I would like to expand a little more on idea that I have talked about earlier over at the coffee table about my uneasiness with black hole entropy.

As you know, Ockham's razor refers to the idea that if you have two theories that make identical predictions and you have no way to tell the difference between the two, you should stick to the simpler of the two. For example if theory A is that angels do not exist and theory B is that on every empty chair sits an angel but you cannot see or smell or in any other way determine his presence you should prefer theory A. A similar example would be what Sean Carol would describe as Religion Light: There is a god, but she does not influence anything in the physical universe.

A slightly different perspective would be that at least operationally A and B are the same theory.

To formalise slightly: In theory B, a state would be the tensor product W x A where W describes the state of the physical world and A is the state of the angels. Let's assume for simplicity that A can take k different values (or is vector/density matrix in a k dimensional angel Hilbert space). The fact that the angels are not observable means that all observables are of the form O x 1.

However, the entropy is different in the two theorie: In B, it is always higher by log(k). For ordinary thermodynamics this does not matter as there one always measures only differences in entropy and thus log(k) cancels.

But let's now turn on semi-classical gravity. Then we go to our favourite black hole and make a precision measurement of the horizon area. Then we drop a chair into the black hole and then measure the area again. The difference divided by 4 tells us the entropy of what we have thrown in. If it is just the entropy of a chair, there are no angels, not even invisible ones. If we find the entropy is the entropy of a chair plus log(k) we know that there are invisible angels!

Thus black holes can work as Ockham's razor to get rid of invisible degrees of freedom!

## Thursday, February 10, 2005

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## 3 comments:

Presumably, the effective 4D M_{pl} depends on the dimension of the Angel Hilbert Space.

If the dependence is the one I'm thinking of, then, no, you don't learn anything about the value of k from this experiment.

(I suppose that's an interesting constraint on how the effective M_{pl} can depend on hidden variables.)

I don't think you can fix this by adjusting the gravitational constant (or Planck area), as that sets the scale of entropy (if you like the base of the logarithm). log(k) however is a shift. You can test this by dumping different chairs (whith similar angels).

I suspect that your angels will not contribute to the black hole entropy (or to the surface area of the black hole) unless they are coupled to gravity. This would not be the case for a simple "tensor product of theories". Hence I am not sure there is a puzzle to be resolved.

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