one needs to include quantum effects into consideration and also radiative degrees of freedom (if there are no gravitons or photons or any other '-ons' than nothing would radiate).

Let us start with case of 3d (that is 2+1). The Einstein equations in 2+1 spacetime without any matter fields would simply imply that spacetime is flat, that is 'constructed' from pieces of Minkowski spacetime. It may have nontrivial topology, so 2+1 gravity is a topological theory, but no black hole solutions exist. This model (in mathematical sense) is exactly solvable.

To introduce non-trivial 2+1 solutions we can add matter or cosmological constant (which could be considered the simplest form of matter). It turns out that the spacetimes with negative cosmological constant (which would locally be composed of pieces of anti-de-Sitter spacetimes) do admit the black hole solution: BTZ black hole (name after authors of original paper). This solution shares many of the characteristics of the Kerr black hole: it has mass and angular momentum, it has an event horizon, an inner horizon, and an ergosphere; it occurs as an endpoint of gravitational collapse (for that, of course, we need to include matter beyond cosmological constant in the consideration); and it has a nonvanishing Hawking temperature and interesting thermodynamic properties (see, for instance, paper by S. Carlip). The Hawking temperature of BTZ black hole $T\sim M^{1/2}$ which, in contrast to the (3+1)-dimensional case, goes to zero as $M$ decreases. Additionally, the simplicity of the model allows quantum treatment of it including statistical computation of the entropy (see references in paper by E.Witten).

There are many other variations of solutions in 2+1 gravity theories (for instance by including dilaton and EM fields, scalar fields etc.) but all of them require negative cosmological constant. This is because dominant energy condition forbids the existence of a black holes in 2+1 dimensions (see here).

Now to 1+1 dimensions. Locally all GR models in 1+1D are flat. So to include nontrivial spacetime geometry we need to modify gravity. This can be done by including dilaton field. The resulting models often admit nontrivial geometries with black holes (see paper by Brown, Hennaux, Teitelboim, wiki page on CGHS model, paper by Witten on BH in gauged WZW model, and this review). These black hole solutions also admit nontrivial thermodynamics and Hawking radiation. In particular the Hawking temperature is proportional to mass, so as the black hole evaporates it becomes colder (unlike 4D case where $T \sim M^{-1}$).

Now to higher dimensional gravity. Gravity itself is much richer than in lower dimensional cases, so analogues of all 4D black holes also exist in higher dimensions, as well as some new black hole-like solutions such as black strings and black p-branes. There are also multi-black hole configurations where multiple black holes are placed along the ring or line such that the total force on each of them is zero, resulting in equilibrium configuration. Since many uniqueness theorems for black holes only work for 3+1 dimensions there are even solutions with nontrivial horizon topologies such as black rings.

I suggest to look at the Living Review recommended by Ben Crowell or to this lectures by N. Obers.

The simplest black hole would be Schwarzschild–Tangherlini solution (analogue of Schwarzschild black hole) which is vacuum solution to Einstein field equations:

Here $\mu = R_s^{d-3} = \frac{16 \pi G M}{(d-2)\Omega_{d-2}}$ is mass parameter. This gives us the relationship between mass and Schwarzschild radius: $R_s \sim M^{1/(d-3)}$. The entropy is given by Bekenstein-Hawking formula:

$$ S = \frac {\cal A}{4G}=\frac 14 \left(\frac{\Omega_{d-2} R_s^{d-2}}{ G} \right). $$ Temperature could be found from the first law $ dS = d M / T $: $$T = \frac{d-3}{4 \pi R_s}.$$

Rotating solution (generalization of Kerr metric) would be Myers-Perry metrics. Note, that rotations in higher dimensions are more complex, so the angular momentum is represented by several parameters.

Also note, that many solutions with horizons elongated in one direction (such as black strings or black rings) turn out to be unstable via the Gregory-Laflamme instability, where the smooth 'tubular' horizon evolve growing perturbations of certain wavelengths. So possibly black strings and black rings would tend to decay into droplets-like black hole along them (the exact mechanics is yet unknown). But of course, the second law of thermodynamics would be observed, meaning that total area of the horizons would increase.