Did the Universe Begin? VI: The Generalized Second Law

Last time I discussed the cosmological implications of the regular-old Ordinary Second Law.  Now I want to discuss what happens if you use the Generalized Second Law instead—this being a generalization of the Second Law to situations involving black holes and other horizons, which seem to have an entropy proportional to their surface area.

I started thinking about this issue after Sean Carroll gave a colloquium at U Maryland about the Carroll-Chen model, and the Second Law, back when I was a grad student.

From my perspective, the important thing about that colloquium was that it got me thinking about refining the classic argument that the Second Law predicts a beginning.  I said to myself something like the following:

"Self, for the past couple years you've been spending all of my time thinking about the Generalized Second Law (GSL), that wild new version of the Second Law which applies to causal horizons.  Well, there are horizons which appear in our own cosmology (because of the accelerating expansion of the universe).  So can we make this argument using the GSL instead of the Ordinary Second Law (OSL)?  And if we do, will it make the argument stronger or weaker?"

Well, when I thought about it a little bit, I realized that you could use the GSL in two distinct ways to argue for a beginning.  One of them is a quantum generalization of the Penrose singularity theorem, which I discussed here.  The other way is a generalization of the Argument from the Ordinary Second Law, described above.  Both of these uses of the GSL are discussed in my article, but it is important to realize that they remain two distinct arguments!

Fine-grained vs. Coarse-grained.  The reason is that there are actually two subtly different ways to formulate the GSL.  You see, the entropy is a measure of our ignorance about a system.  To exactly define it, you need to make a list of the things you are allowed to measure about the system (e.g. the  pressure and temperature of a box of gas), and then the entropy measures how much information content is in the things you can't measure (e.g. the positions or velocities of individual molecules).  The procedure of ignoring the things we can't measure is called coarse-graining (because it's like looking at a grainy photograph where you can't see all of the information in the object).

Technically then, there's some ambiguity in the definition of the entropy, since the intitial step where we list what we can measure is a little bit ambiguous.  Fortunately, since the amount of information we can't measure is much larger than the information we can, this doesn't usually matter very much.  Quantitatively, the different ways of defining entropy give pretty close to the same numerical answers.

But we could pretend that we could measure everything about the box of gas to arbitrary accuracy.  The only uncertainty allowed which could produce a nonzero entropy is uncertainty about the initial conditions.  This is called the fine-grained entropy, and while it has the property that it neither increases nor decreases as time passes.  Since the fine-grained entropy can't decrease, it technically obeys the Second Law, but in a really boring and stupid way.

The distinction becomes important when you start talking about black holes and the GSL.  Suppose you have a star orbiting a black hole.  Matter from the star is slowly getting sucked off the outer layers of the star, and getting sucked into the black hole.  (This is a realistic scenario which is believed to really occur in some solar systems, by the way!)

Well, we have a choice.  We could use a coarse-graining to describe the entropy of the star.  In that case, the entropy would go up for 2 distinct reasons: A) because stuff is falling into the black hole making its area increase, and B) because ordinary thermodynamic processes are happening inside the star, making the entropy increase for usual non-black-holey reasons.

Or, we could take the fine-grained point of view, and pretend we know everything about the matter outside the event horizon.  In that case, the entropy increases only because of (A), things falling across the horizon.  Stuff happening inside the star doesn't make a difference.  This would be the fine-grained GSL, and it is nontrivial—the entropy defined in this way can go up, but not down.  You could say, that the only coarse-graining we use is to forget about anything that fell across the horizon, and this is enough to get a nontrivial result.  (This was pointed out by Rafael Sorkin.)

In my dissertation research, I proved the GSL in the fine-grained sense.  This was very useful since there are still some thorny and unresolved issues of interpretation with the Ordinary Second Law due to the exact meaning of coarse-graining.  The fact that one can avoid this issue in discussing the GSL made my life much easier!

Also, coarse-grained versions of the Second Law are only true if you have a history with a well-defined arrow of time—i.e. a universe that is constrained to begin with low entropy, but has no particular constraint on how it has to end up.  The fine-grained GSL, on the other hand, appears to be true for all states and therefore has no dependence on the arrow of time.  As a result, you can even apply the fine-grained GSL backwards in time if you want to, and this is perfectly OK, even though we normally think of the Second Law as something which only works in one time direction.

The forwards-in-time GSL applys when you have a worldline (an "observer", if you feel like anthropomorphizing) which extends infinitely far to the future.  It says that the boundary of what the observer can see (called a "future horizon") has increasing entropy.  The backwards-in-time GSL, says that if you have a worldline which extends infinitely far to the past (if there are any), then boundary of what they can be seen by (called a "past horizon") has decreasing entropy.  Equally true.

How to apply the GSL.  If you want to use the GSL as a singularity theorem to show that time ends in the middle of a black hole, you'll want to use the forwards-in-time GSL.  But if you want to use it to argue that there was an initial singularity at the Big Bang, you need to use the backwards-in-time GSL.  That's what I did to generalize the Penrose singularity theorem in my paper (like the original, it only works if space is infinite).

But I also considered the possibility that you might use the forwards-in-time GSL to argue for a beginning.  In this case, it would be a substitute not for the Penrose theorem, but for the OSL.  The details are in section 4.2 of my article, but the upshot is pretty much the same as before, that there probably had to be a beginning unless either (i) the arrow of time reverses, or else (ii) the universe was really boring before a certain moment of time.

Why even bother?  Other than the fact that the GSL has deeper connections to quantum gravity, the main technical advantage of using the (forwards) GSL is that it is more clear that the entropy reaches a maximum value in our universe (due to the accelerating expansion of the universe at late times, there is a de Sitter horizon at late times whose entropy is about $10^{120}$).  This makes it harder to play games with infinities.

Also, if we want to reverse the arrow of time, the GSL indicates that the universe probably has to also be finite in size.  That's because, if the universe is finite in size, there is the possibility that before a certain time $t_0$, everything is visible to an observer, and before that time there would be no horizon at all.  That would make the fine-grained GSL trivial, forcing us to use the coarse-grained GSL.  That's important because it's only the coarse-grained GSL which depends on the arrow of time.

But mainly I just wanted to see if the standard argument from thermodynamics would still work when I rephrased it in terms of horizon thermodynamics.  Not surprisingly, it does.

(On the other hand, the connections to the Penrose singularity theorem are much more surprising, and I believe that it is telling us something deep about the laws of quantum gravity.)