Category Archives: Physics

Did the Universe Begin? X: Recapitulation

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from http://www.xkcd.com/1352/.

We have now come to the end of my series about whether or not the universe had a beginning.  This is part of a longer series dissecting the debate between St. William Lane Craig and Sean Carroll.  I started out with some general reflections on the debate:

Thoughts on the Carroll-Craig Debate
God of the Gaps  (see also: Gaps at the Dinner Table)

Then I started talking specifically about possible evidence from physics for and against the universe having a beginning.  For ease of understanding I’m going to label each main new argument with FOR or AGAINST to define its main orientation, but the posts also deal with the various counterarguments (that’s the tire swing going back and forth above…). I’ve provided an executive summary of each of these posts, so that you can easily see the main thrust of what I said.  Minus all the caveats, hedging, and detailed explanations my scientific training tends to encourage.

(I’ve heard that politicians hate talking to scientists because, like the Elves in Tolkien, we seldom give a straight answer to a question.  In scientific cultures, we show “sincerity” by discussing all the problems and caveats with our ideas, whereas in political circles this sounds like insincere waffling designed to please too many people…)

Did the Universe Begin? I: Big Bang Cosmology (FOR, as far as it goes…)
– the classical Big Bang Model predicts an initial singularity where time began
– tentative because quantum effects were important and invalidate our usual geometrical notions
– also tentative because we don’t really know how inflation began

Did the Universe Begin? II: Singularity Theorems (FOR)
– classical General Relativity theorems by Hawking and Penrose
– assumptions of Hawking theorem invalid during inflationary epoch
– Penrose theorem says that if space is infinite, there was a beginning
– Penrose theorem invalid in quantum situations, but my work suggests that it might be extendable to quantum gravity, if horizons always obey the 2nd law of thermodynamics.

Did the Universe Begin? III: BGV Theorem (FOR)
– if the universe has a positive average expansion, then “nearly all” geodesics cannot be extended infinitely to the past
– implies that inflation had to have a beginning in time, at least in some places
– can evade theorem by a “bouncing” cosmology where the universe contracts and then expands

Did the Universe Begin? IV: Quantum Eternity Theorem (AGAINST)
– if the usual rules of QM hold at all times, you can calculate what the state would be at any time to the past or future.
– in realistic cosmologies the energy is probably either zero or undefined, making the theorem inapplicable.

Did the Universe Begin? V: The Ordinary Second Law (FOR)
– given reasonable assumptions, 2nd law of thermodynamics requires a beginning
– most plausible way to evade this is to postulate that the “arrow of time” reverses
– such models would have a “thermodynamic beginning” but no “geometrical beginning”

Did the Universe Begin? VI: The Generalized Second Law (FOR)
– second law of thermodynamics also seems to apply to cosmological horizons
– can be used like ordinary 2nd law to argue for beginning
– can also be used as singularity theorem (see II above)
– this closes certain loopholes, but if the universe is finite and the arrow of time reverses, a bounce may still be possible.

Did the Universe Begin? VII: More about Zero Energy
– a more technical explanation of why the energy of the universe can be zero

Did the Universe Begin? VIII: The No Boundary Proposal (AGAINST/FOR)
– a beautiful set of speculative ideas which unify the “laws of physics” with the “initial conditions”, by providing a rule for what the state of the universe is.
– contrary to popular conceptions, the Hartle-Hawking proposal has no beginning in time
– the Vilenkin tunnelling proposal is similar in spirit but does have a beginning.
– unclear whether these proposals are well defined, and Hartle-Hawking appears to give wrong predictions.

Did the Universe Begin? IX: More about Imaginary Time
– a more technical explanation about the notion of imaginary time used by Hartle-Hawking

If you put all of the physics information together, the conclusion I would draw is that: We don’t know for sure whether the Universe began, but to the extent that our present-day knowledge is an indicator, it probably did.  However, as Carroll correctly says, we can also construct models where it doesn’t have a beginning.  Taking into account known results from geometry and thermodynamics, the most plausible such models are 1) spatially finite, and 2) have a reversal of the arrow of time (e.g. the Aguirre-Gratton model).

I also noted that models like AG still have a low entropy “initial condition” somewhere in the middle of time.  One might think that this type of “thermodynamic beginning” still calls out for some type of explanation.

Then I wrote a more theologically-oriented post about whether the Hartle-Hawking no boundary proposal leaves any room for God to have created the universe:
Fuzzing into Existence
– short answer: yes, if you think of God as a storyteller, not a mechanic.

I also discussed the possibility of Reparameterizing Time; is it even meaningful to ask whether time is infinite or finite when you can change coordinate systems?  In this post I also argued that the main theological question of whether the universe needs an explanation seems to me much the same whether the universe has finite or infinite time.

Now, let me make another observation about the tire swing.  Although the weight of the evidence is that the universe probably had some sort of beginning—and even more likely that there was some sort of low entropy “initial condition” even if geometrically time stretches past before that—this cannot be said to be certain.  There is always the possibility that new scientific data or methods could radically change our picture of the very, very early universe.  Similarly, while a finite past seems more in accordance with traditional Christian theology than an infinite past, there appears to be no strictly logical connection between the two ideas, once the act of Creation is viewed in a more timeless, “authorial” way.  Thus one might conceivably have a theist who thinks time is infinite, or an atheist who thinks time was finite.

Should the argument for God’s existence really rest on such a slender foundation as the ultimate decision of physicists about Big Bang Cosmology?  Well, one thing is clear.  In ages past it didn’t depend on it.  Obviously, Sts. Abraham and Sarah, David and Solomon, the prophets and apostles, and all the men and women who followed in their footsteps up through the 19th century, including eminent scientists such as St. Faraday and St. Maxwell: these cannot have believed in God because of the Big Bang Theory, because—guess what?—nobody knew about it yet!  What does the Bible say about these people?

Now faith is confidence in what we hope for and assurance about what we do not see.  This is what the ancients were commended for.  By faith we understand that the universe was formed at the word of God, so that what is seen was not made out of what was visible.  (Hebrews 11:1-3)

Our belief that God is the Creator does not depend on the vicissitudes of scientific progress, the swinging back and forth of the tire swing (or is it accelerating?)  It doesn’t matter, because in this case we have a more certain source of knowledge than Science.

By faith!  The skeptic may scoff here, and say that faith is belief without evidence, but that is not the definition used in the passage above.  It says that faith is confidence about what we hope for, but do not see.  Unless we identify sight (conceived broadly as anything which can be directly experienced in terms of our 5+ senses) with evidence (things which allow us to conclude something about the world)—an identification which would incidentally also make Science impossible—the passage does not say that the ancients were commended for believing without evidence.  But the example of the biblical heroes does give some pointers about what type of evidence was relevant to them.

The ancients did not believe that God was the Creator because they had a detailed scientific theory about where it comes from.  (Indeed, if we take our minds off Genesis for a moment and read the Wisdom literature of the Bible: Job and Psalms and Ecclesiastes and Proverbs, the Scriptures seem to emphasize more our lack of knowledge about the details of creation, then any detailed programme of events…)  On the contrary, the ancient Jews and Christians knew God, by personal acquaintance as it were, and therefore knew him to be creative and powerful, mighty in word and deed.  Thus they could take him at his word that he is the Creator of all that we see.

The glory of Creation does indeed point to the glory of the Creator, so that it is possible for ordinary human reasoners to come to know that there is a Creator intellectually.  But this sort of Theism, by itself, isn’t what Christians mean by faith.  Once we come to know God personally, we learn the more important fact that we can trust him, and know with confidence that there is nothing in existence which does not depend on him.

And therefore, although we see in this world visible things emerging from other visible, material things, we know that ultimately their origin comes from “God’s invisible qualities—his eternal power and divine nature” (Rom 1:20).  He created everything through his Word, Jesus Christ, from whom we have come to know what God is like.  This way of knowing does not seem to depend very strongly on the details of past, present, or future scientific knowledge.

One could definitely argue that the Bible teaches that there was a Beginning (whatever this means from God’s perspective).  For example, the quotation above from Hebrews speaks of the formation of the visible universe.  But whether or not this fact has been revealed by God, it is not obvious to me that the most important theological aspects of Creation really depend essentially on time being finite, or even well-defined.  (Admittedly, if you believe that time is infinite, it might be easier to slip into a false notion whereby matter exists independently of God, who is merely the Chief Organizer of the cosmos.  That would be a heresy—a false belief which may seriously obstruct your ability to relate to God or others properly—but it does not follow necessarily from time being infinite.)

The main point of the doctrine of Creation, I think, is that God is real, and that everything else is derived from his power and will.  We know this doctrine is true because we know God.  Not because of the Big Bang, as natural as it is to connect the two ideas.

Reparameterizing Time

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In recent posts I’ve been discussing whether the universe began or not.

Perhaps the most important issue which I have not yet discussed, is the idea (I think originally due to Charles Misner, first pointed out by St. Edward Arthur Milne, and independently by St. Charles Misner) that it may not be well-defined whether time has a beginning or not.  That is, suppose you have a model in which there is a time coordinate \(t\), and time has a beginning in the sense that the only allowed times are \(t > 0\).  Well, in General Relativity we are free to use whatever time coordinate we like, and nothing stops us from defining a new time coordinate in terms of the old one, let’s say \(\tau = \log(t)\).  If you look at a plot of the log function, you’ll see that \(\tau\) ranges from \(-\infty\) to \(+\infty\).

However, this type of time reparameterization may not be very physical once you get down to the Planck time, about \(10^{-43}\) seconds, when quantum gravity effects become important.  Times less than that might not be well-defined.  In any case, the Misner argument suggests that we need to be more careful to define what we mean by time having a beginning.

Similarly, atheist philosopher Quentin Smith has argued that the standard Big Bang Model is inconsistent with divine creation, due to it not really having a beginning, even though the past is finite.  Smith argues that because the time \(t = 0\) is singular, technically it shouldn’t be included in the spacetime, so actually only times with \(t > 0\) exist.  That means that there is no initial moment of creation, and therefore, he claims, God cannot have created the universe.

This is somewhat reminiscent of Hawking’s claim that the no boundary proposal doesn’t have the right sort of beginning, and it seems to me that my Fuzzing into Existence post is also applicable.  If God is like an author, then he can make a story in which time works in whatever way he pleases.

According to Smith, each time \(t\) exists because the preceding times exist, and indeed the laws of physics hold at a given time \(t\) (according to him) because they hold at earlier times.  Since each moment of time is fully explained by those before, he claims that the universe is therefore self-caused and therefore fully explained, with no more explanation possible.  (Of course, if time is continuous, then we could make a similar infinite regress of times going back closer and closer to any finite time \(t\).  Smith has to struggle a bit to explain why his argument doesn’t apply there…)

Now to me, this seems like the sort of explanation which is really no explanation at all.  A satisfying worldview should explain as much as possible with as few assumptions as possible.  If the laws of physics have some property \(X\) (e.g. having an electron field, or whatever) now because they were like \(X\) a minute ago, and so on all the way back arbitrarily close to the beginning, that doesn’t in any way satisfy my curiosity about why they are like \(X\) instead of some other way \(Y\) (say, having no charged particles).  For if they had been \(Y\) for all time, I could have made the same argument.  So it seems that there is a potentially meaningful question “Why are the laws of physics like \(X\) rather than like \(Y\)”, which Smith’s statements do not really explain.  Maybe there is no explanation, and we have to take \(X\) being the way it is as a fundamental fact.  But to say that there could not possibly be an explanation seems rather dogmatic.

And if God exists, then he can explain this fact.  God’s will chooses what the laws of physics will be for all time.  So he can choose for the universe to be like \(X\) instead of like \(Y\).  This would be the fundamental explanation.  Whether or not it is a useful explanation for us as human beings, would depend on whether our puny minds can identify the actual reasons why God might prefer \(X\) over \(Y\).

The Kalam argument has some intuitive appeal if you think that the universe could not have begun without some causal reason.  Evaluating this claim requires an analysis of what causation is, and why one would think in various situations that a cause is necessary.  But the first preliminary question is whether there are any facts to be explained by the putative cause.  It seems to me that there are.

All of the same reasoning about \(X\) and \(Y\) would also apply if time stretches back to \(-\infty\).  There would still be various timeless facts about the universe which would not really be explained by the infinite regress.  This suggests that the Kalam argument may be misguided to the extent that it attempts to prove God from a temporal beginning a finite time in the past.  The most important issues are the same whether time goes back finitely or infinitely.

But having said all this, it does seem a little bit weirder that the universe should exist for a finite amount of time with no external explanation, than that it should exist for an infinite time with no explanation.   Historically, many materialists (such as Lucretius) have believed that time is infinite, due to their belief that it is impossible for something to come from nothing.  Conversely, monotheists have mostly believed that the universe has a beginning, either for philosophical reasons or because the Bible says so.  (St. Thomas Aquinas argued that God could have created an infinite past, but that divine revelation tells us he didn’t.) To that extent, Big Bang cosmology appears to vindicate the standard religious view over the standard nonreligious one.

(Of course, the same cannot be said if—unlike St. Thomas or St. Augustine—one also takes the 6 day creation about 6,000 years ago literally.  Some fundamentalists have argued that this problem can be solved by reparameterizing our coordinate system, but that just seems silly to me.  Also, the days are not in the right order to correspond to the scientific chronology.)

But a Theist could believe that God created time going back infinitely, without contradicting themselves, so long as they are prepared to be flexible about what “creation” means.  Similarly, an Atheist could believe that the universe just started existing 13.8 billion years ago for no reason, without contradicting themselves, so long as they are prepared to be flexible when deciding when explanations are called for.  All four views are logically consistent; the real question is which viewpoint explains the most with the least.

Fuzzing into existence

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In the last couple of posts, I’ve discussed the Hartle-Hawking proposal and the math behind it.  Now let’s discuss the theological implications.

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 there 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 the way he wants them 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.

Did the Universe Begin? IX: More about Imaginary Time

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In this post, I’ve put some more technical details about what the concept of imaginary time means, to help clarify the previous post about the Hartle-Hawking No Boundary Proposal.  If you don’t want to have to understand equations, skip this.

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 \(z = x + iy\).  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 \(f(z) = z^3 + z / (z^2 – 1)\), 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 $$e^{iy} = \cos(y) + i \sin(y).$$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 complex numbers, they actually make your life simpler!

So when you see something in a scientific equation like \(e^{ix}\), 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 \(x\), 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 \(|x|\), are not.  \(|x\) changes unpredictably at \(x = 0\); if someone told you what it looks like for \(x < 0\), and you tried to extrapolate it to \(x > 0\) 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 \(H\) of the state, as told to us by Schrödinger’s equation:$$H \Psi = i \hbar (d/dt) \Psi.$$If you solve this equation, you find that a state with definite energy \(H = E\) spins around as time passes like \(\Psi(t) = \Psi(0) e^{iEt/\hbar}\), where \(\hbar\) 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 \(p = e^{-E/T}/Z\), where \(T\) is the temperature, and \(Z\) 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: $$1/2T = it,$$where the factor of 2 comes from the fact that the probability is the absolute value squared of the wave function.

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: $$H \Psi = 0,$$also known as the Wheeler-DeWitt equation.  This equation seems to say that nothing changes with time, but it really means that the choice of time slice is arbitrary and has no coordinate-invariant meaning.

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 \(\Sigma\)) 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 \(\Sigma\),
2. For each geometry, integrate the total value of the Ricci scalar \(R\) over the 4 dimensional geometry, call that the action \(S\), and assign to that geometry the value \(e^{-S}\).
3. Figure out how to integrate \(e^{-S}\) 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 \(\Sigma\) 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!

 

Did the Universe Begin? VIII: The No Boundary Proposal

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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.