## In Peyresq

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

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

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

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

Posted in Blog | 1 Comment

## Did the Universe Begin? V: The Ordinary Second Law

The next piece of evidence we will consider for the beginning of the universe is the Second Law of Thermodynamics.  I sometimes call this the "Ordinary Second Law" to distinguish it from the "Generalized Second Law" which involves black holes (or other types of causal horizons).

The Second Law of Thermodynamics is a rather special law of nature because it distinguishes the past from the future.  It says that a quantity called the "entropy" always increases as time passes.  I've already written some articles explaining (1) what the entropy is (hint: it does not measure the amount of evil), (2) why it increases, and (3) how it has an interesting generalization to situations involving black holes (the "Generalized Second Law" mentioned earlier in this series).  Rather than repeat myself, I will link to these articles here:

If you're back from reading those—or if you're willing to take my word for it that there's a number called the "entropy" which has to increase as time goes on—then let's start by using it to make an argument that there had to be a Beginning.  Then we can see if there are any loopholes that can be used to evade the argument.

At each time $t$ the entropy $S(t)$ takes some particular numerical value.  As you go back in time, the Second Law says that $S$ gets smaller and smaller, until eventually it reaches its minimum value.  (Because of the way entropy is defined, it normally can't be a negative number, so the smallest it can get is 0.)

Now, either (a) there was a beginning of time, or (b) the entropy remained more or less at the same value for an infinite amount of time prior to some particular moment; let's call this $t_\mathrm{early}$ since it would have to be at least 13.8 billion years ago (since we know the entropy has been increasing since then).  But in that case, the universe would have had to be really boring for the first "half" of eternity $t < t_\mathrm{early}$, since interesting processes tend to produce entropy.  It's unclear what mechanism would cause the universe to suddenly become interesting.  Since scenario (b) seems implausible (though not necessarily impossible), we conclude that probably (a) is right, and there was a first moment of time.

Now, how might we evade this conclusion?  Here's three possible ways, although the first one doesn't really work, and the second one seems to run into some problems as well...

1. Thermal Fluctuations.  One way might be to take advantage of the fact that the Second Law is not an exact law of Nature.  Because it is statistical in Nature, the entropy can decrease, it's just very unlikely for it to decrease by large amounts.  But if you keep a system at maximum entropy for an very very long amount of time, eventually there will be thermal "fluctuations" in which the entropy gets down to arbitrarily small amounts.

So could our universe be a thermal fluctuation?  No, because a thermal fluctuation is unlikely to produce a whole cosmology filled with low entropy galaxies.  It would be much more likely for the fluctuation to produce the minimum amount of matter necessary to support a (briefly existing) intelligent life form (this is called a Boltzmann brain, by the way).  Since fluctuations are totally random, every possible matter configuration (with a given energy) would be equally likely, whereas elementary sanity says that this is not the case.

2. Shell Games with Infinity.  Another possible loophole is that actually none of this is well-defined because space is infinite and so $S = +\infty$.  Entropy could be produced both to the past and the future, but it wouldn't matter since the total amount is always infinity.

This loophole is used (e.g.) in the ekpyrotic scenario, a rather wild alternative to inflation in which there are membranes living in a 5th dimension which periodically collide with each other, causing Big Bounces (supposedly—this was really just a guess about what might happen).  The bounces are supposed to happen on a cyclic basis, so that the model is eternal in both time directions.  From the perspective of the 4 ordinary spacetime dimensions, the universe is infinite and expanding on average, which makes it so that the entropy "thins out" and prevents the universe from dying of heat death when its entropy reaches a maximum value.  Hence the BGV theorem tells us that the spacetime would have a beginning for most geodesics, even though some of them go back in time infinitely.

(Also, If the BICEP2 measurement of primordial gravity waves is right, that's also inconsistent with the ekpyrotic scenario.  Although there's some doubt now about whether BICEP2 properly screened for alternative sources of CMB polarization due to intervening dust.  Anyway the ekpyrotic scenario is just an example, not necessarily the only model like this.)

3. Arrow of Time Reversal.  This exploits the fact that we don't know the real reason why the Second Law is true in the first place.

Here is a paradox: the fundamental Laws of Physics are (more or less) symmetric between the past and future.  That is, if you replace $t \to -t$ in the equations, everything stays the same, more or less (*).  Yet, in the actual universe the past and future are quite different because of the Second Law, which says that the entropy is increasing.  And yet, the Second Law is regarded not as a fundamental law of Nature, but merely an effective statistical measure of what is most likely to happen given the fundamental laws of Nature?  So what gives?—How can you get a time asymmetric Law to pop out of time symmetric Laws.

The best people can tell is that the universe just started in a low entropy state.  It's a matter of the "initial conditions", not the Laws of Physics themselves.  (Although later we will discuss the Hartle-Hawking proposal, which blurs the boundaries between "initial conditions" and "Laws of Nature".)

Since we don't really understand why the universe began in a low entropy state, we are free to build (equally perplexing) models in which the entropy of the universe is small somewhere in the middle of time, rather than at the beginning.  If we assume the entropy was small at some time (let's make an arbitrary coordinate choice and call it $t = 0$), and then evolve that low-entropy state in both time directions, we typically find that the entropy will increase in either time direction.  Thus, for times $t < 0$, we find that $S(t)$ is a decreasing function of $t$.  We then say that the thermodynamic arrow of time is reversed.

This occurs in the Aguirre-Gratton model, in which the entropy decreases during a period of contraction, and then when the universe reaches its smallest size, there is a "bounce" instead of a singularity, after which the universe expands and entropy increases.  This model is symmetric under $t \to -t$.  Any people living in that time would (un)die and then shrink than be (un)born, but it would all seem just the same to them, because they'd also remember things backwards in time!

Sean Carroll and Jennifer Chen have also suggested a model like that, which involves many baby universes being created from an original inflating mother universe, whose arrow of time reverses.  (**)

During the debate, Craig focused most of his fire on the Carroll-Chen model, although Carroll modestly wanted to talk about the Aguirre-Gratton model instead:

So, I want to draw attention not to my model but to the model of Anthony Aguirre and Steven Gratton because this is perfectly well defined. This is a bouncing cosmology that is infinite in time, it goes from minus infinity to infinity, it has classical description everywhere. There is no possible sense in which this universe comes into existence at some moment in time. I would really like Dr. Craig to explain to us why this universe is not okay.

When Carroll says that there is "no possible sense in which this universe comes into existence at some moment of time", I think he is neglecting to consider that the thermodynamic arrow of time itself defines a notion of past and future.  There is a very real sense in which, in the Aguirre-Gratton or Carroll-Chen models, the time $t = 0$ is a beginning of time (due to being the lowest entropy state), and that as one travels away from $t = 0$ to either positive or negative values of $t$, one is travelling to the future in the sense that actually matters to us living and breathing creatures.  As I said in the concluding section of my own paper:

This kind of bounce evades both the singularity and thermodynamic arrow constraints, but still has in some sense a thermodynamic ‘beginning’ in time at the moment of lowest entropy [$t_0$]. That is, both the past and the future would be explained in terms of the low entropy state at $t_0$, while the state at $t_0$ would itself have no explanation in terms of anything to the future or the past. (Thus the moment $t_0$ would seem to raise the same sorts of philosophical questions that any other sort of beginning in time would.)

The Aguirre-Gratton model has no beginning in a geometrical sense, but it still has a beginning in a thermodynamic sense of unexplained "initial conditions".  Thus, I stand by my comments that an Aguirre-Gratton bounce raises the same sorts of questions as a more traditional "beginning" would.

Indeed, one could argue that the low entropy conditions of Aguirre-Gratton would be even more mysterious than in the traditional Big Bang model with a singularity.  In the latter case, there's a mysterious low entropy state, but it emerges from a singularity, and we don't know what laws of physics might exist at that singularity which cause the low entropy condition to emerge.  To some extent the mysteries cancel and make each other less mysterious, since it's not surprising that unknown causes should have unknown effects.

Whereas, if the low entropy condition occurs at a bounce, and the laws of physics there are by stipulation perfectly normal and comprehensible—and even so there is an additional low-entropy condition there, without any explanation in terms of anything else in the universe, either before or after it—then to me that suggests a need to find some sort of philosophical explanation for this strange phenomenon.

This would include potential Cosmological Arguments for the existence of God, although such arguments obviously have philosophical premises as well as physics premises.  This is made abundantly clear by the fact that Carroll doesn't accept the Cosmological Argument even on the assumption that there was a first moment of time.  One wonders therefore why he spent so much time trying to rebut Craig's claims that the universe probably did have a beginning, if it doesn't actually matter in the end.  (For purposes of the debate about God, I mean.  Obviously the cosmological origin of time is a fascinating question, which merits discussion even apart from any theological considerations!  Speaking as a physicist myself, I can certainly sympathize with Carroll getting sidetracked by interesting physics questions, as I've been doing myself throughout this series.)

Endnotes:

(*) Except for some tiny effects associated with the weak force which may not be relevant here, and even these are invariant under CPT, the combination of time reversal $t \to -t$ (T), a spatial reflection $x \to -x$ which switches left and right (P), and switching matter & antimatter (C).  Since the phrasing of the Second Law doesn't care about the distinction between matter/antimatter or left/right, one still has the question: why is the CPT-asymmetric Second Law true?

(**) For some reason in their paper Carroll and Chen wanted to have space be infinitely large even at $t = 0$, which runs into potential issues with the Penrose singularity theorem.  I wrote a paper with Alex Vilenkin slightly extending the classical singularity theorem in this context.  We showed that even if black holes form, the resulting singularities (inside the black holes) are not enough to satisfy the singularity theorem.  You need more of a "cosmological" singularity which is extended through space.  A bounce is not possible unless any observer that escapes to infinity is at least "completely surrounded" by an event horizon, beyond which there are singularities.

Posted in Physics, Reviews | 22 Comments

## The Name

In Ps. 110:4 the word LORD is spelled in all capitals. Why?

It stands for YHWH, the proper name of God which was revealed to Moses at the Burning Bush (Exodus 3:14-15), which is in turn short for the divine proclamation I AM WHO I AM.  This title declares God's eternal self-existence.

Thus, the words "the Lord" are not actually a translation, rather it is a substitution performed throughout translations of the Old Testament in order to avoid writing the Sacred Name.  In some of these translations, the word LORD is placed in all capital letters so that you know when this was done.  (In some Jewish versions of the Bible, they instead substitute HaShem (which means "the Name").  There is another Hebrew word "Adoni" which actually means a lord (this word could be used of a human ruler as well as of God), so that way you can tell which word is being used.  (These types of things can be learned by reading the "translator's preface" in the uninspired pages of your Bible located before Genesis 1:1.)

In the first verse of Psalm 110, we actually have both types of "Lord" appearing together.  King David prophesies of a greater king than himself when he says:

YHWH said to my Lord, sit at my right hand until I make your enemies a footstool for your feet. (Psalm 110:1)

Since Hebrew was written at the time written with consonants only, the vowels in between these four letters are unknown, but modern Hebrew scholars tend to think it was pronounced something like "Yahweh".  The old-fashioned pronunciation "Jehovah" was based on inserting the vowels in "Adonai".

The Third Commandment says "You shall not misuse the name of YHWH your God, for YHWH will not hold anyone guiltless who misuses his name." (Ex. 20:7).  God made himself more tangibly present to the Israelites by giving them his Name, but in doing so he opened up the possibility that his Name could be abused and used flippantly or as a curse, instead of as a blessing.  For this reason, God commanded that his Name, which made the Israelites holy, must be held with respect.  Vulnerability is the flip side of intimacy.  As it is written:

I will walk among you; I will be your God, and you will be my people. (Lev 26:12)

Some people might think that misuse of God's name can't really matter very much, but that would mean that using it doesn't matter either.  If reverent use of the Name sanctifies God's people, then irreverent use will, conversely, coarsen and corrupt the sense of being dedicated to God.  For this reason Jesus taught us to pray, "Hallowed be your Name" (Matt. 6:9).

This is one of several ways in which the Torah says that placed aspects of himself among the Israelites, others being the Holy Spirit, the Shekhinah [Glory Cloud], and the Temple.  Somewhat surprisingly, the Temple is often referred to as the place where God's Name dwells.  For example, when God promises King David that he will have a descendent who will reign forever, he says:

He is the one who will build a house for my Name, and I will establish the throne of his kingdom forever.  (2 Samuel 7:13)

Here "my Name" is a metonymy: it stands for the presence of the God whose Name it is.  As it is written:

Then I will dwell among the Israelites and be their God.  They will know that I am YHWH their God, who brought them out of Egypt so that I might dwell among them.  I am YHWH their God.  (Ex 29:43)

It was always part of God's plan to find a place to rest on the Earth.  Not that he can really be confined to a Temple built with human hands, for as Solomon says:

But will God really dwell on earth?  The heavens, even the highest heaven, cannot contain you. How much less this temple I have built!  Yet give attention to your servant’s prayer and his plea for mercy, Lord my God.  Hear the cry and the prayer that your servant is praying in your presence this day.  May your eyes be open toward this temple night and day, this place of which you said, ‘My Name shall be there,’ so that you will hear the prayer your servant prays toward this place.  Hear the supplication of your servant and of your people Israel when they pray toward this place.  Hear from heaven, your dwelling place, and when you hear, forgive.  (1 Kings 8:27-30)

Nevertheless, his Name can be spoken, and in this way he can be honored or dishonored.

At first, the Jews freely used the name YHWH to refer to God, even in ordinary conversation, as can be seen from the Old Testament.  However, as time went on, this came to be regarded as unsafe and presumptuous.  Eventually, the rabbis decided that the best way to satisfy the Third Commandment was never to say the Name at all.  That way there would be no chance of accidental blasphemy.  The High Priest could use the Name on Yom Kippur [The Day of Atonement], but that was it.  After the Temple was destroyed, the Name was never spoken again by halakha-compliant Jews.

This was the opposite mistake from careless or flippant use of the Name.  Instead they played it so safe that God may as well not have revealed the Name in the first place.  God had put himself on a first name basis with Israel, so to speak, and they declined to exercise the privilege.

So what did God do?  Did he rectify this situation by telling people they were missing out?  Did he put things back the way they were before?  No, instead, in his divine wisdom he had a completely different plan.  As it is written:

Shout and be glad, Daughter Zion. For I am coming, and I will live among you,” declares YHWH. “Many nations will be joined with YHWH in that day and will become my people.  I will live among you and you will know that YHWH Sabaoth has sent me to you.  YHWH will inherit Judah as his portion in the holy land and will again choose Jerusalem.  Be still before the Lord, all mankind, because he has roused himself from his holy dwelling.” (Zech. 2:10-13)

Rather than restore the old intimacies, God gave new ones.  He himself came and lived among us.  Greater intimacy implies greater vulnerability.  The new name he has given us to speak aloud is the name of "Jesus",

Who, being in very nature God,
did not consider equality with God something to be used to his own advantage;
by taking the very nature of a servant,
And being found in appearance as a man,
he humbled himself
by becoming obedient to death—
even death on a cross!
Therefore God exalted him to the highest place
and gave him the name that is above every name,
that at the name of Jesus every knee should bow,
in heaven and on earth and under the earth,
and every tongue acknowledge that Jesus Christ is Lord,
to the glory of God the Father.  (Phil. 2:6-11)

When Paul says that God gave to Jesus "the name that is above every name", there is only one Name which a first century rabbinically educated Jew could have in mind.  The first century Christians might not have ever spoken the name YHWH, but in passages like this there is an implicit allusion to the Name of God.

When you see the name Lord in the New Testament, this is the Greek word kyrios.  In principle, this word, like Adoni, could itself refer to an ordinary human being.  However, the New Testament writers continually quote from the Greek Septuagint, which uses kyrios as its substitution for YHWH.  This puts an important overtone into the word.

People argue about whether the New Testament really refers to Jesus as God.  What they don't realize is that in the Greek linguistic context, calling Jesus theos is actually a surprisingly ambiguous statement of Jesus' divinity (and many of the passages calling Jesus theos can be explained away).  What is quite unambiguous is calling Jesus kyrios and then applying to him Old Testament passages in which kyrios is a substitution for YHWH.  For example "Everyone who calls on the name of the Lord will be saved (Rom 10:13)"

To those of my readers who are baptized, remember that you have been dedicated into "the name of the Father, the Son, and the Holy Spirit"!   What name is this?  It is a singular name.  "The Father, the Son, and the Holy Spirit" is grammatically plural, not a single name.  So the "name" must instead refer to some unmentioned title which belongs to all three persons equally.  The mystery of our faith is that all three persons are one YHWH, the eternal existence.  This passage is another example of an implicit reference to that Name which no first century Jew would ever say.

This is the Name into which you were baptized, making you into God's Temple.  If you defile this Temple, you also dishonor the Name which God placed in you.   But if you honor him, he will honor you, by coming to live with you.

Posted in Theology | 10 Comments

## Did the Universe Begin? IV: Quantum Eternity Theorem

Having pointed out that the BVG theorem presupposes the existence of a classical spacetime, Carroll goes on to cite some evidence that the universe did not have a beginning, based on quantum mechanics (QM):

If you need to invoke a theorem, because that’s what you like to do rather than building models, I would suggest the quantum eternity theorem. If you have a universe that obeys the conventional rules of quantum mechanics, has a non-zero energy, and the individual laws of physics are themselves not changing with time, that universe is necessarily eternal. The time parameter in Schrödinger’s equation, telling you how the universe evolves, goes from minus infinity to infinity. Now this might not be the definitive answer to the real world because you could always violate the assumptions of the theorem but because it takes quantum mechanics seriously it’s a much more likely starting point for analyzing the history of the universe. But again, I will keep reiterating that what matters are the models, not the abstract principles.

First of all, some background.  In QM, there's a gizmo called the wavefunction $\Psi$.  This is the thing that tells you what are the probabilities for any particular thing to be happening, at any given moment.  It involves specifying a complex number $z = a + bi$ for each possible configuration of the universe.  Complex numbers are two-dimensional, so they have both an absolute value (or magnitude) $|z| = \sqrt{a^2 + b^2}$ and a phase (or direction) in the two dimensional plane.  The square of the absolute value $|z|^2$ gives you the probability to be in that state, while the phase (or direction) of the complex number is an additional weird extra piece of information which is special to QM.  (There's some deep conceptual issues about what the wavefunction "really" means, but let's not get into that here.)

In ordinary QM, the wavefunction of the universe changes with time.  If you want to work out how it changes with time, you need to know the formula for the total energy of the universe, written out as a function of the positions and momenta of all the particles or fields.  Once you know what $H$ is, you can then use Schrödinger's equation:

This equation tells you that if your state $\Psi$ is in a state with a specific energy $H \Psi = E \Psi$ (this is called an energy eigenstate), then its phase just spins around and around, at a rate proportional to the energy $E$ divided by Planck's constant $\hbar$.  That's rather boring, since it would mean that none of the probabilities actually change at all.  On the other hand, if you have a state where the energy has quantum uncertainty (meaning that it is actually a superposition of states with definite energy) then more interesting things can happen due to interference patterns between the different energy eigenstates.

So, if you know what H is (that specifies the dynamics of your theory) and you know what the wavefunction $\Psi$ is at some specific time $t_1$, and if you assume that this theory is valid at all moments of time, then you can work out what $\Psi$ is at any other moment of time, past or future.  And it particular, you know what it would have been at a time $t_2$ which is arbitrarily earlier than $t_1$ is.  Hence—so Carroll's argument goes—the universe cannot have had a beginning.

That's all the Quantum Eternity Theorem (QET) says.  It's a little bombastic for Carroll to even refer to this as a "theorem", since it's just an elementary restatement of one of the most basic principles of QM.   As Carroll said in his post-debate reflections:

For convenience I quoted my own paper as a reference, although I’m surely not the first to figure it out; it’s a fairly trivial result once you think about it.

You could still imagine that God miraculously created the universe at a given moment of time $t = 0$, and that the laws of physics only apply after that moment of time.  Then physics as such would have nothing to say about the actual Beginning, but only what happens after that.  There's no logical contradiction in saying that, but it might make some people uncomfortable if—so far as we can tell from Science—the universe has to have lasted forever.  In some ways, this is the position Christians were in prior to Modern Science, when the study of the heavens seemed to indicate that the universe just kept going on and on, like a clock that never needs winding up.  Back then, Christians mostly believed there was a Beginning for philosophical reasons, or else because it said so in the Bible.  We now know that the Universe developed from a simpler form, and that it has only existed in its currently observable form for about 13.8 billion years.  The scientific case for a Beginning is certainly much more conclusive now than it was then, since back then there wasn't much of a scientific case at all!

But if Carroll's QET does apply, then no matter how many fireworks there were at the "Big Bang", it could only really have been the universe passing from one form to another.  So is he right?

Probably not.  Carroll himself states the important loophole in his reasoning, although he does it in a kind of a cryptic way so that only another physicist like me knows what it really means.  Let's have it again:

If you have a universe that obeys the conventional rules of quantum mechanics, has a non-zero energy, and the individual laws of physics are themselves not changing with time, that universe is necessarily eternal.

What Carroll neglected to say during the debate, is that there's very good reason to believe that the energy of the universe is zero (if it is defined at all).

It's actually rather tricky to make precise the concept of "energy" in General Relativity.  The reason is that energy is defined with respect to how things change with time, and time is a rather slippery concept in GR.  There isn't just one notion of time, but rather any choice of "t" coordinate you might choose is equally valid.  If there's no well-defined concept of time, then there's also no well-defined concept of energy, and the QET won't apply.

So when people do refer to energy in GR, they need to be some type of special situation that allows them to invoke the concept.  Here are the cases people talk about most often:

1. If we zoom in close to one point, we can adopt a particular local inertial reference frame and define the energy of an object using that local coordinate system.  But Special Relativity tells us there are several equally good notions of time , and even those are only good in the neighborhood of a single point, so this won't work for the QET.
.
2. If you have a spacetime which is approximately unchanging with respect to some special time coordinate "t", you can define the energy of objects with respect to that time coordinate, as long as their gravitational field is small.  This is called the Killing energy, but this is also inapplicable in cosmology since the universe is not anywhere close to static (it is expanding).
.
3. If you have a system of objects sitting by themselves inside an otherwise empty infinite space, then you can use the notion of time defined by a clock which is very far away from the system.  This is called the ADM energy, and it tells you the effective gravitational mass of the system as measured from far away.  But this is also inapplicable to cosmological settings, since so far as we know the universe is not a clump of matter in an empty space.
.
4. Finally, if you have a closed universe (one with no boundary) then there is an unambiguous notion of energy associated with the gravitational Hamiltonian $H$.  However, it is exactly zero for all physically allowed states: $H = 0$!

The conventional view of researchers in quantum gravity—with, apparently, the exception of Carroll himself—is that the same thing is likely to be true in quantum gravity.  That is, instead of the usual Schrödinger's equation, the dynamics of the theory are encoded in the Wheeler-DeWitt equation:

Now since $H$ tells us how $\Psi$ changes with time, the Wheeler-DeWitt equation tells us that the quantum state does not change with time at all!  That's weird, since we all know that things do change with time.

Does that mean that Zeno was right and time is an illusion?  Well, we have to be very careful with interpretation here.  The real reason why this happens in gravitational theories is because the choice of spacetime coordinates is arbitrary—you are free to label your spacetime points with any $(x,y,z,t)$ coordinates you like: there is not one "best" way to do it.  (Although I've been focusing on General Relativity, physicists expect similar issues to pop up in almost any decent theory of gravity.  So long as it does not reintroduce a notion of absolute Newtonian time, there will necessarily be a "Hamiltonian constraint" saying that the only physically allowed states of a closed universe are those for which $H = 0$.)

So when we say that the wavefunction doesn't change with time, what this really means is that the choice of time coordinate is arbitrary.  "Time" needs to be measured relative to some physical clock.  There is no absolute "t" coordinate relative to which everything else moves,  So, I think I would say that in this case, the QET "applies", but in a totally trivial way, and when you unpack its real meaning, it doesn't tell us anything about whether or not there was any time before the Big Bang.  Thus the formalism of ordinary QM is not applicable.

To summarize, in a closed cosmology, the energy is zero, and in an open cosmology it might not even be defined.  Thus Carroll's appeal to the QET probably doesn't make sense.

As I said to Carroll in the comments to his post-debate reflections:

Regarding the QET, to my mind the most conservative belief about quantum gravity is that it is—like GR—governed by a Hamiltonian constraint rather than an ordinary Hamiltonian (as in standard QM). In this setup, it’s not obvious that the QET applies.

What’s more, since there is no “absolute time” in GR, there are lots of different, inequivalent ways to evolve space forwards in time. As Wheeler put it, time is many fingered. This concept of time evolution will be much more subtle to quantize, and it’s far from obvious (to me, at any rate) that it’s forbidden for time to begin or end. In any case, this is quantum gravity, so none of us really know what we’re talking about!

And he replied:

Aron– That’s certainly a respectable point of view. It’s basically choosing the option that the energy is zero, which is definitely a possibility. And if that does turn out to be the case, time can certainly “end,” but in a very funny sense, since “time” was only emergent to begin with.

But the other option, that the energy is not zero and the ordinary time-dependent Schrödinger equation applies, is at the very least equally reasonable (perhaps more so). Our best-understood example of quantum gravity is the AdS/CFT correspondence, where the theory is most carefully defined in terms of the Hamiltonian of the boundary theory — in which perfectly conventional Schrödinger evolution applies. My suspicion is that quantum gravity will work similarly in other cases as well. But as you say, it’s quantum gravity, so we’re allowed to speculate but not allowed to act like we know the answer.

AdS/CFT is a famous duality between an ordinary QM theory (the CFT) and a gravitational (string) theory with a negative cosmological constant.  In this case, there is a well-defined nonzero $H$, but that is because you have a bunch of matter sitting in an otherwise empty AdS space, so you can use the ADM definition of the Hamiltonian.  (This duality tells us very interesting things about general aspects of quantum gravity, but it probably doesn't apply directly to our own universe, which has a positive cosmological constant, among other considerations.)

GR predicts (A) that $H \ne 0$ for matter sitting in empty AdS space, and (B) that $H = 0$ for closed universes.  It doesn't make any sense to me to say that because string theory agrees with GR about (A), it probably disagrees with GR about (B).  To me, the most conservative thing to say is that both of these facts continue to be true.  Furthermore, case (B) is far more likely to describe the real universe than (A) is.

Although, as we both said to each other, no one really knows for sure how the correct theory of quantum gravity is going to be formulated.  Of course, there is nothing wrong with Carroll putting forward his personal opinion in the debate—I can hardly complain about that after Craig put forward my opinions!  But I think he could have been more clear that it was his personal opinion, and that, given more "conventional" beliefs about quantum gravity, the QET probably can't be applied in cosmological settings.

[9/22/14: a few minor wording changes—AW]

Posted in Physics, Reviews | 18 Comments

## Did the Universe Begin? III: BGV Theorem

There is a theorem due to Borde, Guth, and Vilenkin which might be taken as evidence for a beginning of time.

Roughly speaking, this theorem says that in any expanding cosmology, spacetime has to be incomplete to the past.  In other words, the BGV theorem tells us that while there might be an "eternal inflation" scenario where inflation lasts forever to the future, inflation still has to have had some type of beginning in the past.  BGV show that "nearly all" geodesics hit some type of beginning of the spacetime, although there may be some which can be extended infinitely far back to the past.

If we assume that the universe was always expanding, so that the BGV theorem applies, then presumably there must have been some type of initial singularity.

The fine-print (some readers may wish to skip this section):
[BGV do not need to assume that the universe is homogeneous (the same everywhere on average) or isotropic (the same in each direction on average).  Although the universe does seem to be homogeneous and isotropic so far as we can tell, they don't use this assumption.

More precisely, let $H$ be the Hubble constant which says how rapidly the universe is expanding.  In general this is not a fully coordinate-invariant notion, but BGV get around that by imagining a bunch of "comoving observers", one at each spatial position, and defining the Hubble constant by the rate at which these observers are expanding away from each other.  The comoving observers are assumed to follow the path of geodesics, i.e. paths through spacetime which are as straight as possible, that is without any acceleration.

Now let us consider a different type of geodesic—the path taken by a lightray through spacetime.  Now if the average value $H_\mathrm{avg}$ along some lightlike geodesic is positive, then BGV prove that it must reach a boundary of the expanding region in a finite amount of time.  In other words, these lightlike geodesics reach all the way back to some type of "beginning of time" (or at least the beginning of the expanding region of spacetime which we are considering).

We can also consider timelike geodesics, describing the motion of particles travelling at less than the speed of light.  For nearly all timelike geodesics, if $H_\mathrm{avg} > 0$ then that geodesic also begins at a beginning of time.  However, the theorem only applies to geodesics which are moving at a finite velocity with respect to the original geodesics which we used to define $H_\mathrm{avg}$.  The original set of observers is allowed to extend back infinitely far back in time.

As an example of this, one can consider a spacetime metric of the following form:

If we set the "scale factor" to be exponentially inflating:

then such a universe extends infinitely far to the past from the perspective of an observer who remains at a fixed value of $(x,\,y,\,z)$.  But nevertheless, observers travelling at a finite velocity relative to those hit a beginning of time (or else exit the region of spacetime where this metric is valid).

Since the BGV theorem only refers to the average value of the expansion, it applies even to cosmologies which cyclically oscillate between expanding and contracting phases, so long as there is more expansion (during the expanding phases) then there is contraction (during the contracting phases).

On the other hand, in certain cases even an expanding cosmology may have 0 average expansion, due to the fact that we are averaging over an infinite amount of time.  So the BGV theorem does not rule out e.g. a universe where the scale factor $a(t)$ approaches some constant value in the distant past.]
The fine print is now over.

All right, everyone who skipped the details section is back, yes?

The BGV theorem is sometimes referred to as a "singularity theorem", but it is not really very closely connected to the others, because it doesn't use an energy condition or any other substantive physical assumption.  It's really just a mathematical statement that all possible expanding geometries have this property of not being complete.

Carroll correctly observes that the BGV theorem relies on spacetime being classical:

So I’d like to talk about the Borde-Guth-Vilenkin theorem since Dr. Craig emphasizes it. The rough translation is that in some universes, not all, the space-time description that we have as a classical space-time breaks down at some point in the past. Where Dr. Craig says that the Borde-Guth-Vilenkin theorem implies the universe had a beginning, that is false. That is not what it says. What it says is that our ability to describe the universe classically, that is to say, not including the effects of quantum mechanics, gives out. That may be because there’s a beginning or it may be because the universe is eternal, either because the assumptions of the theorem were violated or because quantum mechanics becomes important.

It is quite true that the BGV theorem is proven only for classical metrics, although I see no particular reason to believe that its conclusion (if the universe is always expanding, than it had an edge) breaks down for quantum spacetimes.

However, Carroll's secondary point that the assumptions of the theorem might not hold seems even more devastating.  It says that there must be a beginning if the universe is always expanding.  So maybe have it contract first, and then expand.  That's an easy way around the BGV theorem, and (as Carroll points out) there are a number of models like that.  On this point I agree with Carroll that the BGV theorem is not by itself particularly strong evidence for a beginning.

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