# Black hole entropy versus entropy of normal matter [duplicate]

It has been established that the entropy of a black hole is equal to:

$$\frac{1}{4} \frac{c^3 A}{ G \hbar} k$$

Which if one substitutes for A the surface area of the event horizon:

$$16\pi\frac{G^2M^2}{c^4}$$

One gets for the entropy:

$$8\pi^2\frac{GM^2}{hc}k$$

My question is the following:

Entropy for ordinary matter very roughly scales with the number of particles in the system. Which for stars mostly composed of fully ionized hydrogen is proportional to the mass of the system. However for black holes this seems to scale with the mass squared.

• Why is this?

• Does this seem to imply that matter falling into a black hole is heated to such an extreme that more particles are created (from energy gained by gravity or by decomposition of nuclei and nucleons into constituent particles) in such high numbers that entropy seems to have this proportionality to the mass squared?

• Do neutron stars have a similar relationship?

• Possible duplicate of Why is black hole entropy not an extensive quantity? – heather Jul 30 '16 at 22:13
• I'm voting to leave open, and then voting the newer question be closed as a duplicate. Both are well-written, and it seems unfair to punish (in the sense that closed questions will probably never get new upvotes) someone with 11 month's worth of priority. – user10851 Aug 2 '16 at 2:05
• @ChrisWhite, I can understand what you are saying, but the newer question has a bounty on it and is getting a lot of attention - that was my reasoning. I upvoted this question, for what it's worth. – heather Aug 2 '16 at 19:32