Did the Universe Begin? VI: The Generalized Second Law

Last time I discussed the cosmological implications of the regular-old Ordinary Second Law.  Now I want to discuss what happens if you use the Generalized Second Law instead—this being a generalization of the Second Law to situations involving black holes and other horizons, which seem to have an entropy proportional to their surface area.

I started thinking about this issue after Sean Carroll gave a colloquium at U Maryland about the Carroll-Chen model, and the Second Law, back when I was a grad student.

From my perspective, the important thing about that colloquium was that it got me thinking about refining the classic argument that the Second Law predicts a beginning.  I said to myself something like the following:

"Self, for the past couple years you've been spending all of my time thinking about the Generalized Second Law (GSL), that wild new version of the Second Law which applies to causal horizons.  Well, there are horizons which appear in our own cosmology (because of the accelerating expansion of the universe).  So can we make this argument using the GSL instead of the Ordinary Second Law (OSL)?  And if we do, will it make the argument stronger or weaker?"

Well, when I thought about it a little bit, I realized that you could use the GSL in two distinct ways to argue for a beginning.  One of them is a quantum generalization of the Penrose singularity theorem, which I discussed here.  The other way is a generalization of the Argument from the Ordinary Second Law, described above.  Both of these uses of the GSL are discussed in my article, but it is important to realize that they remain two distinct arguments!

Fine-grained vs. Coarse-grained.  The reason is that there are actually two subtly different ways to formulate the GSL.  You see, the entropy is a measure of our ignorance about a system.  To exactly define it, you need to make a list of the things you are allowed to measure about the system (e.g. the  pressure and temperature of a box of gas), and then the entropy measures how much information content is in the things you can't measure (e.g. the positions or velocities of individual molecules).  The procedure of ignoring the things we can't measure is called coarse-graining (because it's like looking at a grainy photograph where you can't see all of the information in the object).

Technically then, there's some ambiguity in the definition of the entropy, since the intitial step where we list what we can measure is a little bit ambiguous.  Fortunately, since the amount of information we can't measure is much larger than the information we can, this doesn't usually matter very much.  Quantitatively, the different ways of defining entropy give pretty close to the same numerical answers.

But we could pretend that we could measure everything about the box of gas to arbitrary accuracy.  The only uncertainty allowed which could produce a nonzero entropy is uncertainty about the initial conditions.  This is called the fine-grained entropy, and while it has the property that it neither increases nor decreases as time passes.  Since the fine-grained entropy can't decrease, it technically obeys the Second Law, but in a really boring and stupid way.

The distinction becomes important when you start talking about black holes and the GSL.  Suppose you have a star orbiting a black hole.  Matter from the star is slowly getting sucked off the outer layers of the star, and getting sucked into the black hole.  (This is a realistic scenario which is believed to really occur in some solar systems, by the way!)

Well, we have a choice.  We could use a coarse-graining to describe the entropy of the star.  In that case, the entropy would go up for 2 distinct reasons: A) because stuff is falling into the black hole making its area increase, and B) because ordinary thermodynamic processes are happening inside the star, making the entropy increase for usual non-black-holey reasons.

Or, we could take the fine-grained point of view, and pretend we know everything about the matter outside the event horizon.  In that case, the entropy increases only because of (A), things falling across the horizon.  Stuff happening inside the star doesn't make a difference.  This would be the fine-grained GSL, and it is nontrivial—the entropy defined in this way can go up, but not down.  You could say, that the only coarse-graining we use is to forget about anything that fell across the horizon, and this is enough to get a nontrivial result.  (This was pointed out by Rafael Sorkin.)

In my dissertation research, I proved the GSL in the fine-grained sense.  This was very useful since there are still some thorny and unresolved issues of interpretation with the Ordinary Second Law due to the exact meaning of coarse-graining.  The fact that one can avoid this issue in discussing the GSL made my life much easier!

Also, coarse-grained versions of the Second Law are only true if you have a history with a well-defined arrow of time—i.e. a universe that is constrained to begin with low entropy, but has no particular constraint on how it has to end up.  The fine-grained GSL, on the other hand, appears to be true for all states and therefore has no dependence on the arrow of time.  As a result, you can even apply the fine-grained GSL backwards in time if you want to, and this is perfectly OK, even though we normally think of the Second Law as something which only works in one time direction.

The forwards-in-time GSL applys when you have a worldline (an "observer", if you feel like anthropomorphizing) which extends infinitely far to the future.  It says that the boundary of what the observer can see (called a "future horizon") has increasing entropy.  The backwards-in-time GSL, says that if you have a worldline which extends infinitely far to the past (if there are any), then boundary of what they can be seen by (called a "past horizon") has decreasing entropy.  Equally true.

How to apply the GSL.  If you want to use the GSL as a singularity theorem to show that time ends in the middle of a black hole, you'll want to use the forwards-in-time GSL.  But if you want to use it to argue that there was an initial singularity at the Big Bang, you need to use the backwards-in-time GSL.  That's what I did to generalize the Penrose singularity theorem in my paper (like the original, it only works if space is infinite).

But I also considered the possibility that you might use the forwards-in-time GSL to argue for a beginning.  In this case, it would be a substitute not for the Penrose theorem, but for the OSL.  The details are in section 4.2 of my article, but the upshot is pretty much the same as before, that there probably had to be a beginning unless either (i) the arrow of time reverses, or else (ii) the universe was really boring before a certain moment of time.

Why even bother?  Other than the fact that the GSL has deeper connections to quantum gravity, the main technical advantage of using the (forwards) GSL is that it is more clear that the entropy reaches a maximum value in our universe (due to the accelerating expansion of the universe at late times, there is a de Sitter horizon at late times whose entropy is about 10^{120}).  This makes it harder to play games with infinities.

Also, if we want to reverse the arrow of time, the GSL indicates that the universe probably has to also be finite in size.  That's because, if the universe is finite in size, there is the possibility that before a certain time t_0, everything is visible to an observer, and before that time there would be no horizon at all.  That would make the fine-grained GSL trivial, forcing us to use the coarse-grained GSL.  That's important because it's only the coarse-grained GSL which depends on the arrow of time.

But mainly I just wanted to see if the standard argument from thermodynamics would still work when I rephrased it in terms of horizon thermodynamics.  Not surprisingly, it does.

(On the other hand, the connection to the Penrose singularity theorem are much more surprising, and I believe that it is telling us something deep about the laws of quantum gravity.)

Posted in Physics, Reviews | 2 Comments

Help with leaving comments

I have moved my comments policy, with help entering equations to its own page, accessible from the top bar above.

Recently, a few people have had trouble leaving comments on the site, due to incorrectly formatting equations, and some capricious comment mulilation by the WordPress software.  It took me a little while to figure out the exact rules since apparently the comment box which you would enter text into, has slighly different rules than the ones I enter text into as a logged-in user.

So here's the deal.  WordPress uses < and > to enclose html tags.  For example, if you write "<b>this is bold</b>" in a comment, you'll see "this is bold".  Unfortunately, this means that if you include an < followed by an > in your comment, WordPress will interpret whatever is in between as an html tag, and—even if it is not a valid html tag—will simply delete everything in between them!  Even I will be unable to see what you originally wrote.  So do NOT use > or < to mean greater than or less than (unless you use just one kind, or only >'s followed by <'s... but the safest rule is just to avoid them entirely...).

If you want to include greater than or less than symbols, you can write them as {\mathrm \$\$\verb|\|\mathrm{gt}\$\$} or {\mathrm \$\$\verb|\|\mathrm{lt}\$\$}.  That will look like > or < respectively.  Or better still, put your entire equation inside of the double dollar signs using LaTeX notation.

[quietfanatic points out in a comment below that you can also use the html escapes \&gt; or \&lt; if you don't want to use the double dollar signs.]

Also, do NOT try to use wordpress.com latex notation (which has a single dollar signs, and the magic word "latex").  It won't work!  There is a difference between wordpress.com and wordpress.org.  The former is a website used to host WordPress blogs, while the latter is where you download software to host your own WordPress blog.  This is a WordPress.org blog which is hosted on my family server, wall.org.

In order to put LaTeX in a wordpress.org blog, you have to install a special plugin to do so.  You might think that this plugin would use the same notation as wordpress.com, but no it doesn't.  Instead you type \$\$\mathrm{E = mc\verb|^|2}\$\$ to get E = mc^2, and type \$\$!\mathrm{E = mc\verb|^|2.}\$\$ to break it out into a separate line like so:

E = mc^2.

  It's best not to put any spaces after the dollar signs in this case, or they'll make a weird indentation in the next line, as shown here.

Got it?  Good.

Posted in Blog | 4 Comments

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:

One Way Streets: Black Holes and Irreversible Processes
An Introduction to Horizon Thermodynamics for Non-Physicists

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 | 16 Comments

The Name

A reader named Ken Murphy asks:

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;
rather, he made himself nothing
by taking the very nature of a servant,
being made in human likeness.
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 singular.  So the "name" must refer to some unmentioned name 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:

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 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:

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 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.A

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]

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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:

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.

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Did the Universe Begin? II: Singularity Theorems

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.

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