Undivided Looking

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.)

The next piece of evidence we will consider for the beginning of the universe is the Second Law of Thermodynamics.  I sometimes call this the "Ordinary Second Law" to distinguish it from the "Generalized Second Law" which involves black holes (or other types of causal horizons).

The Second Law of Thermodynamics is a rather special law of nature because it distinguishes the past from the future.  It says that a quantity called the "entropy" always increases as time passes.  I've already written some articles explaining (1) what the entropy is (hint: it does not measure the amount of evil), (2) why it increases, and (3) how it has an interesting generalization to situations involving black holes (the "Generalized Second Law" mentioned earlier in this series).  Rather than repeat myself, I will link to these articles here:

One Way Streets: Black Holes and Irreversible Processes
An Introduction to Horizon Thermodynamics for Non-Physicists

If you're back from reading those—or if you're willing to take my word for it that there's a number called the "entropy" which has to increase as time goes on—then let's start by using it to make an argument that there had to be a Beginning.  Then we can see if there are any loopholes that can be used to evade the argument.

At each time $t$ the entropy $S(t)$ takes some particular numerical value.  As you go back in time, the Second Law says that $S$ gets smaller and smaller, until eventually it reaches its minimum value.  (Because of the way entropy is defined, it normally can't be a negative number, so the smallest it can get is 0.)

Now, either (a) there was a beginning of time, or (b) the entropy remained more or less at the same value for an infinite amount of time prior to some particular moment; let's call this $t_\mathrm{early}$ since it would have to be at least 13.8 billion years ago (since we know the entropy has been increasing since then).  But in that case, the universe would have had to be really boring for the first "half" of eternity $t < t_\mathrm{early}$, since interesting processes tend to produce entropy.  It's unclear what mechanism would cause the universe to suddenly become interesting.  Since scenario (b) seems implausible (though not necessarily impossible), we conclude that probably (a) is right, and there was a first moment of time.

Now, how might we evade this conclusion?  Here's three possible ways, although the first one doesn't really work, and the second one seems to run into some problems as well...

1. Thermal Fluctuations.  One way might be to take advantage of the fact that the Second Law is not an exact law of Nature.  Because it is statistical in Nature, the entropy can decrease, it's just very unlikely for it to decrease by large amounts.  But if you keep a system at maximum entropy for an very very long amount of time, eventually there will be thermal "fluctuations" in which the entropy gets down to arbitrarily small amounts.

So could our universe be a thermal fluctuation?  No, because a thermal fluctuation is unlikely to produce a whole cosmology filled with low entropy galaxies.  It would be much more likely for the fluctuation to produce the minimum amount of matter necessary to support a (briefly existing) intelligent life form (this is called a Boltzmann brain, by the way).  Since fluctuations are totally random, every possible matter configuration (with a given energy) would be equally likely, whereas elementary sanity says that this is not the case.

2. Shell Games with Infinity.  Another possible loophole is that actually none of this is well-defined because space is infinite and so $S = +\infty$.  Entropy could be produced both to the past and the future, but it wouldn't matter since the total amount is always infinity.

This loophole is used (e.g.) in the ekpyrotic scenario, a rather wild alternative to inflation in which there are membranes living in a 5th dimension which periodically collide with each other, causing Big Bounces (supposedly—this was really just a guess about what might happen).  The bounces are supposed to happen on a cyclic basis, so that the model is eternal in both time directions.  From the perspective of the 4 ordinary spacetime dimensions, the universe is infinite and expanding on average, which makes it so that the entropy "thins out" and prevents the universe from dying of heat death when its entropy reaches a maximum value.  Hence the BGV theorem tells us that the spacetime would have a beginning for most geodesics, even though some of them go back in time infinitely.

(Also, If the BICEP2 measurement of primordial gravity waves is right, that's also inconsistent with the ekpyrotic scenario.  Although there's some doubt now about whether BICEP2 properly screened for alternative sources of CMB polarization due to intervening dust.  Anyway the ekpyrotic scenario is just an example, not necessarily the only model like this.)

3. Arrow of Time Reversal.  This exploits the fact that we don't know the real reason why the Second Law is true in the first place.

Here is a paradox: the fundamental Laws of Physics are (more or less) symmetric between the past and future.  That is, if you replace $t \to -t$ in the equations, everything stays the same, more or less (*).  Yet, in the actual universe the past and future are quite different because of the Second Law, which says that the entropy is increasing.  And yet, the Second Law is regarded not as a fundamental law of Nature, but merely an effective statistical measure of what is most likely to happen given the fundamental laws of Nature?  So what gives?—How can you get a time asymmetric Law to pop out of time symmetric Laws.

The best people can tell is that the universe just started in a low entropy state.  It's a matter of the "initial conditions", not the Laws of Physics themselves.  (Although later we will discuss the Hartle-Hawking proposal, which blurs the boundaries between "initial conditions" and "Laws of Nature".)

Since we don't really understand why the universe began in a low entropy state, we are free to build (equally perplexing) models in which the entropy of the universe is small somewhere in the middle of time, rather than at the beginning.  If we assume the entropy was small at some time (let's make an arbitrary coordinate choice and call it $t = 0$), and then evolve that low-entropy state in both time directions, we typically find that the entropy will increase in either time direction.  Thus, for times $t < 0$, we find that $S(t)$ is a decreasing function of $t$.  We then say that the thermodynamic arrow of time is reversed.

This occurs in the Aguirre-Gratton model, in which the entropy decreases during a period of contraction, and then when the universe reaches its smallest size, there is a "bounce" instead of a singularity, after which the universe expands and entropy increases.  This model is symmetric under $t \to -t$.  Any people living in that time would (un)die and then shrink than be (un)born, but it would all seem just the same to them, because they'd also remember things backwards in time!

Sean Carroll and Jennifer Chen have also suggested a model like that, which involves many baby universes being created from an original inflating mother universe, whose arrow of time reverses.  (**)

During the debate, Craig focused most of his fire on the Carroll-Chen model, although Carroll modestly wanted to talk about the Aguirre-Gratton model instead:

So, I want to draw attention not to my model but to the model of Anthony Aguirre and Steven Gratton because this is perfectly well defined. This is a bouncing cosmology that is infinite in time, it goes from minus infinity to infinity, it has classical description everywhere. There is no possible sense in which this universe comes into existence at some moment in time. I would really like Dr. Craig to explain to us why this universe is not okay.

When Carroll says that there is "no possible sense in which this universe comes into existence at some moment of time", I think he is neglecting to consider that the thermodynamic arrow of time itself defines a notion of past and future.  There is a very real sense in which, in the Aguirre-Gratton or Carroll-Chen models, the time $t = 0$ is a beginning of time (due to being the lowest entropy state), and that as one travels away from $t = 0$ to either positive or negative values of $t$, one is travelling to the future in the sense that actually matters to us living and breathing creatures.  As I said in the concluding section of my own paper:

This kind of bounce evades both the singularity and thermodynamic arrow constraints, but still has in some sense a thermodynamic ‘beginning’ in time at the moment of lowest entropy [$t_0$]. That is, both the past and the future would be explained in terms of the low entropy state at $t_0$, while the state at $t_0$ would itself have no explanation in terms of anything to the future or the past. (Thus the moment $t_0$ would seem to raise the same sorts of philosophical questions that any other sort of beginning in time would.)

The Aguirre-Gratton model has no beginning in a geometrical sense, but it still has a beginning in a thermodynamic sense of unexplained "initial conditions".  Thus, I stand by my comments that an Aguirre-Gratton bounce raises the same sorts of questions as a more traditional "beginning" would.

Indeed, one could argue that the low entropy conditions of Aguirre-Gratton would be even more mysterious than in the traditional Big Bang model with a singularity.  In the latter case, there's a mysterious low entropy state, but it emerges from a singularity, and we don't know what laws of physics might exist at that singularity which cause the low entropy condition to emerge.  To some extent the mysteries cancel and make each other less mysterious, since it's not surprising that unknown causes should have unknown effects.

Whereas, if the low entropy condition occurs at a bounce, and the laws of physics there are by stipulation perfectly normal and comprehensible—and even so there is an additional low-entropy condition there, without any explanation in terms of anything else in the universe, either before or after it—then to me that suggests a need to find some sort of philosophical explanation for this strange phenomenon.

This would include potential Cosmological Arguments for the existence of God, although such arguments obviously have philosophical premises as well as physics premises.  This is made abundantly clear by the fact that Carroll doesn't accept the Cosmological Argument even on the assumption that there was a first moment of time.  One wonders therefore why he spent so much time trying to rebut Craig's claims that the universe probably did have a beginning, if it doesn't actually matter in the end.  (For purposes of the debate about God, I mean.  Obviously the cosmological origin of time is a fascinating question, which merits discussion even apart from any theological considerations!  Speaking as a physicist myself, I can certainly sympathize with Carroll getting sidetracked by interesting physics questions, as I've been doing myself throughout this series.)

Endnotes:

(*) Except for some tiny effects associated with the weak force which may not be relevant here, and even these are invariant under CPT, the combination of time reversal $t \to -t$ (T), a spatial reflection $x \to -x$ which switches left and right (P), and switching matter & antimatter (C).  Since the phrasing of the Second Law doesn't care about the distinction between matter/antimatter or left/right, one still has the question: why is the CPT-asymmetric Second Law true?

(**) For some reason in their paper Carroll and Chen wanted to have space be infinitely large even at $t = 0$, which runs into potential issues with the Penrose singularity theorem.  I wrote a paper with Alex Vilenkin slightly extending the classical singularity theorem in this context.  We showed that even if black holes form, the resulting singularities (inside the black holes) are not enough to satisfy the singularity theorem.  You need more of a "cosmological" singularity which is extended through space.  A bounce is not possible unless any observer that escapes to infinity is at least "completely surrounded" by an event horizon, beyond which there are singularities.