The Bekenstein entropy for a black hole is proportional to the surface area $A$ of the black hole $$ S_{BH} = \frac{k_B}{4 l_P^2} A $$ with the Planck length $l_P = \sqrt{\frac{\hbar G}{c^3}}$.
The area is the surface of a sphere with Schwarzschild radius $r_s = \frac{2 M G}{c^2}$, so $$ A = 4 \pi r_s^2 = 16\pi \left(\frac{G}{c^2}\right)^2 M^2 $$ and the black hole entropy is therefore proportional to the mass of the black hole $M$ squared: $$ S_{BH} = \frac{4 \pi k_B G}{\hbar c} M^2. $$ But this quite unusual for an entropy. In classical thermodynamics entropy is always supposed to be an extensive quantity, so $S\sim M$. But the black hole entropy $S_{BH} \sim M^2$ is obviously a non-extensive quantity. Isn't a non-extensive entropy inconsistent within the framework of thermodynamics? Why is it, that the entropy of a black hole must be a non-extensive quantity? Shouldn't we better define an entropy for a black hole from e.g. the ratio of Schwarzschild radius to Planck length, which would give us an extensive entropy like $$ S_{BH, ext} \sim k_B \frac{r_s}{l_P} \sim k_B\sqrt{\frac{4G}{\hbar c}} M $$
No comments:
Post a Comment