In his Brief History of Time (written 1988; I'm just going to be engaging with *this* book and not with any of his more recent pronouncements), Hawking has the following famous saying about the Hartle-Hawking state:

The idea that space and time may form a closed surface without boundary also has profound implications for the role of God in the affairs of the universe. With the success of scientific theories in describing events, most people

[!]have come to believe that God allows the universe to evolve according to a set of laws and does not intervene in the universe to break these laws. However, the laws do not tell us what the universe should have looked like when it started—it would still be up to God to wind up the clockwork, and choose how to start it off. So long as the universe had a beginning, we could suppose it had a creator. But if the universe is really completely self-contained, having no boundary or edge, it would have neither a beginning nor end: it would simply be. What place, then, for a creator?

The first question to ask here is who counts as "most people"?

The majority of people in the world believe in some type of God or gods capable of supernatural intervention. Even in the Western world, the majority* *of people believe in God (as Hawking indicates), and the majority of *those *believe in a religion called Christianity which teaches that God does produce miracles from time to time.

If Hawking means the English or the Europeans, then admittedly has been a marked decline in religious faith in Europe (much less so in the US) and many "Christians" there have a merely nominal or cultural affiliation. But belief in miracles is still far from nonexistent.

In any case, *I *am obviously not the target demographic, since I believe that God has done some remarkable things since that moment, perhaps 13.8 billion years ago, when he set the ball rolling. Or was there such a moment?

Hawking suggests that (if his model is correct) there was no such moment of creation. Not, according to him, because the universe goes infinitely far back in time—he says that it doesn't. Rather, because the geometry of spacetime is rounded off like a sphere, so that there is no *special *beginning point, but rather a whole region of points none of which would be any better or worse as a beginning. As he says:

The universe would be completely self-contained and not affected by anything outside of itself. It would just BE.

Now this only works if you go to imaginary time to describe the universe. With respect to real time, the Hartle-Hawking state *does *go back forever in time (with high probability). So if real time is what is important, then what Hawking says about the absence of a beginning is still true, although for a different reason.

If the Hartle-Hawking proposal is right, this could itself be taken as good reason to endorse an "imaginary time" view of the universe, although I'm not sure that's a consistent thing to do given that *we* at any rate seem to live in real time. But Hawking himself expresses a more ambivalent view:

So maybe what we call imaginary time is more basic, and what we call real is just an idea that we invent to help us describe what we think the universe is like. But, according to the approach I described in Chapter 1, a scientific theory is just a mathematical model we make to describe our observations: it exists only in our minds. So it is meaningless to ask: which is real, "real" or "imaginary" time? It is simply a matter of which is the more useful description.

Yet on this more positivistic view where the model is only aiming to be a "useful description", how could one use it to draw the metaphysical deductions Hawking wants to make, about there being no "place" for a Creator? But let's leave that aside, and accept the "imaginary time" point of view for purposes of our theological excursion, since it doesn't much matter whether the universe lacks a beginning because it's closed off like a sphere, or because it goes back in time forever.

Now when Hawking asks rhetorically whether there is a "place" for a Creator, the context suggests that he's not so much asking whether there's *good reason *to believe in a Creator, but whether there *even could be *a Creator, given the absence of a clear first moment of time. What would there be left for him to do? Aside from deciding *that *there should be a universe, selecting the laws of physics for said universe, deciding that the Hartle-Hawking state is the prettiest state for it to be in, and then (according to Hawking) deciding not to intervene even if it turns out we could use some help. Other than that, it seems like there is nothing left for God to do!

Really, Hawking is assuming (quite explicitly) that Science has *already* displaced God to such an extent that the only "place" that could be left for him is to push the button to make everything go, and then "sit back and watch". (This view is often *called *Deism nowadays, although historically Deists actually had a much more robust view of divine providence, and merely rejected the miracles and special revelations of particular religions.)

This rather limited God is the type of bad theology which makes religious people throw around the phrase "God of the Gaps", although I still believe that this term is highly misleading and should be retired. I tried to express a better set of points in that post:

1. Any time we ever believe in anything rationally, we do so because there is some kind of "gap" in our understanding of how the universe works, which is filled by postulating the existence of that thing.

2. All phenomena which occur in Nature do so because God sustains the world in being, thus (at least indirectly) causing everything.

Hawking allows no role for God as the Sustainer of all existence. But God's role in "sustaining" the world is not really a different type of act from his act of "creating" it. Hawking invites us to look at the world from a 4-dimensional perspective; in this perspective *all *points of spacetime exist because God gives them the power to exist, delineating the role that each one plays in the bigger scheme of things. From that perspective, Creation is something which is happening NOW, not just something which happened (or didn't happen) 13.8 billion years ago. Stated in a tenseless way, for all the things that exist, they exist because God chooses for them the conditions of their existence. (One of those conditions being that they are causally related in particular ways to the events before, after, or around them.)

God's role in creation is not a "mechanical" one, providing the initial impetus or force to get the machine working, which can then run for a while on its own. God is more like an Author writing a story. An Author stands outside the time-stream of their own story. As my dad said in a Slashdot interview:

Once you see the universe from that point of view, many arguments fade into unimportance, such as Hawking's argument that the universe fuzzed into existence at the beginning, and therefore there was no creator. But it's also true that the Lord of the Rings fuzzed into existence, and that doesn't mean it doesn't have a creator. It just means that the creator doesn't create on the same schedule as the creature's.

If God is creating the universe sideways like an Author, then the proper place to look for the effects of that is not at the fuzzy edges, but at the heart of the story. And I am personally convinced that Jesus stands at the heart of the story. The evidence is there if you care to look, and if you don't get distracted by the claims of various people who have various agendas to lead you in every possible direction, and if you don't fall into the trap of looking for a formula rather than looking for God as a person.

To think that God creates the universe and then stands back to watch it, is like thinking that an Author only has to write the first sentence, and then they can read the rest. Bad news for aspiring fiction writers: you have to write the whole thing. Maybe once the plot gets into full swing, the characters will start having a "mind of their own", and fail to act in the way the Author originally intended. But the Author is still in charge.

Nor does he have to "intervene" in order to get things to come out they way they want to: everything in the book is subject to the control of the Author, both the parts which follow naturally and inevitably from the previous scenes, and the parts where the Author does something totally unexpected. In any case, the main "point" of the story is seldom found right at the beginning, but develops as the story progresses.

Traditionally, books have a fixed and determinate sequence of letters, but if the Author wants to start out with something which doesn't have a definite time order (say a map on the first page) then that doesn't impugn their authorship of the rest of the book. And if the Author wants to make their book be infinitely long in both directions....well, that would probably be easier for God than for a human writer, wouldn't it!

So I think that belief in the creation of the universe does not really depend on there being a first moment of time. Conversely, this might also make one suspicious of the kalam argument championed by St. William Lane Craig in the debate. If the doctrine of Creation is not *about *there being a first moment of time, then there's something dubious about arguing for it as though it were. This doesn't automatically imply that St. Craig's argument is unsound, but it does suggest that it might not be the best way of looking at things.

Of course, we should also keep in mind what I said in my original post, that the Hartle-Hawking proposal is a speculative idea. It is a very beautiful idea, but it is difficult to make well-defined, and there is no direct evidence for it. While there was originally some reason to think it might predict inflation, the current indications seem to be that it predicts the wrong type of universe.

I remember my surprise when, several years ago, I read an article by the atheist philosopher Quentin Smith, showcasing the Hartle-Hawking state as an argument for Atheism. Never mind his actual argument, which makes no sense. In a talk given to some atheist club, he stated that his argument "is the strongest scientific argument there is against theism. I think it's even stronger than Darwin's theory of evolution."

Oh my! Neither Stephen Hawking nor Jim Hartle would make the claim that the Hartle-Hawking state is anywhere *near* as solidly supported as Darwinian evolution; in fact Jim told me just the other day that he isn't particularly committed to it being true. (People often assume that if a scientist *thinks *of an interesting, publishable idea, they must believe in it, but they might only think it is worth considering!) In fact, I think that only an outsider to the field of quantum gravity could take the "no boundary proposal" as anything other than a provisional, interesting idea worth exploring, which at best *might* be true.

I've discussed a lot of speculative physics in these last several posts, and I wouldn't want anyone walking away thinking that the physics is more clearly established than it is. In our current state of knowledge, any statements about the beginning of the universe are necessarily speculative, and if we rest our theological beliefs (for or against Theism) on that shaky foundation, we are setting ourselves up for trouble.

]]>First of all, a bit of remedial math. There are a lot of functions which (even if they teach them to you in school as being functions of *real* numbers) actually make sense when extended to complex numbers of the form . I already had to say something about complex numbers earlier in this series. If you know how to add, subtract, multiply, and divide complex numbers, you can pretty easily make sense out of polynomial fractions like , but you can also make sense out of things like sines and cosines and exponentials. For example, if we take an exponential of an imaginary number we get

This formula allows you to turn all sines and cosines into exponentials, enormously simplifying trigonometry by making it so you don't have to memorize a bunch of weird trig identities. So even though they call them

So when you see something in a scientific equation like , that looks like an exponential, but the power is imaginary, that's really something that's spinning around in the complex plane as you change , without growing or shrinking in its absolute size. It is a general rule that things which *oscillate *in the real direction correspond to things which exponentially grow and/or shrink in the imaginary direction, and vice versa.

This process of extending functions to the complex plane is called *analytic continuation*, and functions which can be so continued are called (wait for it!) *analytic*. (Not all functions are analytic: those which suffer from abrupt changes, like the absolute value function , are not. changes unpredictably at ; if someone told you what it looks like for , and you tried to extrapolate it to you'd guess wrong.

Now it turns out that there is a close mathematical connection between *quantum mechanics *and *thermodynamics* (a.k.a. statistical mechanics). Quantum mechanics is all about how the phase of a wavefunction oscillates around as time passes. The rate at which the phase spins around is proportional to the energy of the state, as told to us by Schrödinger's equation:

If you solve this equation, you find that a state with definite energy spins around as time passes like , where is Planck's constant.

On the other hand, statistical mechanics is all about thermal equilibrium states, and the rule of thermal equilibrium is that the probability to be in a given state *falls off exponentially* with the energy. The probability is proportional to , where is the temperature, and is an extra random thing called the "partition function'' you throw in to normalize the probabilities so they add up to 1. It turns out that states like these *maximize the entropy *given how much entropy they have. If you squint these two exponentials they start looking quite similar to each other, if only you can accept the mystical truth that *inverse temperature **is like imaginary time*:

where the factor of 2 comes from the fact that the probability is the absolute value

If you start with an initial condition where all states have equal probability, and "evolve'" for a finite quantity of "imaginary'" time, you end up with a thermal state ( after normalizing the total probabilities to be 1 at the end). Better still, if you start with (almost any) state and evolve for an *infinite *amount of imaginary time, you end up with the "vacuum" state of lowest energy, all other states being exponentially damped by comparison to that one.

Well, this may seem like a bit of mumbo-jumbo, but with the help of that complex number math I mentioned above, you can actually put it on a fairly rigorous footing, for ordinary QM systems, and even for quantum field theories. So of course, Hartle and Hawking had to be more bold than that, and try to apply this idea in the context of *quantum gravity.*

In quantum gravity (to the extent that we understand it), the dynamics are not governed by an ordinary Hamiltonian. Instead they are governed by a Hamiltonian constraint:

also known as the Wheeler-DeWitt equation. This equation seems to say that

Now the Hartle-Hawking prescription is really just a clever way to calculate one particular state which (at the level of formally manipulating equations that we can't really make sense of) solves the Wheeler-DeWitt equation.

It tells us the wavefunction of the universe, expressing the "quantum amplitude" for any possible metric of space at one time to exist. (The quantum amplitude is just a term for the complex number saying what the wavefunction is for a particular possibility to occur. Take the absolute value squared and you get the probability.) Since there are many ways to slice spacetime into moments of time, all of them have to exist side-by-side in this wavefunction, late moments in time no less than early ones. That's what it means to solve the Wheeler-DeWitt equation!

It's not the only solution to the Wheeler-DeWitt equation, but it's an especially nice one. In some ways it is like a "vacuum" state of the theory, one especially nice state to which others may be compared. (In other ways, it's more like a thermal state, due to the fact that there is only a finite amount of imaginary time evolution, before one reaches the end of imaginary time).

In order to calculate the Hartle-Hawking amplitude that a given geometry for 3 dimensional space (call it ) will appear *ex nihilo *(as it were), all you have to do is this:

1. Consider the space of all 4 dimensional curved *spatial* geometries whose only boundary is ,

2. For each geometry, integrate the total value of the Ricci scalar over the 4 dimensional geometry, call that the *action* , and assign to that geometry the value .

3. Figure out how to integrate over the infinite dimensional space of all possible 4 dimensional geometries. This requires choosing a measure on this space of possibilities, which is quite tricky for infinite dimensional spaces,

4. Cleverly dispose of several different kinds of infinities which pop up, and

5. Consider all possible choices of and figure out how to normalize it so that the total probability adds to 1 (nobody knows how to do this properly either).

Good luck!

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[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.

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 . The supposed "beginning of time" actually occurs at values of the time parameter that are imaginary! If you only think about values of 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 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 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.

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 wholeproportions 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.

]]>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:

Here is the electric field vector at any given point, and is the

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 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 , 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:

Here is the total charge inside the region , is fancy-schmancy notation for the boundary of , is the number of electric field lines poking out

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 to be the whole universe, then is the empty set, and so the Gauss Law just says .

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

The symbol is called the

A key point here is that 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 ). When the cosmologists tell you about the "energy budget" of the universe, they are only really talking about . 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 ) 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 slice , the total energy ends up being given by something like the following integral over the volume of space:

(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 .) The boundary term is an integral 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 . 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.

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 ). 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 , 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.)

]]>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 or . 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 > or <, 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 to get , and type to break it out into a separate line like so:

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.

]]>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.]

]]>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 the entropy takes some particular numerical value. As you go back in time, the Second Law says that 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 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 , 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 . 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 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 ), 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 , we find that is a decreasing function of . 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 . 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 *is *a beginning of time (due to being the lowest entropy state), and that as one travels away from to either positive or negative values of , 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

[]. That is, both the past and the future would be explained in terms of the low entropy state at , while the state at would itself have no explanation in terms of anything to the future or the past. (Thus the moment 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), a spatial reflection 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 , 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.

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