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I was looking with a friend of mine through time travel related stuff, and for some reason we ended up at the Hawking equation for black hole entropy which is:

$$S= \frac{\pi Akc^3}{2hG}$$

where:

A: Area of Black hole

k: Boltzmann's constant-$1.38*10^{-23}JK^-1$

c: speed of light-$3*10^8ms^{-1}$

h: Planck's constant-$6.626*10^{-34}Js$

G: Gravitational Constant: $6.67*10^{-11}Nm^2kg^{-2}$

The only variable in this equation is the Area of a Black Hole, so I tried calculating the value of all of the other constants. You get

$S\approx A*2.64*10^{46}$

Since all the mass is concertared in a singularity, intuitivly I would have thought for there to be very little disorder to the system, everything is in the singularity, and so the entropy would be low!(or close to the singularity in theories like quantum loop gravity). Why is the value so high; is it just coincidence are black holes highly disordered systems?

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  • $\begingroup$ This entropy is the entropy contained on the surface of the event horizon, not on the singularity (hence the $A$ term, meaning the surface area of the event horizon). In a very non-rigorous sense, as matter falls into the black hole, the black hole grows. In order to prevent the entropy of the universe from decreasing, when the black hole grows, the entropy contained on its surface must increase. In a sense, the entropy of a black hole is the entropy of all of the matter that has fallen into it. $\endgroup$ – probably_someone Feb 18 '18 at 21:52
  • $\begingroup$ Oh yes of course that does make a lot of sense. What would be a more rigorous way to see it? Because in this logic, it would mean that the entropy of the black hole depends on how much entropy has fallen into it, which would imply black holes with same areas would have different entropies. $\endgroup$ – Francesco Feb 18 '18 at 22:16
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First of all, the answer to your question is that : $We\space are\space not\space certain\space as\space to\space why\space this\space is\space so\space!$

Now, this is too broad, but let me highlight a few points that I have come across regarding black hole entropy, and then tell you the leading theory for the high entropy black holes.

It will be a bit long. Here we go :

In ordinary statistical mechanics, the entropy $S$ is a measure of the multiplicity of microstates that hide behind one particular macrostate. A special case of this is Boltzmann's famous formula $S=lnW$ where $W$ stands for the number of equally probable microstates of a particular macrostate. Since black hole entropy plays a role quite analogous to that of ordinary entropy, e.g. it participates in the generalized second law, many have wondered what the microstates that are counted by black hole entropy are. Some of the interpretations I remember are the following:

$- Black\ hole\ entropy\ counts\ the\ number\ of\ internal\ states\ of\ matter\ and\ gravity$

This interpretation takes into account all the ways a black hole of a given mass, charge and spin can form. I'ts a strenuous calculation but can produce the desired results.

$- Black\ hole\ entropy\ is\ the\ entropy\ of\ entanglement\ between\ degrees\ of\ freedom\ inside\ and\ outside\ the\ horizon$

This method says that the quantum degrees of freedom outside the event horizon must be entangled to those inside it. But since the inside is unavailable to the observer, the degrees to be counted should have their internal parts removed. Not a very successful method, it can bring the proportionality part but the constants have to be put by hand.

$- Black\ hole\ entropy\ is\ a\ conserved\ quantity\ connected\ with\ coordinate\ invariance\ of\ the\ gravitational\ action$

To quote Sir Wald, "black hole entropy is the Noether charge of the diffeomorphism symmetry". It comes from Noether's theorem which states that every conservation law is coupled to a symmetry. This method is so powerful that it not only predicts the exact formula for black hole entropy, it even modifies it for higher order corrections!

$- Black\ hole\ entropy\ counts\ the\ number\ of\ states\ or\ excitations\ of\ a\ fundamental\ string$

The first (maybe second... the first was i guess when a spin 2 tensor boson was found lurking in the equations) major accomplishment of string theory and it came at the perfect time to salvage the theory (At that time, string skeptics were growing powerful and criticizing string theorists for not doing "real" physics - this accomplishment silenced them ! ) This is also a very powerful tool because this is the closest we get to have a parallel with classical entropy. Classically we count the number of microstates of the system; in string theory, the dual of a black hole is first found(turns out to be a few branes and strings), and then count them. The result almost drops out of the counting process!

There a few more I don't recall now. But lets see as to why is this entropy, which has been shown from various methods to be extremely large, actually is large.

(It obviously leaves a very very small window of doubt that the entropy could be low... I needed to show this first, so i put those points above)

We sometimes define entropy as randomness or disorder in a system (Not quite true, but its quite close). But what is this disorder? Is it the information contained in a system that gets disordered over time , or is it something else? Turns out that the former is correct. Entropy is the measure of the amount of information in a system (the so-called Shannon Entropy concept... But we might talk about that later.). And with time, this information content actually increases as more and more particles interact with each other, making the information and thus the system, more chaotic or highly entropic.

By the No Hair Theorem, we also know that the information content of a black hole is defined by its three parameters - mass, charge and spin- the same three parameters defining the surface area of a black hole. This is no coincidence, it can be proved that the Information content of a black hole is proportional to its surface area (Apparently you can judge a black hole by its cover... ). And whats the information content of a black hole? Theoretically speaking, it is defined as the number of Planck-Length Squares that can fit the surface of a black hole . A Planck length is almost equal to $1.6*10^{-35} m$. So the information content is proportional to the inverse of this value squared. And i hope you can see that its a hefty number!

As to why Planck dimensioned squares, no one can give a satisfactory answer. String theory predicts quantum behavior at this length, and loop quantum gravity uses this definition of entropy and information to produce meaningful results. Maybe, someday this definition will be justified. Till then, lets take it as a face value definition.

And this again leads us to the statement that no one is certain.

Cheers!!

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You are right to be puzzled, indeed from the No Hair Theorem you should expect an entropy equal to zero. This follows from the statistical interpretation of the entropy as the log of the number of microstates and the trivial identity $log(1) = 0$.

Nobody really know why the entropy is so big. My view is that black holes manage to excite the quantum-gravitational degrees of freedom, so to really understand them you should work in quantum-gravity. In a such a theory it should be possible to identify a number of microstates proportional to the exponential of the area. For instance, these microstates have been identified in strings and branes in string theory.

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