Friday, May 23, 2014


OK, so last time I claimed that our best, experimentally successful model of the early universe is one that is infinitely old and has no initial singularity. If you are savvy about these things (from listening to Craig debates, for instance) you are wondering, "But what about the Borde-Guth-Vilenkin theorem? Doesn't it prove that an inflating universe must have had an initial singularity?" The answer is, "No, it doesn't."

There has been a lot of confusion about this, with clip quotes from one or another of the paper's authors being traded to "prove" that the theorem does, or doesn't, prove the universe had a beginning. So let's look at what the theorem actually says.

"Any theorem is only as good as its assumptions," writes Alexander Vilenkin in a letter to Lawrence Krauss. So let's start with the assumptions of the theorem. Roughly speaking, there are two:
  1. Spacetime is classical.
  2. Spacetime is expanding on average.
(I say "roughly speaking" because there are all sorts of technical issues about spacetime congruences and so forth that one needs to make these assumptions precise, but I think these short versions are sufficient to understand the main philosophical issues involved.)

Now let's jump to the conclusion, which I quote from their paper:
... we see that if Hav > 0 along any null or noncomoving
timelike geodesic, then the geodesic is necessarily
Some translation: a "timelike geodesic" is simply the path that an object will travel on if it is not subject to any forces other than gravity. Similarly, a "null geodesic" is the path that a light ray will travel. "Hav > 0" is the mathematical statement of assumption (2.): the universe is expanding on average. "Past-incomplete" means that if you try to follow one of these paths backwards in time, you can only do so for a finite amount of time.*

OK, so here's my first point: the BGV theorem is not a singularity theorem!

The conclusion says nothing at all about singularities: it only says that certain paths cannot be extended infinitely backward in time. One way that this might happen is if the path encounters a singularity. But that is not the only way it can happen.

The other thing that can happen is that, as we trace the path backward in time, we encounter a region where one (or both) of the assumptions of the theorem is no longer valid.

Start with assumption (2.). The path can enter a region in which spacetime is static, or contracting, or cyclically expanding and contracting. Then assumption (2.) is violated and the theorem's conclusion is avoided. In spacetimes like these, the BGV theorem simply doesn't apply.

What about assumption (1.)? Classical spacetime is a pretty basic assumption in any sort of cosmology. But it is expected to break down in the quantum gravity regime, where we encounter "spacetime foam" of some sort. This is difficult to discuss, since in the absence of a good theory of quantum gravity, no one has any idea what spacetime foam should look like. But it's possible that the BGV theorem is pointing us to a place where quantum gravity comes into play.

So the BGV theorem does have something very interesting to tell us about the early universe: namely, that the infinitely old, infinitely expanding inflationary period that I discussed in the previous post cannot be the end of the story. Not just because of vague speculations about the Planck epoch, but because of the properties of classical spacetime itself. Here's what the authors said in the paper itself:
Whatever the possibilities for the boundary [where the geodesics come to an end - RNO], it is clear that unless the averaged expansion condition can somehow be avoided for all past-directed geodesics, inflation alone is not sufficient to provide a complete description of the Universe, and some new physics is necessary in order to determine the correct conditions at the boundary. This is the chief result of our paper.

But the BGV theorem does not say that spacetime must be singular; still less does it say that there was an initial singularity from which the entire universe arose. This is why I say that the theorem is irrelevant to the Kalam Cosmological Argument: it doesn't say anything about whether the universe had a beginning or not.

* A few more technical points: by "backward in time" I mean in the opposite time direction from the direction in which the universe is expanding. And by "finite amount of time" I mean time as measured by a clock carried along with the moving object (proper time). In the case of null (light) rays, we have to use an "affine parameter" rather than the proper time.


  1. Hey Dr Oerter.
    I think you're attacking a straw man. Where did someone say BGV was a singularity theorem? To my knowledge no proponent of Kalam has claimed that.
    Kalam does not depend in any way on the universe having a singularity rather than some other beginning see for example
    William Lane Craig & James D. Sinclair ,On Non-Singular Space-times and the Beginning of the Universe
    where Craig discusses non-singular models
    I might be mistaken about this but I think Vilenkin says that BGV will hold whatever modifications to Einstein's gravity we make with quantum gravity.
    “A remarkable thing about this theorem is its sweeping generality. . . . We did not even assume that gravity is described by Einstein’s equations. So, if Einstein’s gravity requires some modification, our conclusion will still hold. The only assumption that we made was that the expansion rate of the universe never gets below some nonzero value” [Vilenkin,Many Worlds in One pg 175]

    I agree that the BGV theorem does *prove* the universe had a beginning , but its a very general theorem that applies to a wide class of models , so it does have teh implication that many contemporary models of the universe (including ones like the Turok cyclic model) which were once thought to avert a beginning , do have a beginning.

  2. Thanks for your comment, Obsidian. As far as your "straw man" claim, note that I didn't accuse Craig of claiming BGV was a singularity theorem. In his writings, at least, he seems to be careful to avoid this mistake. Still, I think some of the things he says/writes might mislead people into thinking that BGV requires a singularity.

    For instance, on this page, Craig writes

    "But the Borde-Guth-Vilenkin theorem is independent of any physical description of that moment. Their theorem implies that even if our universe is just a tiny part of a so-called “multiverse” composed of many universes, the multiverse must have an absolute beginning."

    And later in the same article,

    "The first of these string cosmologies, Ekpyrotic cyclic models, is subject to the Borde-Guth-Vilenkin theorem and so is admitted to involve a beginning of the universe. The second group, Pre-Big Bang models, cannot be extended into the infinite past if they are taken to be realistic descriptions of the universe. The third group, the string landscape models, feature the popular multiverse scenario. They are also subject to the Borde-Guth-Vilenkin theorem and so imply a beginning of the universe."

    [Emphasis added.]

    It is clear that Craig is claiming that the BGV theorem requires a beginning of the universe. This is simply false: as I explain in the OP, all the BGV theorem implies is a beginning to the region of space-time that satisfies the expansion assumption. Craig has either misunderstood the implications of the theorem, or is begin deliberately misleading.

    I agree that BGV is a remarkable theorem in that it doesn't rely on Einstein's equations. This does not mean, however, that it applies in the quantum gravity realm. Quantum gravity is not a well-understood area, but many physicists expect that in the quantum gravity regime, not only will Einstein's equations be violated, but also the assumptions of classical spacetime will no longer hold true. BGV of course relies on the assumptions of classical spacetime, so it would not apply in this case.