# 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 connection 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.)

## About Aron Wall

I am a postdoctoral researcher studying quantum gravity and black hole thermodynamics at the Institute for Advanced Study in Princeton. Before that, I read Great Books at St. John's College (Santa Fe), got my physics Ph.D. from U Maryland, and did my first postdoc at UC Santa Barbara.
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### 2 Responses to Did the Universe Begin? VI: The Generalized Second Law

1. Edger Sterjo says:

Hi Aron,

I recently read an article you wrote on Scientific American where you apply the GSL to show that a time machine contradicts this law, and so can't actually be built according to the GSL. http://blogs.scientificamerican.com/critical-opalescence/2014/05/23/time-machines-would-run-afoul-of-the-second-law-of-thermodynamics-guest-post/#respond

However, in your counter example you assume that the the horizon of the Close Timelike Curve would be infinite in the past (thus having infinite entropy which can't be matched by the finite entropy in the future). My question is what allows you to make this assumption?

Sincerely,

Edger

2. Aron Wall says:

Dear Edger,

If you are just trying to build a time machine in the laboratory, then to a good approximation you can think of the system (e.g. the Earth or perhaps the Solar System) as just sitting inside of an infinite space. It would follow that as the horizon goes out into infinite empty space, it goes back infinitely far in time. This is just an approximation: in reality space outside the Solar System is not empty, afte that you would eventually hit the Big Bang singularity. But it's still a very good approximation for a long way out, so the entropy of the horizon would get really really big.

I answered a similar question in the comments section to my original post on this topic:

Since, as you mention, entropy is proportional to surface area, wouldn’t the decrease in entropy of the shrinking sphere bounding the CTC always be finite? In order for it to be infinite, the sphere would need to have an infinite size at some time. But hasn’t the universe been finite at all points in time between the big bang and right now?

We physicists tend to prove theorems in idealized situations, which may not always correspond exactly to the real world, but which we hope will nonetheless be illuminating. In the classical theorem of Hawking which I am extending, he assumes that spacetime is asymptotically flat (i.e. no Big Bang). And so do I. This is a pretty good approximation if you are just trying to create a time machine in a laboratory here in the solar system.

In the case of the real universe, what you say is quite right. The area of the past lightcone of a point today is finite (it is 0 at the present day tip, 0 at the Big Bang, and has a maximum area roughly of order $(10\,\mathrm{billion\,lightyears})^2$ before that. (I'm just estimating by using the fact that the current age of the universe is 13.8 billion years old). Hence the difference in entropy is finite: a mere $10^120$ or so. (I'm setting Boltzman's constant to 1, so that entropy has no units). This is much, much larger than any conceivable increase in the matter entropy inside of the lightcone (and in fact, in a homogeneous universe like ours the matter entropy would also be decreasing as the lightcone shrinks, since the entropy of the stuff inside is proportional to the volume).

Now it is true that in statistical mechanics the entropy is allowed to fluctuate downwards at times. However, a downward fluctuation of this size happens with a probability no greater than $10^{?10^120}$, so I wouldn't wait around for it if I were you.