Buzz Bloom said:
Hi kimbyd:
I believe you, but the I find it difficult to get my head around the concept of how entropy is theoretically calculated for an assumed finite universe as a whole during contraction.
The finiteness is not necessary. Rather, it's the assumption of large-scale homogeneity that matters. The idea is that you break up the universe into large boxes which are each close to identical in their large-scale properties, and observe how those boxes evolve over time. These finite-size boxes won't exchange entropy with the neighboring boxes, because the neighboring boxes have the same properties (i.e., they'll expel just as much heat as they absorb).
Buzz Bloom said:
Assume a small enclosed room say, 2 m x 2m x 2m. Compare the entropy in the room between (1) a gas uniformly distributed throughout the room, and (2) all the gas molecules bunched into one corner, say 1m x 1m x 1m. Which has more entropy? My naive thought is that (1) has more entropy than (2), and that since (2) is unstable, after a while the (2) gas will gain entropy as the gas fills the whole room like (1). Is this correct?
Yes, this is correct.
Buzz Bloom said:
Now, consider (1) if we add the assumption that there is a black hole (BH) just outside of the room at one corner. Since the gas cannot get into the BH, the dynamics would have the gas move from state (1) towards state (2). So the presence of gravity changes something about the entropy.
Yes. Gravity changes entropy dramatically. In fact, nobody knows how to actually calculate the entropy for a generic gravitational system. We know a few extreme cases (like black holes), but that's about it.
But the underlying theory of how entropy works is independent of how entropy is calculated. We know that entropy must increase over time (or stay the same, if it's in equilibrium already) because of how entropy works, no matter the physical system. So when we know that our theories of gravity predict that the universe will get more lumpy in a collapsing universe as time goes on, then that means that that must represent an increase in entropy, because that's how entropy works.
This does lead to an interesting puzzle: how did entropy get so low in the early universe? There are lots of ideas, but so far no good answers to that question.
For example, a number of years ago Sean Carroll and Jennifer Chen proposed an idea where in the far future of an expanding universe with a cosmological constant, there would be rare events which would kick off a new inflating region. This is caused by a small local drop in entropy (which do happen, though are quite rare), followed by a rapid increase as the new pocket universe expands. The paper describing it is
here.
Sean Carroll later showed that this particular model doesn't actually work because those local drops in entropy are far more rare than the original model proposed.
Buzz Bloom said:
BTW, I have read somewhere that Hawking's idea about Hawking radiation (HR) was motivated for seeking a resolution of the problem that entropy was decreased as stuff fell into a black hole, and the only way he could think of to restore the reduced entropy was for the BH to evaporate. That is, without HR there would be a permanent reduction of entropy.
Not quite.
The first issue was that it was noticed that the behavior of a black hole could be described with thermodynamics-like laws. Just as entropy doesn't decrease in a closed system, the area of a black hole's event horizon couldn't decrease (so far as anybody knew). And the mathematical way of describing that looked suspiciously liked entropy. Jacob Beckenstein came up with the idea that maybe this apparent similarity was real: that the properties that mathematically looked similar to thermodynamics were genuinely thermodynamics. This meant that the non-decreasing nature of the area meant that the area of the horizon was proportional to the entropy of the black hole. Hawking took that idea and worked out the mathematical consequences of it, eventually concluding that if the black hole had an entropy, then it also had to have a temperature. And if it has a temperature, then it will radiate based upon that temperature. He made use of some quantum field theory to show physically how that could occur without violating the notion that nothing can escape a black hole, in particular showing that the quantum field theory prediction of radiation was exactly the radiation that would be expected from the black hole being a thermodynamic body at a specific temperature.
The information stuff came later, where there was a concern that because black holes evaporate, their entropy drops. It was shown that the entropy of the outgoing radiation is even higher, so entropy continues to increase throughout this whole process.
