Undivided Looking

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:

$$H \Psi = i \hbar \frac{d}{dt} \Psi.$$

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 in 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 in 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.
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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).
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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.
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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:

$$H \Psi = 0.$$

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 in 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]

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:

$$ds^2 = dt^2 - a(t)^2 (dx^2 + dy^2 + dz^2).$$

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

$$a(t) = e^{Ht},$$

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.

Our best theory of gravity is classical General Relativity.  "Classical" is physics-speak for not taking into account quantum mechanics.  So we know that classical GR has to break down during the Planck era, if not later.

Classical Big Bang cosmology predicts that there is an initial singularity at the first moment of time.  In fact, there are some theorems to that effect.  These are the Penrose and Hawking singularity theorems, which will be the subject of this post.

In GR, attractive gravity is caused by the energy or pressure of matter.  Tension (which is negative pressure) produces antigravity (repulsion rather than attraction).

Very crudely speaking, the singularity theorems say that if you assume that matter obeys some energy condition restricting the amount of energy and/or pressure, then you can deduce that under certain conditions there has to be a place where your spacetime has an edge and cannot be extended any further.  This we call a singularity.  Typically, some component of the curvature becomes infinite at the singularity.

There are several different singularity theorems, pioneered by Hawking and Penrose.  One of them says that singularity theorem says that all expanding cosmologies like our own have to begin with a singularity.  Roughly speaking, it says that if there is only gravity and no antigravity, then tracing the universe backwards in time there is no way to stop it from crunching down to zero size.   Hence there must exist an initial singularity (at least somewhere, perhaps everywhere).

However, this Hawking-Penrose theorem uses something called strong energy condition, which says that the repulsive antigravity from tension is not allowed to be greater than the gravity from energy.  It turns out that the strong energy condition can be violated by lots of different types of otherwise reasonable physics theories.  In particular, it was violated during inflation, and it is violated by the cosmological constant today.  So no one really takes this theorem very seriously anymore.

There is another singularity theorem (proven originally by Penrose) which is better, because it only uses the null energy condition, which says that nearby lightrays are always focused by gravity.  This turns out to be a much weaker condition, which is satisfied by most respectable classical matter theories (although it is violated quantum mechanically).  However the Penrose singularity theorem only says that there has to be a singularity if space at one time is infinite.

If space at one time is finite in size (for example, if it is shaped like a 3-sphere) then there might be a "bounce" where the universe contracts to a small size and then starts expanding again.  The de Sitter cosmology is an example of this, although there are also examples of finite cosmologies that begin with singularities.  We don't really know whether space is finite or infinite, since inflation stretched it out so much that even if it were a giant sphere, the radius is so large that it seems flat today.

A few years ago I wrote an article in which I argued that the conclusions of the Penrose singularity theorem should continue to hold in quantum gravitational situations.  Even though the null energy condition can be violated by quantum fields, it turns out that you can get the same conclusions if you instead assume something called the "Generalized Second Law" (GSL), which says that the Second Law of thermodynamics applies to black holes and similar types of horizons.

(I described the application of this result to time travel in a  recent Scientific American blog post.  Technically, you have to use the time-reverse of the GSL, which I mentioned in the comments here, but if the GSL is true, its time-reverse should also be.  This may seem weird because normally we think of the Second Law as something which only works in one time direction, but I promise you that one can make sense of it.)

The advantage of using the GSL is that it makes it more plausible that the conclusions of the Penrose singularity theorem apply even in fully quantum-gravitational situations, e.g. during the Planck era.  In my article, I showed that the results apply "semiclassically", meaning when the quantum corrections to spacetime are small but still taken into account.   I then argued (and not everyone would find this part of my article persuasive) that under certain assumptions one might expect the result to hold even in full quantum gravity, when these quantum fluctuations are large.  But remember, all statements about quantum gravity are speculative.

I am a little reluctant to even bring up my own work, since personally I think it is more persuasive that clearly established (but incomplete) physics predicts a beginning, than that speculative new physics says this.  I think of it more as laying the groundwork for a possible future understanding, then a totally conclusive result.  Still, I think that the Penrose theorem is connected to enough other deep principles of physics that something like it will probably be true and important in the final theory of physics.  Other physicists think that singularities are so disturbing that any "complete" theory of physics should eliminate them.

Funny story.  One time I was arguing with an atheist grad student about God and the question of the universe's beginning came up.  I mentioned my own work (and I am quite sure that I put in some caveats about the potential limitations, since I always do this).  A few weeks later I found him posting on some atheist website cocky statements along the lines of "Theists believe that the universe had a beginning because of the GSL, but this is silly for the following reasons...".  And this at a time when practically no one else had even heard of my work!  So just in case it isn't clear: many smart people believed in God before I came along, and the case for Theism is hardly dependent on my tiny contributions to physics!

In conclusion, to the extent that the singularity theorems are relevant, they tend to point to a Beginning, although it might be possible to evade this conclusion either by (a) having space be finite, or else (b) through quantum gravity effects, if my speculative arguments for a quantum singularity theorem are wrong.

The next topic from the Carroll-Craig debate which I wish to discuss is what Science has to say about whether or not there was a beginning.  Was there a first moment of time, before which the universe did not exist?  What does Modern Cosmology have to say about this question?

I think that Modern Cosmology gives a fairly clear answer: probably, but not almost certainly.  But, rather than try to argue only for one particular conclusion, I will instead try to provide the evidence in both directions, on which my opinions are based.

The reason why I say probably is that, given our current best theories of the universe, there are some decent reasons to think that the universe had some type of beginning at the so-called "Big Bang".  However, once you get to an early enough moment of time, we don't really understand anything anymore, so really anything might have happened.  That is why the term "Big Bang Model" refers to the (very well-confirmed) theory of the expansion of the universe after the Big Bang, rather than to the Big Bang singularity itself.

Given our current best understanding of particle physics, we think we can describe fairly well the history of the universe starting at around $10^{-6}$ seconds after the Big Bang.  We're certainly on-base in the period from about 10 seconds to 20 minutes, since this is when Big Bang nucleosythesis occurred (creating the first atomic nuclei), and we can check that the current abundances of H, He, and Li atoms are in agreement with what our theory of nucleosynthesis predicts.

Inflation (which would have happened at a much earlier time) is somewhat less certain, but it makes pretty good predictions so almost everyone believes in it these days.  The recent BICEP2 results indicate that the energy scale of inflation was just a couple orders of magnitude below the Planck scale seem to have been contaminated by too much dust to be reliable, although most models of inflation still place it at a ridiculously high energy scale.  This is a much higher energy scale than anything else we can measure in physics, although it is comparable to the GUT scale (where most particle physicists, but not I, believe that the forces probably unify into one force).  During the inflation era, the universe grew in an extremely rapid way, stretching out and diluting any information about what the universe was like before inflation.

The Planck era was approximately the first $10^{-43}$ "seconds after" the "Big Bang".  This is the era where strong quantum gravity effects become important.  In other words, the quantum uncertainty in concepts of "space" and "time" become so large that our classical concepts break down.  That's why I put scare-quotes around things in this paragraph—we no longer know what on earth (or in the heavens) we are talking about.  This is the point when everything is pretty much up for grabs.

So, even if we can say there appears to have been a beginning based on an extrapolation of the Big Bang Model to early times, there are also reasons why we can't be completely sure, so long as we don't completely understand quantum spacetime (or the initial conditions for inflation).  Certainly the universe as we know it began, but we cannot completely eliminate the possibility of a pre-Big-Bang stage.

Nevertheless, in the next few posts I will discuss the limited evidence which we do have, especially those points which were mentioned in the debate.  In particular I will cover singularity theorems, the BGV theorem, the 2nd Law of Thermodynamics, the quantum eternity theorem.  Oh, and the Hartle-Hawking no-boundary proposal.  That too.

[Updated description of BICEP2 results]