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

O Teorema da Eternidade Quântica

Um Universo do Nada?

Apresentando a Buracos Negros #1 - Relatividade Especial
Apresentando a Buracos Negros #2 - Relatividade Geral

Apresentando a Buracos Negros #3 - Buracos Negros

Os Pilares da Ciência - Introdução
Os Pilares da Ciência #1 - Observação Repetida
Os Pilares da Ciência #2 - Hipóteses Elegantes
Os Pilares da Ciência #3 - Modelos Aproximados
Os Pilares da Ciência #4 - Descrições Precisas
Os Pilares da Ciência #5 - Integridade Ética

Os Pilares da Ciência #6 - Exame da Comunidade

Os Pilares da Ciência - Sumário
[abridged]

Depressão

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, which I have added to my sidebar, 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 update the list on this post (without making any other announcement).  Thanks so much to Felipe for his endeavers.

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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!

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Slightly Less Random Links

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!

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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 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.

Posted in Physics, Reviews, Theology | 35 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

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!

 

Posted in Physics, Reviews | 10 Comments