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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?

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    $\begingroup$ Possible duplicate of Why is black hole entropy not an extensive quantity? $\endgroup$ – heather Jul 30 '16 at 22:13
  • $\begingroup$ 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. $\endgroup$ – user10851 Aug 2 '16 at 2:05
  • $\begingroup$ @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. $\endgroup$ – heather Aug 2 '16 at 19:32
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This is basically a consequence of the fact the BH entropy scales with the surface area of the event horizon, and not the volume of the hole. Not that that makes things any less weird.

The microphysical basis of black hole entropy, if there is one, is not known. Coming up with a compelling explanation is one of the major lines of attack of any self-respective quantum gravity theory.

The explanation you give in your second point (heating of particle pairs), however, is definitely not the reason. This is because the entropy law holds for any black hole regardless of its size. Should the hole be very large then as far as infalling particles are concerned the local spacetime geometry near the horizon is flat and no heating need take place. Because of this, an explanation for black hole entropy cannot resort to standard local physics.

The entropy of a neutron star will depend on the volume of the star (among other things), so basically the answer to your third question is no. However the BH entropy law is sometimes interpreted as the maximum entropy of a volume of spacetime, so perhaps there will be a (small) contribution from a term proportional to the surface area.

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