Did the Universe Begin? VIII: The No Boundary Proposal

The last bit of evidence from physics which I'll discuss is the "no-boundary" proposal of Jim Hartle and Stephen Hawking (and some related ideas).  The Hartle-Hawking proposal was described in Hawking's well known pop book, A Brief History of Time.  This is an excellent pop description of Science, which also doubles as a somewhat dubious resource for the history of religious cosmology, as for example in this off-handed comment:

[The Ptolemaic Model of Astronomy] was adopted by the Christian church as the picture of the universe that was in accordance with Scripture, for it had the great advantage that it left lots of room outside the sphere of fixed stars for heaven and hell.^{[citation\,needed!]}

Carroll, after making some metaphysical comments about how outdated Aristotelian metaphysics is, and how the only things you really need in a physical model are mathematical consistency and fitting the data—this is Carroll's main point, well worthy of discussion, but not the subject of this post—goes on to comment on the Hartle-Hawking state in this way:

Can I build a model where the universe had a beginning but did not have a cause? The answer is yes. It’s been done. Thirty years ago, very famously, Stephen Hawking and Jim Hartle presented the no-boundary quantum cosmology model. The point about this model is not that it’s the right model, I don’t think that we’re anywhere near the right model yet. The point is that it’s completely self-contained. It is an entire history of the universe that does not rely on anything outside. It just is like that.

Temporarily setting aside Carroll's comment that he doesn't actually think this specific model is true—we'll see some possible reasons for this later—the first thing to clear up about this is that the Hartle-Hawking model doesn't actually have a beginning!  At least, it probably doesn't have a beginning, not in the traditional sense of the word.  To the extent that we can reliably extract predictions from it at all, one typically obtains an eternal universe, something like a de Sitter spacetime.  This is an eternal spacetime which contracts down to a minimum size and then expands: as we've already discussed in the context of the Aguirre-Gratton model.

This is because the Hartle-Hawking idea involves performing a "trick", which is often done in mathematical physics, although in this case the physical meaning is not entirely clear.  The trick is called Wick rotation, and involves going to imaginary values of the time parameter t.  The supposed "beginning of time" actually occurs at values of the time parameter that are imaginary!  If you only think about values of t which are real, most calculations seem to indicate that with high probability you get a universe which is eternal in both directions.

Now why is the Hartle-Hawking model so revolutionary?  In order to make predictions in physics you need to specify two different things: (1) the "initial conditions" for how a particular system (or the universe) starts out at some moment of time, and (2) the "dynamics", i.e. the rule for how the universe changes as time passes.

Most of the time, we try to find beautiful theories concerning (2), but for (1) we often just have to look at the real world.  In cosmology, the effective initial conditions we see are fairly simple but have various features which haven't yet been explained.  What's interesting about the Hartle-Hawking proposal is that is a rather elegant proposal for (1), the actual initial state of a closed universe.

One reason that the Hartle-Hawking proposal is so elegant is that the rule for the initial condition is, in a certain sense, almost the exact same rule as the rule for the dynamics, except that it uses imaginary values of the time t instead of real values.  Thus, in some sense the proposal, if true, unifies the description of (1) and (2).  However, the proposal is far from inevitable, since there is no particularly good reason (*) to think that this special state is the only allowed state of a closed universe in a theory of quantum gravity.  There are lots of others, and if God wanted to create the universe in one of those other states, so far as I can see nothing in that choice would be inconsistent with the dynamical Laws of Nature in (2).

(Hawking has a paragraph in his book asserting that the proposal leaves no room for a Creator, but I'll put my comments on that into a later post!)

In the context of a gravitational theory, imaginary time means that instead of thinking about metrics whose signature is (-, +, +, +), as normal for special or general relativity, we think about "Euclidean" (or "Riemannian") signature metrics whose signature is (+, +, +, +).  So we have a 4 dimensional curved space (no longer spacetime).

The assumption is that time has an imaginary "beginning", in the sense that it is finite when extended into the imaginary time direction.  However, because there is no notion of "past" or "future" when the signature of spacetime, it's arbitrary which point you call the "beginning".  What's more, unlike the case of the Big Bang singularity in real time, there's nothing which blows up to infinity or becomes unsmooth at any of the points.

All possible such metrics are considered, but they are weighted with a probability factor which is calculated using the imaginary time dynamics.  However, there are some rather hand-waving arguments that the most probable Euclidean spacetime looks like a uniform spherical geometry. The spherical geometry is approximately classical, but there are also quantum fluctuations around it.  When you convert it back to real time, a sphere looks like de Sitter space: hence the Hartle-Hawking state predicts that the universe should look have an initial condition that looks roughly like de Sitter space, plus some quantum fluctuations.

I say handwaving, because first of all nobody really knows how to do quantum gravity.  The Hartle-Hawking approach involves writing down what's called a functional integral over the space of all possible metrics for the imaginary-time goemetry.  There are an infinite-dimensional space of these metrics, and in this case nobody knows how to make sense of it.  Even if we did know how to make sense of it, nobody has actually proven that there isn't a classical geometry that isn't even more probable than the sphere.  Worst of all,  it appears that for some of the directions in this infinite dimensional space, the classical geometries are a minimum of the probability density rather than a maximum!  This gives rise to instabilities, which if interpreted naively give you a "probability" distribution which is unnormalizable, meaning that there's no way to get the probabilities to add up to 1.

So Hartle and Hawking do what's called formal calculations, which is when you take a bunch of equations that don't really make sense, manipulate them algebraically as if they did make sense, cross your fingers and hope for the best.  In theoretical physics, sometimes this works surprisingly well, and sometimes you fall flat on your face.

Unfortunately, it appears that the predictions of the Hartle-Hawking state, interpreted in this way, are also wrong when you use the laws of physics in the real universe!  The trouble is that there are two periods of time when the universe looks approximately like a tiny de Sitter space, (a) in the very early universe during inflation, and (b) at very late times, when the acceleration of the universe makes it look like a very big de Sitter space.  Unfortunately, the Hartle-Hawking state seems to predict that the odds the universe should begin in a big de Sitter space is about 10^{120} times greater than the odds that it begins in the little one.  That's a shame because if it began in the little one, you would plausibly get a history of the universe which looks roughly like our own.  Whereas the big one is rather boring: since it has maximum generalized entropy, nothing interesting happens (except for thermal fluctuations).  St. Don Page has a nice article explaining this problem, and suggesting some possible solutions which even he believes are implausible.

Alex Vilenkin has suggested a different "tunnelling" proposal, in which the universe quantum fluctuates out of "nothing" in real time rather than imaginary time.  This proposal doesn't actually explain how to get rid of the initial singularity, and requires at least as much handwaving as the Hartle-Hawking proposal, but it has the advantage that it favors a small de Sitter space over a big one.  From the perspective of agreeing with observation, this proposal seems better.  And it has an actual beginning in real time, something which (despite all the press to the contrary) isn't true for Hartle-Hawking.

(*) There is however at least one bad reason to think this, based on a naive interpretation of the putative "Holographic Principle" of quantum gravity, in which the information in the universe is stored on the boundary.  A closed universe has no boundary, and therefore one might think it has no information, meaning that it has only one allowed state!  (The argument here is similar to the one saying the energy is zero.)  At one time I took this idea seriously, but I now believe that such a strong version of the Holographic Principle has to be wrong.   There are lots of other contexts where this "naive" version of the Holographic Principle gets the wrong answer for the information content of regions, and actual calculations of the information content of de Sitter-like spacetimes give a nonzero answer.  So I'm pretty sure this isn't actually true.

Posted in Physics, Reviews | 19 Comments

Separation of Physics and Theology?

Down in the comments section of this post, reader St. TY has the following kind thing to say about me:

What an excellent blog. I have been looking for one like this for a long time. I tell what I like about it: Although we all know St. Aron’s Christian bias, but he does not let it intrude into his physics and, as one with a mathematical background, I like that separation of Church and State.

As for the format I’m old fashioned and I like the written word because good writing demands clarity and coherence I must add honesty, and so I like reading Aron’s pieces and the comments.

I would like Aron to put all of this meaty stuff in a book.
Would you, Aron?
Thank you.

Thanks so much for your gracious compliments about my blog!  It's too bad really, that I must strongly disagree with you when you say that

Although we all know St. Aron’s Christian bias, but he does not let it intrude into his physics and, as one with a mathematical background, I like that separation of Church and State.

Your proposal that I keep a separating wall is not really very undivided, is it?  I expressed a different aspiration in my About page:

"Undivided Looking" expresses the aspiration that, although compartmentalized thinking is frequently helpful in life, one must also step back and look at the world as a whole. This involves balancing specialized knowledge with common sense to keep both kinds of thinking in perspective.

So in response I would say, that one's physics views can and should be influenced by one's theological views (or vice versa), if there is a legitimate reason why it should do so.  There is, after all, only one universe, and therefore no compartments can be kept completely watertight.  For example, most economists don't need to know much about chemistry, but if they're talking about buying things that might explode then there needs to be some cross-talk.

Christianity is not a "bias", but a "belief", one which happens to be true.  Deducing things from one's beliefs is not bias unless it is done in an irrational and capricious manner.  But perhaps you were speaking in a semi-humorous way, in the way that we might say that all scientists seek to be biased towards the truth!

Reasonable physicists will probably have similar intuitions about how physics should be done (I'm excluding unreasonable people like Young Earth Creationists), regardless of whether they are atheists or theists.  Or rather, people have different intuitions about physics but they mostly don't correlate with religious views!  But if on a particular matter (e.g. the universe having a beginning in time) somebody happens to be influenced by their religion (or lack thereof) to think that one viewpoint is more likely than another, I don't think that should be taboo.

Far from corrupting the scientific process, I think science usually works better when people explore a variety of intuitions and options.  As I said in discussing the importance of collaboration in science:

Healthy scientific collaboration encourages reasonable dissent.   Otherwise group-think can insulate the community from effective criticism of accepted ideas.  Some people say that scientists should proportion their beliefs to the evidence.  However, there's also some value in diversity of opinion, because it permits subgroups to work on unpopular hypotheses.  I suppose things work best when the scientific community taken as a whole proportions its research work to the evidence.

It doesn't necessarily matter whether the source of the original intuition is something that could be accepted by all scientists.  What matters is that the resulting idea can be tested.  Sometimes, the original motivation for a successful scientific theory is rather dubious (e.g the Dirac sea motivation for antimatter), but nevertheless the resulting theory is confirmed by experiment and later is motivated by a different set of considerations.

So I don't believe in the complete separation of Physics and Theology, hence the blog.  But maybe I believe in something else which has some similar effects on my writing.  You must after all be detecting something about what I am doing which provoked your favorable statement.

Perhaps it is this: I believe in being honest.  I must to the best of my ability weigh the evidence on fair scales, and be open about what I am doing.  It would be dishonest if, because I want to prove the truth of Theism, I were to report the relevant Physics data in an imbalanced way, playing up anything which might seem to help my case and playing down anything which does not.  People often do this kind of thing reflexively when they argue, even to the extent of first deceiving themselves before they deceive others.  But it's still unfair tactics, especially when deployed by the expert against the layman.

It is not dishonesty for me to have my own views about what's important in Physics and what's not, but it would be dishonest if I implied that all physicists agreed with me about that when they don't.  Nor would it be dishonest if my views about speculative physics are influenced to some extent by my theological views—I think this is inevitable, and possibly not even fully conscious—but to pretend that a view is based on purely physical considerations when it is not, or to distort the data about Physics to match a preconceived agenda (theological or otherwise) is repugnant to me.

So I'll do the best I can to be honest, and hopefully that will tilt the scales in the right direction.

Once upon a time, a college friend and I planned to write a book about Science-and-Religion topics, but that never got off the ground.  A few of the ideas from that time are being recycled here.

I originally started this blog because an elder Christian whom I respect back in Maryland told me (and gave me to understand that it was a divine revelation to him, and I trust him to know the difference) that I should not neglect my gift of teaching when I went to Santa Barbara.  At first I tried to start a Bible study with my church, but it already had lots of other groups, and it kept not working out for various reasons; then I thought of the idea of blogging instead.

Once I reach a critical mass on the blog, perhaps some of them could be organized into book format.  But I don't need to decide that yet.  For the time being, the informal blogging environment seems more fruitful for developing ideas.

Posted in Blog, Ethics, Scientific Method, Theological Method | 14 Comments

Server back up

The blog was down for several days due to computer troubles.  There was a power failure in Mountain View while my parents were out of town, and then when it ended the wall.org computer booted up with the wrong operating system.  Just wanted to let everyone know that the problem is fixed now.

Posted in Blog | 1 Comment

Did the Universe Begin? VII: More about Zero Energy

A reader who wishes to be anonymous writes in with the following question:

I heard your paper referenced in the Carroll vs Craig debate, attempted to read it, then looked you up and found your blog (which I really like!!).  I’m fascinated by the origin of the universe and think it is a great argument for a creator.  I have a question I’m hoping you can help me with, or better yet, do a blog post on so I have something to reference!

Frequently when I debate an atheist online, they will bring up the argument that the net energy of the universe is zero and so the First Law of Thermodynamics was not violated at the origin of the universe since energy was still conserved.  As they explain it, the positive energy of matter is countered by the negative energy of gravity.  Our universe formed from a freak quantum fluctuation and is the ultimate free lunch.  I understand this at a very simple level, but what I do not understand is how a zero-energy universe matches what we observe.  If matter only makes up ~5% of the universe, 30% if you include dark matter, then how does the universe have a net energy balance of zero if 70% of it is dark energy pushing the universe apart through repulsive gravity?  It seems the expansion of the universe indicates a net positive energy.  Could you please give a simple layperson explanation for why folks like Hawking, Krauss, Guth, etc claim the universe has a net energy of zero?  It feels like there is a slight-of-hand going on and dark energy is being excluded, but I don’t know enough or have any sources to point to that say otherwise.

Dear Reader, thanks for your question.  I notice there's an interesting inversion here from the Carroll-Craig debate.  In that debate, St. Craig was trying to argue that the universe had a beginning, and Carroll was trying to outmaneuver him with the "Quantum Eternity Theorem", saying that the universe couldn't have begun if its energy isn't zero.  He then opened himself up to the retort that the energy probably is zero.

On the other hand, in your debate, it's the atheist who seems to be championing the position that the energy of the universe is zero.  Presumably this is because he wants to say that the universe emerged from a Nothing somewhat like the one Krauss' has in mind (though all this talk of Nothing doing things as if it were Something keeps reminding me of "The Nothing" in The Neverending Story...) and therefore `no room for a Creator' etc.  In this case the theist might argue that Energy Conservation makes this impossible (absent a miracle), opening herself up to the retort that the energy probably is zero.

So perhaps if you and Craig were locked in a room together, you might discuss whether a physics-type beginning of the universe is helpful or unhelpful, when arguing for Theism.  Alternatively, there could be a Krauss-Carroll debate about whether there's less "room" for a Creator with or without a beginning of time (both of them granting that the idea is absurd either way).  One could more or less construct such a debate just from their remarks directed against Theism already linked to on this blog.  Carroll could argue that in models like Aguirre-Grattan:

There is no room in such a conception [an eternal universe with the entropy lowest in the middle] for God to have brought the universe into existence at any one moment.

and Krauss could respond that:

It has become clear that not only can our universe naturally arise from nothing, without supernatural shenanigans, but that it probably did.

and Carroll could retort that:

That is not what the universe does even in models where the universe has a beginning, a first moment. Because the verb popping, the verb to pop, has a temporal connotation, is the word I'm looking for. It sounds as if you waited a while, and then, pop, there was the universe. But that's exactly wrong. The correct statement is that there are models that are complete and consistent in which there is a first moment of time. That is not the same as to say there was some process by which the universe popped into being.

Apologies to Krauss and Carroll for wrenching their remarks totally out of context, but I believe I have not done any violence to their actual views.  If you'd rather see what the real Carroll actually said about Krauss' conception, you can find that on his blog here.

But that wasn't your question.  Setting aside which team benefits more from it, what does physics say about whether the energy is zero?

As I said when discussing the "Quantum Eternity Theorem", there are lots of different concepts of energy in General Relativity, and even the experts sometimes find the relationships between them tricky to think about.  It's no wonder laypeople get confused when the "experts" make definitive sounding pronouncements about the subject.  If the energy at every point in the universe is positive, how could it possibly be true that the total adds to zero?

Well, the ``simple layperson'' explanation is that in cosmology, there's contributions to the energy both from 1) matter (baryons, dark matter, dark energy, etc.) and 2) from spacetime, stored in the gravitational fields.  There's a notion of energy density where you only count category #1, and then the energy density is positive.  But this notion isn't very useful for discussing things like energy conservation, since it isn't conserved in situations where space is changing with time (e.g. expanding).  There's another notion where we count both #1 and #2, and then it turns out that the contribution from #2 is negative and (in a finite sized "closed" universe).

That's the best I can do without launching into technicalities.  But I can't resist trying to say more about the real story, even if what follows may not really count as a simple layperson explanation.

Perhaps it would be easiest to explain if we start with a theory that's simpler than GR.  GR is in many ways quite similar to an easier theory of physics, namely Maxwell's equations.  Like the gravitational field, the electromagnetic field is sourced by a particular type of matter.  Gravitational fields are produced by the flow of energy and momentum through a spacetime, while electric and magnetic fields are produced by the flow of charge.

Let's just think focus on one of the Maxwell equations right now, the Gauss Law.  This is a special type of Law of Physics called a constraint.  That means, instead of telling you how things change with time, it places restrictions on what is allowed to be the case at a single moment of time.

The Gauss Law is written in equations like this:

\nabla \cdot E = \rho.

Here E is the electric field vector at any given point, and \rho is the rate at which charge is flowing through time at a given point.  Which is a really fancy way of saying, the charge density.  \nabla \cdot E means \nabla_x E^x + \nabla_y E^y + \nabla_z E^z, where \nabla_i means taking the derivative with respect to the i-th spatial coordinate.

But maybe you hate equations: if so you are in good company.  When I was at St. John's College we read a funny letter in which St. Faraday wrote to St. Maxwell, saying that he loved his work, but why did he have to write it using math?   St. Faraday, you see, lived in the time where you could still be a respectable scientist and explain everything using words.  Very carefully chosen words, expressing precise quantitative relationships.

Anyway, Faraday figured out this brilliant way to visualize the Gauss Law, which we still use as a crutch today.  Instead of thinking of E as a vector, you can think of it as a density of electric field lines passing through a point.  The direction of the vector says which direction the lines are going in, and the magnitude says how many there are.  I'm sure you've seen electric and magnetic field lines before, but if not, here are some pretty pictures on Google.

The Gauss Law says that electric field lines can only begin or end on charges.  The number of electric field lines coming out of (into) a charge, is proportional to the positive (negative) charge of the particle.  (We say "number" to make it easy to visualize, but in fact the field lines form a continuum.)

This means that if you have a region of space R, you can do a census of the total charge in that region, simply by measuring the total amount of electric field lines coming into or out of that region.   One can write this as an equation too:

Q_R = \int_{\partial R} E_n\,dA.

Here Q_R is the total charge inside the region R, \partial R is fancy-schmancy notation for the boundary of R, E_n is the number of electric field lines poking out per unit area, and \int dA tells you to integrate that over the whole area to get the total number of electric field lines poking out.  (Faraday would have said, why work so hard to invent these silly symbols when you could just say "count the number of electric field lines poking out"?)  We physicists call an integral like this a boundary term, because—go figure—it's the integral over a boundary of a region.

We are now in a position to appreciate the following interesting truth.  Suppose the universe is closed.  (That means, finite in size but without any boundary.  For example, space at one time could be shaped like a giant hypersphere; as we all know a sphere is finite in size but has no end.  Or like one of those video games where if you go off the edge of the screen on one side, you "wrap around" and appear on the other side, so that there isn't really an edge there.)  In a closed universe, the total electric charge is always EXACTLY ZERO.

If you're Faraday, that's because each electric field line has to either circle around in loops, or else begin on a positive charge and end on a negative charge.  So everything has to balance out.  If you're Maxwell, it's because if you take the region R to be the whole universe, then \partial R is the empty set, and so the Gauss Law just says Q_R = 0.

This doesn't necessarily have to be true if space is infinitely big.  You could just have a single electric charge sitting in infinite empty space, and this would be OK because the field lines beginning at the charge would go out to infinity, so they don't need another endpoint.

Now what about GR?  It turns out that things work in a very similar way, only using energy instead of charge.  If the universe were a single star or a galaxy sitting in an otherwise empty infinite space, then the gravitational ``field lines'' coming out of the mass extend out to infinity.  This allows the total "ADM" energy of the spacetime to be nonzero.  In fact, there is a Positive Energy Theorem in GR which says that, for reasonable types of matter, this energy is always positive for any state besides the vacuum (which has 0 energy).

On the other hand, if the universe is closed, then the total energy is zero because there's no boundary for gravitational field lines to go off to.  But how can this be, when the cosmologists tell us that the universe consists of about 5% ordinary matter, about 25% dark matter and 70% dark energy, and each of these components of energy is positive?

(I hate the term ``dark energy'', by the way, since it makes people think it's related to dark matter.  The two are nothing alike.  Dark matter is just some other kind of stuff, which clumps into structures.  The so-called dark energy is most likely just a cosmological constant, i.e. a constant positive energy density throughout all of space.)

To answer this, I need to remind you of how Einstein's equation of GR works.  The Einstein equation says how energy and momentum lead to spacetime curvature.  It can be written like this

G_{ab} = T_{ab}.

The symbol G_{ab} = R_{ab} - \tfrac{1}{2} g_{ab} R is called the Einstein tensor; basically it's a 4x4 symmetric matrix which encodes certain properties of the curvature of spacetime.  On the other hand, T_{ab} is the stress-energy tensor of matter.  This is also a 4x4 symmetric matrix, which encodes the rate at which momentum in the a-direction is flowing in the b-direction.  (The T_{tt} component, where both indices are chosen to be time, is just the energy density, since energy is momentum in the time direction.)

A key point here is that T_{ab} only counts the energy and momentum in matter.  It does not count the energy and momentum stored in the gravitational field (although by convention, these days most people include the cosmological constant or ``dark energy'' in T_{ab}).  When the cosmologists tell you about the "energy budget" of the universe, they are only really talking about T_{tt}.  They are ignoring the contribution from the gravitational field, which also contributes to the total energy of the universe.  It turns out that in a closed universe, the gravitational part (due to G_{tt}) counts negatively and this exactly cancels the matter contribution.

Defining the total energy of the universe is, as I said, quite tricky, since in the Hamiltonian formalism energy is related to time, and you have to make an arbitrary decision about what counts as the ``time'' direction.  You have to decide this separately for every single point, so there's actually a lot of arbitrariness here.  Once you've picked a time coordinate, if you want to evaluate the total energy on t = \mathrm{constant} slice \Sigma, the total energy H ends up being given by something like the following integral over the volume V of space:

H = \int_R (T^t_t - G^t_t)\,dV + \mathrm{boundary\,term}.

(If you don't know about tensor notation, just don't worry about the fact that one of the t's moved upstairs.  If you do, I've raised an index using the inverse metric g^{ab}.)  The boundary term is an integral \int_{\partial R} of something I'm not bothering to write down.

Now the t-t component of the Einstein equation, a.k.a. the Hamiltonian constraint, tells us that T^t_t = G^t_t.  So the whole thing boils down to a boundary term, and in a closed universe that has to be zero.  Thus, the ambiguity about time doesn't matter in the end, since "0" is conserved no matter what.

Posted in Physics, Reviews | 7 Comments

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.)

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Help with leaving comments

I have moved my comments policy, with help entering equations to its own page, accessible from the top bar above.

Recently, a few people have had trouble leaving comments on the site, due to incorrectly formatting equations, and some capricious comment mulilation by the WordPress software.  It took me a little while to figure out the exact rules since apparently the comment box which you would enter text into, has slighly different rules than the ones I enter text into as a logged-in user.

So here's the deal.  WordPress uses < and > to enclose html tags.  For example, if you write "<b>this is bold</b>" in a comment, you'll see "this is bold".  Unfortunately, this means that if you include an < followed by an > in your comment, WordPress will interpret whatever is in between as an html tag, and—even if it is not a valid html tag—will simply delete everything in between them!  Even I will be unable to see what you originally wrote.  So do NOT use > or < to mean greater than or less than (unless you use just one kind, or only >'s followed by <'s... but the safest rule is just to avoid them entirely...).

If you want to include greater than or less than symbols, you can write them as {\mathrm \$\$\verb|\|\mathrm{gt}\$\$} or {\mathrm \$\$\verb|\|\mathrm{lt}\$\$}.  That will look like > or < respectively.  Or better still, put your entire equation inside of the double dollar signs using LaTeX notation.

[quietfanatic points out in a comment below that you can also use the html escapes &gt or &lt, if you don't want to use the double dollar signs.]

Also, do NOT try to use wordpress.com latex notation (which has a single dollar signs, and the magic word "latex").  It won't work!  There is a difference between wordpress.com and wordpress.org.  The former is a website used to host WordPress blogs, while the latter is where you download software to host your own WordPress blog.  This is a WordPress.org blog which is hosted on my family server, wall.org.

In order to put LaTeX in a wordpress.org blog, you have to install a special plugin to do so.  You might think that this plugin would use the same notation as wordpress.com, but no it doesn't.  Instead you type \$\$\mathrm{E = mc\verb|^|2}\$\$ to get E = mc^2, and type \$\$!\mathrm{E = mc\verb|^|2.}\$\$ to break it out into a separate line like so:

E = mc^2.

  It's best not to put any spaces after the dollar signs in this case, or they'll make a weird indentation in the next line, as shown here.

Got it?  Good.

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In Peyresq

I'm going to be in a tiny village in France called Peyresq, for a quantum gravity workshop.  There won't be any internet access (except possibly for a hotel at the beginning and end), so don't expect much posting during that time.

Feel free to talk amongst yourselves, and/or wait patiently until I get back to the USA on the 21st.

When I do get back, I'll start cooking up the next post in my current series, probably about what happens if you use horizon thermodynamics (the GSL) in place of the ordinary Second Law of thermodynamics (OSL) to argue that the universe had a beginning, and maybe say some more about models like Aguirre-Gratton which don't have a beginning. Then maybe I'll take a more philosophical turn and start discussing different types of Cosmological Arguments for the Existence of God from a broader perspective.  There will be some important differences from Craig's approach as well as from Carroll's, so hopefully it should be interesting.

[By the way, Craig also has some post-debate reflections which I've linked to on my first post on the debate.]

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Did the Universe Begin? V: The Ordinary Second Law

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.

Posted in Physics, Reviews | 16 Comments