What Is the Entropy of a Black Hole?

[jackle]

Does a black hole have entropy? How do we know? What does it "do"?

 

[Labguy]

Does a black hole have entropy? How do we know? What does it "do"?

Everything "has entropy" at some given state or another. Entropy is not a state of being, it is a measure of a change.

Entropy is defined as the "capacity for change" of a system. If the state of a system is changed but the entropy is not changed (Del S = 0), then the process was reversible (able to be changed back to the original state without added energy). If the state of a system is changed and the entropy increases (Del S > 0), then the process was irreversible, or spontaneous. NOTICE, the Del S (change in entropy) for a state change cannot be less than zero! Since most processes are irreversible, it is said that universal entropy is always increasing - since entropy is the driving force behind equilibrium (not chaos), this means that the universe is constantly moving toward a less dynamic state.

So, the question should be whether there is a change in entropy, not just does it have entropy.

 

[SpaceTiger]

Does a black hole have entropy?

We think so, yes. In natural units, the entropy of a black hole is given simply by:

$$S=\frac{1}{4}A$$

where A is the area of the black hole:

$$A=\int \sqrt{g_{\theta\theta}g_{\phi\phi}}d\theta d\phi$$,

which is evaluated at the outer event horizon.

How do we know?

It's a long story, but the basic idea is that black holes seem to follow their own laws of thermodynamics if one makes the replacements, T -> T_{Hawking}, E -> M, and S -> A. For example, in classical theory (prior to the recognition that they could radiate), there was no way to decrease the total area of all black holes in the universe. This is like the second law of thermodynamics, which says that the total entropy of a closed system must always increase.

The recognition of these connections actually predated Hawking's realization that black holes had a temperature, so the temperature analogy was originally made to a constant quantity defined on the event horizon of the black hole. The development of the theory surrounding Hawking radiation only strengthened the connection between black holes and thermodynamics.

Initially, it was treated simply as an analogy to the laws of thermodynamics, but we now believe that the entropy I give above is the actual thermodynamic entropy of the black hole. This is difficult to test with actual experiments, but we have been unable to find any contradictions in these identities when black holes are included in thermodynamic systems.

What does it "do"?

Not quite sure what you mean here.

 

[jackle]

What does it "do"?

Not quite sure what you mean here.

I heard a lecturer saying that black holes were thought to be high entropy singularities and the big bang was thought to be a low entropy singularity.

I am trying to understand what difference it makes.