# Entropy of Schwarzschild black holes

Gold Member
I am trying to reconcile three things:
(1) The entropy of a black hole is proportional to the logarithm of the number of possible states of that object to give the same event horizon.
(2) The only parameter for a S. black hole is its mass, since its electric charge and angular momentum are, by definition, absent.
(3) The entropy of a S. black hole is huge, and definitely non-zero.

So, there are different states for the black hole, but how is that possible, when the entropy is clearly defined by the unique parameter, its mass? I'm missing something elementary here..........

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bcrowell
Staff Emeritus
Gold Member
I am trying to reconcile three things:
(1) The entropy of a black hole is proportional to the logarithm of the number of possible states of that object to give the same event horizon.
(2) The only parameter for a S. black hole is its mass, since its electric charge and angular momentum are, by definition, absent.
(3) The entropy of a S. black hole is huge, and definitely non-zero.

So, there are different states for the black hole, but how is that possible, when the entropy is clearly defined by the unique parameter, its mass? I'm missing something elementary here..........
I think the idea here is that parameters like mass (and charge and angular momentum) are like the macroscopic properties of a cylinder of helium: pressure and volume pretty much do it. A high-entropy object is one that has many microscopic states for a given macroscopic state. When the black hole evaporates, its hidden microscopic degrees of freedom will be made visible again.

Having said that, I get the general impression that nobody really knows for sure how to count the degrees of freedom of a gravitating system.

Chronos
Gold Member
Stephen Hawking showed how the entropy of a black hole was represented by the surface area of its event horizon.

Gold Member
Thank you, bcrowell. In other words, what I did not take into consideration is time: the entropy now depends on events in the future or in the past. Makes sense, so that your definition would have to say "a high-entropy object is one that has, had, or will have many microscopic states for a given macroscopic state."

Also, I guess its microscopic degrees of freedom before evaporation need not be called "hidden", since they are "revealed" in the Bekenstein-Hawking entropy of the horizon. Otherwise one would have to incorporate events with memory into the description.

Gold Member
Chronos: Thanks, this was implicit in my original question, and I was remiss in not making that more explicit. [I do so in my answer to bcrowell to which I answered just before receiving your comment.]