What does the BGV theorem say?

The way I understand it, BGV says there was no space, time, matter, and energy logically prior to the initial event. If such is the case the cause would be a cause beyond the physical realm, and the effect would be spontaneous with the initiation of the event by the cause.

Then you’re wrong I’m afraid. How did you get that impression?

It is a theorem about how far back you can extend the dimension of time, it doesn’t say anything about whether it is coherent to speak about a “before” time itself or what there might have been, or even if the universe really came out of nothingness. It just doesn’t say that.

Apologists who rely on it in debates usually invoke the BGV theorem as an argument against theories that postulate an infinitely old universe. But again, saying that time has a first moment a finite time ago, is different from saying that there was nothingness before time itself and that the universe literally once never even existed. There is no evidence that the universe was ever somehow not in existence.

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@Jim and @Rumraket, the only thing the BGV theorem says is that if you have a particle (which could be massive or massless) traveling in a spacetime that is on average inflating, then when you “play the movie in reverse”, so that the particle is now traveling backwards in time, you will reach a point where general relativity breaks down.

This is to be expected - after all, we know that the beginning of the Universe is a quantum event, and therefore is not captured by general relativity.

It really has surprisingly little to do with the beginning of the universe; it just states that general relativity is insufficient to study it.


Well so much the worse for people who say it somehow means the universe transitioned into existence out of non-being.

These quotes are attributed to Vilenkin:
“It is said that an argument is what convinces reasonable men and a proof is what it takes to convince even an unreasonable man. With the proof now in place, cosmologists can no longer hide behind the possibility of a past-eternal universe. There is no escape: they have to face the problem of a cosmic beginning.”

“There are no models at this time that provide a satisfactory model for a universe without a beginning.”

“All the evidence we have says that the universe had a beginning.”

“The answer to the question, “Did the universe have a beginning?” is, “It probably did.” We have no viable models of an eternal universe. The BGV theorem gives us reason to believe that such models simply cannot be constructed.”

It seems he’s at least convinced that there is a beginning. Doesn’t the BGV theory suggest a singularity regardless of GR or QT? That seems to suggest a beginning is highly probable, doesn’t it?

BGV deals with classical
Spacetime. And I can list a whole bunch of cosmologists (even Christians) who say we don’t know if the universe (all that exists) had a beginning. Listing a bunch of quotes is pointless

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What Vilenkin is convinced of personally is not really relevant. The fact is the BGV theorem only considers GR and where it breaks down (this is what is called the singularity).

No. The BGV theorem only suggests a singularity in GR, not quantum gravity. You can look at the paper itself here. Nowhere in the paper is quantum gravity invoked. Indeed, the fact that the BGV theorem simply states that GR is insufficient to study the Universe beyond the singularity is in the last sentence of the abstract:

Thus inflationary models require physics other than inflation to describe the past boundary of the inflating region of spacetime.

“physics other than inflation” here refers to quantum cosmology models beyond the GR inflationary universe.


Hmm. If there’s expansion, it seems to suggest going back in time it’s highly probable that there would have to be a beginning regardless of QT, don’t you think? What could there conceivably be in quantum gravity that would suggest an alternative?

No. For example, there are bouncing cosmology models where the Universe just bounces off the singularity. The imperfect analogy is a ball that continuously expands and contracts.

Many things. Indeed, it is reasonable to think that a physical singularity i.e. a moment where all of spacetime is compressed to 0 volume is impossible due to the quantum uncertainty principles.


What does Vilenkin mean by a beginning? Does he mean the universe at first did not exist at all and nothing did, and then later it suddenly existed?

Or does he simply mean that the universe has a finite age, and thus a first “moment”, or instance, on the axis of time?

Thanks for clarifying this. This is really important, I think, since due to William Lane Craig the BGV has been mentioned a lot, though most of the time without reference to the actual peer-reviewed text (but instead, a lot of arguing over Vilenkin’s and other people’s personal views). I’m not an expert on GR, but skimming the conclusions of the paper, what you are saying seems consistent with the actual text:

Our argument shows that null and timelike geodesics are, in general, past-incomplete in inflationary models, whether or not energy conditions hold, provided only that the averaged expansion condition Hav > 0 holds along these past-directed geodesics. This is a stronger conclusion than the one arrived at in previous work [8] in that we have shown under reasonable assumptions that almost all causal geodesics, when extended to the past of an arbitrary point, reach the boundary of the inflating region of spacetime in a finite proper time (finite affine length, in the null case).

What can lie beyond this boundary? Several possibilities have been discussed, one being that the boundary of the inflating region corresponds to the beginning of the Universe in a quantum nucleation event [12]. The boundary is then a closed spacelike hypersurface which can be determined from the appropriate instanton.

Whatever the possibilities for the boundary, 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 [20]. This is the chief result of our paper. The result depends on just one assumption: the Hubble parameter H has a positive value when averaged over the affine parameter of a past-directed null or noncomoving timelike geodesic.

@PdotdQ, can you explain a bit further what a “past-incomplete” geodesic is?

In General Relativity, there are many notions of singularities. The existence of incomplete geodesics is one of the most popular. As you are a physicist, I’ll be more liberal with my language:

The easiest way to understand it is in the case of timelike geodesics. I can parameterize a timelike geodesic by its proper time, τ.

A complete timelike geodesic is a geodesic in which the proper time can go to plus or minus infinity. Imagine a 2-dimensional flat surface. Now pick a geodesic on this surface that is just a vertical line that goes through (0,0). Now, I can say for example, τ=0 to be (0,0), and τ=1 to be (0,1), and τ=-1 to be (0,-1). You can see that I can do this for all τ up to plus and minus infinity. This geodesic is complete.

Imagine instead the same 2-dimensional flat surface with a hole at (0,-5). Now pick the same geodesic. I can extend this geodesic all the way to positive infinity, but I can’t do it all the way to negative infinity as it will hit the hole at (0,-5). My geodesic can’t go through this hole as the geodesic equation is not defined on a hole! This geodesic terminates at τ=-5 and cannot be extended all the way to minus infinity. This geodesic is incomplete.

A past-incomplete geodesic is a timelike geodesic that hits something bad when I try to extend it all the way to τ=-infinity that it is incomplete in the negative time direction.

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Probably the later, although it’s hard to say for sure. But either way, it means an initial event which I would say cries out for an explanation.

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6 posts were merged into an existing topic: What is “Nothing”?

Which is why he (Vilenkin) has spent the past several decades (at least?) working on such questions. I’m not sure why you go right from ‘the universe had some sort of beginning as per a spacetime theorem that only applies within the equations of GR and this beginning = only explained by a supernatural being, specifically the being that I already believed in.’

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If I’m wrong about how I understand the issues here, I’m happy to be set straight. But this is how I understand it. Evidently there seem to be some scientists who still don’t agree with Vilenkin that there was a beginning to the universe. But from all I can gather, Vilenkin himself firmly believes, regardless of whatever we don’t know about quantum gravity, and no matter what currently proposed model of the universe is used, including the multiverse, that BGV applies in all situations, and that the universe had to have a beginning. In his mind the issue is completely settled. If you can find anywhere where he says otherwise I’d be interested to hear it. What he does argue, though, is that there is no material cause of the universe, not that it had no beginning.

And even though there evidently are scientists who still disagree with him on the issue of a beginning, the impression I get is there are more scientists that agree with Vilenkin than those that don’t. Lawrence Krauss is on public record in 2012 as saying he wasn’t sure, but if he had to bet, that he would bet on a beginning. So I think it’s perfectly justified to take the stance that the universe had a beginning based on the fact that there seem to be many scientists, including Vilenkin, who hold to that view. But if I’m wrong about any of this, I’m happy to be set straight.

This part is wrong. Vilenkin does not believe that the BGV theorem applies in all situation “regardless of whatever we don’t know about quantum gravity”. Again, what Vilenkin believes is really irrelevant. Look at the paper - it’s completely classical general relativity.

Now, there is another theorem that is somewhat contentious (or at least the physicists that I talked to have some issues with it), that might show that there is a singularity even in quantum gravity as long as it obeys a condition known as “the generalized 2nd law”. However, this is not as strong as the BGV theorem as a singularity does not necessarily imply that the universe has a beginning. I have never seen any apologist cite that theorem - weird since the physicist who derived it is a devout Christian. I suspect it’s because its conclusion is not as strong as the BGV theorem.


Well, I would argue that it is relevant what he thinks, cause he seems to be saying that it’s pretty conclusive regardless of what happens at the last moment of Planck time. Now if that’s not what he means by the following statements, how is one meant to interpret them?

“It is said that an argument is what convinces reasonable men and a proof is what it takes to convince even an unreasonable man. With the proof now in place, cosmologists can no longer hide behind the possibility of a past-eternal universe. There is no escape: they have to face the problem of a cosmic beginning.”

“Now I would like to report on a new theorem, proved in collaboration with Arvind Borde and Alan Guth [42], which appears to close that door completely.”

“All the evidence we have says that the universe had a beginning.”

“This is good news for quantum cosmology. It follows from the theorem that the inflating
region has a boundary in the past, and some new physics (other than inflation) is necessary to determine the conditions at that boundary. Quantum cosmology is the prime candidate for this role. The picture suggested by quantum cosmology is that the universe starts as a small, closed 3-geometry and immediately enters the regime of eternal inflation, with new thermalized regions being constantly formed. In this picture, the universe has a beginning, but it has no end.”

Any information available to the public on this?

I might be wrong in saying that Vilenkin does not personally believe that the BGV theorem fails in quantum gravity. But if he thinks so, then he needs to write down the proof of the generalization of the BGV theorem to quantum gravity.

I am not sure how else to respond to you besides reiterating the point that this:

is not supported by the original paper, which I linked previously. I do not understand why it is relevant at all what he thinks. Quoting him and appealing to his authority is just not how physics works - we only care about what the proof is showing, and it does not show that this classical singularity survives when we consider quantum gravity.

Edit: I forgot this part:

This is Aron Wall’s paper.

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Aron Wall is the one contributed to said theorem (here’s an overview of his blog posts on whether the universe had a beginning). Craig did bring up Wall’s theorem when I asked him about Wall’s critique of his use of the BGV, which is similar to @PdotdQ’s. (I didn’t know much about it at the time.) However, I don’t think Craig has referred to the theorem in public debates much.