Updated to ask readers more directly for their thoughts, if you have any...

A random thought.  Suppose we ask whether the world has a Beginning or an End, or whether it is eternal is one or both directions.  It seems like there are 5 possible views, which I will name by association to various cultural groups who supposedly have had these views:

1.  Norse view: the world began, and it will end.
2.  Greek view: time is infinite in both directions
3.  Hindu view: time goes in a circle
4.  Hebrew view:  the world began, but it will never end.
5.  Nobody ever: time had no beginning, but it will still end!

I find it interesting that the first four views all have some intuitive appeal, to different people, but the fifth view just seems horribly wrong and perverse!  Why do you suppose that is?

My best guess is that there's is a certain obvious symmetry to treating the past and future in the same way, which makes views (1-3) seem reasonable.  And there is also an argument that the past is not like the future, but if so it had better be like (4) rather than like (5)!  I guess we all know deep down (it's really the Second Law of Thermodynamics) that it makes sense for the universe to start from a simple initial condition and then develop complexities from there.  But if we have to deal with infinite regresses AND we don't even get an eternal universe out of it, that seems a bridge too far... but if anyone has any further thoughts on this, I'd be interested.

My cultural names are a brutal oversimplification, and you shouldn't take my assigning these views to different cultures too literally.  For one thing, there were lots of different Greeks and there are lots of different Hindus who believe all sorts of different things.  For another, there is a conceptual difference between the world—in the sense of an ordered cosmology with a history—beginning, and time (a much more abstract notion) having a beginning. It takes a certain amount of intellectual sophistication to think about the latter question.

Norse mythology begins with fire and ice swirling around a bottomless pit for aeons; it is only later that a bit of fire strikes a bit of ice and spontaneously generates a giant and a cow, from whom later the jotun and gods emerge by various removes.  (As you can see, the Norse were ultimately Materialists even about their so-called divinities.)  At the end, the cruel jotun defeat the merry gods and the world is destroyed, plunging back into chaos.  So it's not really clear that time has a beginning or end, just that the story has a beginning and an end.

The Hebrews had the notion of divine Creation in Genesis 1:1 and elsewhere, but it is controversial whether Genesis 1:1 actually teaches the creation ex nihilo of later theology.  St. Augustine is usually credited with the idea that there was not even time before creation, but in fact Philo, a 1st century Hellenistic Jew, got there first.

Similarly, our current best "concordance cosmology" appears to begin with an initial singularity, but has no end in time.  (Well, really we should talk about spacetime, which allows time to end in some places, e.g. inside black holes, but not others.)  This appears at first sight to be like the Hebrew view.  At late times the universe expands exponentially forever, thinning matter out to a very cold but finite temperature.  This is in accordance with the Generalized Second Law of Thermodynamics, which tells us that the universe will reach a boring maximum entropy state at late times.   Thus, the story ends at finite time, and we really have the heroic defiance against inevitable destruction, as in the Norse view.

Even in Hebrew cosmology, there is that little matter of the whole universe being destroyed and then recreated again:

“See, I will create
new heavens and a new earth.
The former things will not be remembered,
nor will they come to mind.
But be glad and rejoice forever
in what I will create,
for I will create Jerusalem to be a delight
and its people a joy.
I will rejoice over Jerusalem
and take delight in my people;
the sound of weeping and of crying
will be heard in it no more.

Never again will there be in it
an infant who lives but a few days,
or an old man who does not live out his years;
the one who dies at a hundred
will be thought a mere child;
the one who fails to reach a hundred
will be considered accursed.
They will build houses and dwell in them;
they will plant vineyards and eat their fruit.
No longer will they build houses and others live in them,
or plant and others eat.
For as the days of a tree,
so will be the days of my people;
my chosen ones will long enjoy
the work of their hands.
They will not labor in vain,
nor will they bear children doomed to misfortune;
for they will be a people blessed by the Lord,
they and their descendants with them.
Before they call I will answer;
while they are still speaking I will hear.
The wolf and the lamb will feed together,
and the lion will eat straw like the ox,
and dust will be the serpent’s food.
They will neither harm nor destroy
on all my holy mountain,”
says the Lord.  (Isaiah 65:17-25)

It's really this "new heavens and new earth" that will last forever.  Christianity is about Death and Resurrection, both for the universe and for each person.  Science can get us as far as the doomed-to-die bit, but it can't get us any farther.  That is Law, the rest is Grace, revealed in Jesus Christ.

Posted in Scientific Method, Theological Method | 2 Comments

Reparameterizing Time

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 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.

Posted in Physics, Reviews, Theology | 6 Comments

Portuguese translations

I'm pleased to announce that some of my blog posts have been translated into Brazilian Portuguese.  St. Felipe of Olhar Unificado (added to blogroll) has translated the following posts:

Castidade: Não é apenas para religiosos

Deus das Lacunas

If you can't tell which posts of mine these are translations of, then this new feature might not be for you!  Felipe also has a bunch of his own writing on Science and Religion on Olhar Unificado, but I can't say much about it, other than that it appears to be composed of words relevant to the topic.  There are, however, pictures.

Regarding the post on chastity, I don't know how it reads to someone who actually understands Portuguese, but just looking at the words, my talk about sex and romance seems ever so much more impressive in a Romance language!

As Felipe comes out with more translations, I will add them to the list on this post.  Thanks so much to Felipe for his endeavers.

Posted in Blog | 1 Comment

Moving to the Institute for Advanced Study

In case you are wondering why I haven't been posting very much recently.... My postdoc at UC Santa Barbara ends this month, and on Sept 1st I will be starting an exciting new postdoc at the Insititute for Advanced Study near Princeton, as announced earlier.  I'm looking forward to some great conversations with people at the IAS and at Princeton.

So I've been rather tied up with finishing projects and packing books, not to mention two physics workshops I went to at McGill and at U British Columbia.  If there's ever a time when one is tempted to forswear possessions all together, it's when one is trying to clear out an apartment for a move.  Maybe we forget sometimes how practical the advice of Jesus is, to store up our treasures in Heaven rather than on Earth.

Really, it's kind of surprising I posted even once this month.  Hopefully things will be more peaceful once I arrive at Princeton, at least before all my stuff catches up with me!

Posted in Blog | 3 Comments

Those who have been following the debate between St. Craig and Carroll, or my own recent posts about it, might also be interested in the viewpoints contained in the following articles:

Carroll on laws and causation (St. Feser)

Cosmology and Theology (Stanford Encyclopedia of Philosophy)

Just to make things a bit more random, about a year ago my brother (St. Lewis) wrote a blog post about data loss and death.  This post was very difficult for me to read.  Death is not just a big problem but also a little problem, and that is why even children instinctively know about it.  Fortunately the Lord is greater than death, and will wipe away every tear from our eyes (Rev. 21:4).

I am reminded also of the words of the Lord, when he appeared to my best friend St. Yoaav three times in a dream, saying "I am a God of little things, and little things are preserved in me." (cf. Zech. 4:10, written when the Jewish Temple was being rebuilt by Zerubbabel in 516 BC.)  This occured at a time when he was getting serious about relating to God personally through prayer, but before his conversion to Christianity (from Judaism) the following year.

But we shouldn't get so distracted by our own little tragedies that we forget that we're destroying our own planet.  Some Christians think that because Jesus is going to come back soon, we don't need to take responsibility for the environment, because "people are more important".  They must not have read the passage of Scripture where Jesus says that "you do not know the day or the hour" (Matt. 25:13), or the place where it says that God will "destroy those who destroy the earth" (Rev. 11:18).  If the Master is a long time coming back, then how are we going to take care of all these people once our natural resources are all shot?  And when he does come back, he will judge how well we have fulfilled our responsibilities, one of which is to take care of God's creation as stewards.

True, God will restore all things in the end.  But that doesn't mean we won't be held responsible.  If you suffocate somebody in their sleep to get their money and then—surprise!—10 minutes later Jesus comes back and all the dead are raised and it's the Final Judgment, do you really think you won't be regarded as a murderer because, after all, you only deprived your victim of a few moments of relaxation?  I don't think so.  It will be the same if we are "saved by the bell" from the consequences of our own foolish decisions.  But we must also prepare for the possibility that we are here for the long haul, in which case the problem becomes all the more urgent.  Lord have mercy!

Fuzzing into existence

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 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.

Posted in Physics, Reviews, Theology | 12 Comments

Did the Universe Begin? IX: More about Imaginary Time

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

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:

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:

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:

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!

Posted in Physics, Reviews | 4 Comments

Did the Universe Begin? VIII: The No Boundary Proposal

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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