Is there a black-hole information equation?

Question

According to some sources, black-hole information is proportionate to the surface area, and, the equation trotted out is $$S = \frac{A k}{4 l_p^2}.$$ But according to another source, the entropy and information aren't the same. I will add that they don't seem to correlate. Entropy always stays the same or increases, but somehow information is lost when it it enters a black hole. So is there an information equation that I'm not aware of?

Answer 1

There are multiple notions of information being used in this question. The main issue is that they are not the same. Let me first notice three points:

1. Yes, entropy is related to information (or at least to the lack of it);
2. Yes, in some sense information is lost in black holes;
3. In the previous two points, I used the word "information" with two completely different meanings.

Entropy and Information

Entropy can be understood, under the light of information theory, as the lack of information of a system, as pointed out in the comments. Systems with high entropy are those about which we have low information about, while systems we have complete information have zero entropy. Notice how this relates nicely with the picture of micro and macrostates: we always know the macrostate of a system, but the specific microstate is unknown. The entropy grows with the number of microstates. Hence, the less we know about what is the specific microstate (i.e., the more microstates there are), the larger the entropy.

In this context, we mean information in the sense of information theory.

For an accessible discussion about this theme, I particularly recommend the review article Entropy? Honest! by Tommaso Toffoli (DOI: 10.3390/e18070247).

Entropy of a Black Hole

The result you quoted, which in Planck units reads $$S = \frac{A}{4},$$ is derived from the laws of black hole thermodynamics in connection with the prediction of Hawking radiation. Hawking radiation establishes the temperature of a black hole to be $$T_H = \frac{\kappa}{2\pi},$$ where $\kappa$ is known as the surface gravity of the black hole. For a Schwarzschild black hole (an uncharged, non-rotating black hole), $\kappa = \frac{1}{4 M}$. From the analogy between the first law of black hole thermodynamics with the usual first law of thermodynamics one then reads the entropy of a black hole to be the formula you quoted.

Information Loss in Black Holes

And then there is information loss in black holes. This is a consequence of Hawking radiation when one trusts the computation (which is done using Quantum Field Theory in Curved Spacetime) up to arbitrarily small sizes. Here's the summary: Hawking radiation leads to black hole evaporation, meaning the black holes get smaller and smaller. The smaller the black hole, the more intense the radiation, and hence it gets smaller even faster. If we trust the computation for arbitrarily small black holes (even when Quantum Gravity effects might come into play), we get the black holes to completely evaporate and vanish.

The thing this time is that black holes are completely characterized by their mass, charge, and angular momentum. So if we pick, for example, a gargatuan amount of neutrinos and make a black hole out of them, and them pick the same mass and angular momentum out of neutrons and make a black hole out of them, then we wouldn't be able to distinguish which black hole is which. Eventually, via Hawking radiation, the black holes could completely evaporate and there would be no way of telling that one of the black holes was made of neutrons while the other one was made of neutrinos. Someone who had only seen the final black holes would have no way of knowing what the black holes were made of in the first place. In this sense, information was lost.

Notice that this time I did not mention information theory nor anything of the sort. The reason is that in this context, the word information is meant in a more colloquial way, just to mean that if you had complete information about the universe at an instant following the black holes' evaporation, you wouldn't have sufficient data to reconstruct what happened in the past. Another way of stating this is saying that time evolution is not unitary.

For more on this theme, I'm particularly fond of the paper Information Loss by W. Unruh and R. Wald (arXiv: 1703.02140 [hep-th]).

Summary

There is no "information equation". Furthermore, the question uses the word "information" in two completely different senses. Unfortunately, terminology in black hole physics can often be difficult to follow.