Bekenstein–Hawking entropy: the proportionality constant and the 2D black hole

Question

The Bekenstein–Hawking entropy is defined by the relation

$$S=\frac{k_BA}{4l_p^2},$$

where $k_B$ is the Boltzmann constant and $A$ is the area of the black hole's event horizon in units of the Planck area $l^2_p$.

I am not familiar with black holes thermodynamics, so I have three relatively simple questions.

1. Where does the proportionality constant $\frac{1}{4}$ come from and how do you interpret it?
   I tried to find a clear answer but always get confused.

2. If you consider the black hole as a 2D disc or circle (not a 3D sphere). Would I have the same expression (in the same unit) and in particular, the same proportionality constant ?

3. In the derivation of the formulae, is there any assumption on the shape of the black hole ? For instance that it is a sphere or whatever the shape.

Answer 1

1. While there are proofs of $S= A/ 4$ (in natural units $\ell_p=1$, $k_B=1$), I think a good interpretation of the $\frac14$ is an open nonproblem (a good interpretation being subjective of course). The proofs are unenlightening as to the $\frac14$.

2. If the black hole horizon has the topology of a $d$-sphere $S^d$ then the area $\sim r^d$ where $r$ is the radius. For a 3D black hole (2 spatial + 1 time dimension) the horizon is an $S^1$, and the entropy is $\frac14 2\pi R = \frac12 \pi R$. There is an example of a 3D black hole known as the BTZ black hole which lives in asymptotically AdS space (not flat space, i.e. not a vacuum solution), and this relation holds true.

3. No, it is quite general. For example, if there is a non-spherically symmetric background matter field, the horizon may deform slightly so that it is no longer a sphere; the entropy is still given by $\frac14$area of horizon. The formula also holds for spinning black holes (Kerr solution) where the horizon is oblate.