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

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.

• With regard to your second question, the dependence on the radius would be the same if the black hole was circular or spherical, since we would only see a disc. The fundamental quantity is the area, because the maximum amount of information that can be stored in a region of space goes as the surface area of that space in Planck units. Jan 9 '20 at 16:54
• Thus you would say the formula for the entropy of a 2D black hole (assimilated as a circle or disc) would be the same $𝑆=k_b A/(4 l^2_p)$ but with $𝐴=\pi R^2$ ?
– Nath
Jan 9 '20 at 20:44
• Black holes don't really exist in 2+1-dimensional gravity. Vacuum solutions are flat in 2+1 dimensions.
– user4552
Jan 9 '20 at 21:25
• @Physics the dependence of the entropy$\sim$area on radius is manifestly different for a circle ($S^1$) and a 2-sphere ($S^2$).@Nath In the former case $S = A/4$ with $A = 2\pi R$. Jan 9 '20 at 23:28

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.
• From your answers, to be sure that I understood correctly, can you confirm the following statement that, the Bekenstein–Hawking entropy as written in my post, that is $S = k_b A/(4l^2p)$, is a general relation that holds for any black hole horizon topology. The only thing that changes is implicitly hidden in $A$ the area, as you showed in your answers 2 and 3.
• That's right, with the minor correction that one must write $\ell_p^d$ for general dimension $d$. Jan 10 '20 at 0:44