There’s a tiny 10^12 entropy black hole for a while in the bath, and then the black hole evaporates and is gone. The baby universe is not located anywhere inside the bath anymore. As far as the bath is concerned, there is no baby universe anymore. It’s gone—finito.

So why does an increase in the entropy of the baby universe now have anything to do with the entropy of the bath?

My point was that the entropy of a subsystem can increase without changing the entropy of the larger system to which it belongs. That happens all the time in quantum mechanics. There’s no need for the entropy increase of the subsystem to be “compensated by a decrease in entropy elsewhere.” There are lots of examples of this.

Back to the specific case of cosmology, the bath sees a tiny cell whose entropy is bounded by 10^12 turn into a tiny 10^12 entropy black hole of roughly the same size, and then the black hole either stays around for a while, still at 10^12 entropy, in thermal equilibrium with its surroundings in the bath or evaporates again, with the 10^12 entropy coming back out. That’s all the bath ever sees, so where is there any marked increase in the entropy of the bath?

At worst, there’s a shift of entropy by 10^12 during this whole process (if that much), but with zillions of cells in the bath and with the bath’s own entropy of order, say, 10^300, that’s not an unlikely fluctuation at all.

So I’m still confused about where there’s a problem here. Where does the bath experience some large change in entropy anywhere? Who cares what happens in the causally disconnected baby universe? It’s fully consistent with dynamically local quantum mechanics for a subsystem to grow in entropy while leaving the full system unchanged in entropy—no compensation needed. And especially so when the subsystem remains causally disconnected from the larger system, as in our case!

So I’m still not certain I see the problem here. Please let me know where I’ve gone wrong.

Thanks!

So your basic point seems to be the following: Suppose that the larger bath cannot grow in entropy, and suppose that our own universe is a subsystem of that bath; then it’s a paradox that our own universe can have a much higher entropy than it does, because that would imply that the bath should be able to have a higher entropy as well, a contradiction. Indeed, as the entropy of our universe gets larger and larger, the entropy of the bath to which it belongs should get larger and larger as well. This seems to be the crux of your argument that there cannot be such a bath and that inflation cannot solve the cosmic entropy problem on its own.

I guess what I’m saying is that I’m not convinced that it’s impossible under unitary time evolution for a subsystem of the larger bath to grow in entropy while leaving the entropy of the bath unchanged, especially if that subsystem, while perhaps open to the larger subsystem at certain times, is otherwise causally hidden from the larger bath—as our inflating baby universe would be.

And besides, in quantum-mechanical theories it’s easy to find simple examples in which a subsystem grows in entropy—under perfectly unitary time evolution—while the larger system to which it belongs stays at constant entropy. As a trivial instance, consider a pure state consisting of two particles that interact and grow mutually entangled—each particle’s state becomes increasing mixed and entropic, but time evolution is always globally unitary. Things get even more interesting when there are horizons that shield subsystems from each other after they’ve become entangled—and horizons abound in our cosmological scenarios. These issues are invisible in any scenario that treats gravity classically, because these kinds of quantum effects are then invisible.

So what am I missing here?

Thanks again!

Indeed, that’s why a tiny inflating patch that starts with only 10^12 units of entropy can unitarily evolve into a vast multiverse with many universes that each have much more entropy than that. Due to the phenomenon of entanglement entropy (which becomes very important when we have lots of horizons around), the entropy of a subsystem in quantum mechanics is by no means whatsoever limited by the entropy of the full system! A pure state, having zero entropy, can easily have subsystems with large entropy, all while not violating unitarity!

And because we’re trying to treat gravity classically here, I’m very, very worried that we’re missing out on the hugely important quantum effects like entanglement entropy that can radically alter our logic.

But I’m not so sure this works. If our whole observable universe is inside a black hole as seen from outside in the bath, then the bath has no clue what’s going on inside our universe. As far as the bath is concerned, the entropy of our little cell is maximal all the time, whatever is happening inside our cell.

The interior of our cell could get blown up exponentially like a balloon so that the entropy is fixed but its density goes way down. But the bath knows none of this, always assigns the same entropy to the tiny black hole (as long as the black hole exists), and stays at maximal entropy all the time.

If the black hole decides to evaporate away, then the entropy originally hidden in the black hole comes back out, and our baby universe pinches off and goes on its merry way, free to go up in entropy to its heart’s desire still without affecting the bath from which it was born. But no significant amount of entropy was ever created or destroyed as far as the bath was concerned.