# Should all black holes have Bekenstein-Hawking entropy especially in string theory?

There are so many confusing descriptions of how Bekenstein-Hawking entropy binds black holes, and thus the question. I previously asked similar questions, but I realized questions were misleading and confusing, so I am asking a new question.

I do know that Bekenstein-Hawking entropy things really on a correct theory of quantum gravity, and thus is only a conjecture. Thus I am assuming that current string-theoretic approaches/interpretations are valid. Or simply say orthodox approaches.

1. One account states that Bekenstein-Hawking entropy does not work when gravity is really strong - this would be when black holes would shrink so much. Would this be correct, or is Bekenstein-Hawking entropy formula universal, at least for some types of black holes?

2. Should all black holes satisfy Bekenstein-Hawking entropy formula?

3. Is Bekenstein-Hawking entropy referring only to state-independent entropic contribution? This is what I am getting out of Ted Jacobson's Entanglement Equilibrium and the Einstein Equation. Look at page 3, and Ted Jacobson literally matches state-independent contribution to Bekenstein-Hawking entropy, and rest of entropic contribution to be determined by the given state, which in this paper is quantum vacuum. However, by Bekenstein bound, it is said that Bekenstein-Hawking entropy is maximal. So it seems that Bekenstein-Hawking entropy refers to entire entropy, not just state-independent ones. How do I reconcile these together? What would be a correct way of understanding these matters?
• Bekenstein-Hawking entropy refers to all radiation modes - UV and IR, right? BH entropy is a different notion than Hawking radiation. For instance in theories without propagating degrees of freedom there are no Hawking radiation, yet there could be black hole and those would have entropy. – A.V.S. Oct 31 '18 at 14:22
• Edited the question as to reflect the comment above. My mistake. – Mark Ripley-AKS Nov 2 '18 at 5:21

## 1 Answer

1. The Bekenstein-Hawking entropy $$S=A/4G$$ (in units $$1=k_B=c=\hbar$$) may only become inaccurate if there are corrections from higher-derivative but gravitating terms in Einstein's equations. And that's only the case when the radius of the black hole is really tiny – comparable to the Planck length

2. In that case, there may be some other dependence of the entropy on the radius, area, or other quantities. Various generalizations of the Bekenstein-Hawking formulae exist, e.g. Wald's formula. They are valid for various classes of theoretical assumptions. But none of them would materially affect the macroscopic black holes, e.g. the real world astronomical ones. For them, the BH formula is basically exact, according to all the theory we think that we know.

3. In any well-defined classical or quantum mechanical theory, the entropy is a measure to count macroscopically similar microstates. For the notion to be meaningful and well-defined, we must specify which state – either a pure state or a mixed state – we are talking about at all. If we don't specify any state at all, we can't talk about "the entropy". So a state-independent entropy is just an oxymoron. One may make "wishes" within GR that some terms "should" be included in some formulae for the entropy. But before there is an actual way to explain this entropy by counting microstates, these proclamations remain a wishful thinking, not well-defined physics.