You all will have to forgive me for making such a technical post as this blog’s first; it just so happens that this has been the topic of my recent research. Sorry.

For those of you who do not know, there was a kinematic incompleteness theorem proven in 2003 that (in very non-technical terms) demonstrates that any universe that has been, on average, expanding throughout its history, cannot be eternal to the past, but must have an past spacetime boundary (i.e., a beginning). It just so happens that our universe has been, by all appearances, expanding throughout its history.

However, there have been many attempts to craft highly speculative cosmogonic models that evade the Borde-Guth-Vilenkin theorem (hereafter, BGV). One of which posits an infinite contraction prior to a bounce, followed by a subsequent expansion phase. It is my contention that such a model does not in fact evade the theorem, and I shall engage in an endeavor to articulate why I believe BGV to conflict with \(H_{av} < 0\) spacetimes in this post. I want to first lay out some basics:

- A spacetime is past-incomplete if there is a null (or timelike) geodesic maximally extended to the past that is finite in length.
- As long as the expansion rate averaged over the affine parameter \(\lambda\) along a geodesic is positive (\(H_{av} > 0\)), BGV proves that there will be causal geodesics that, when extended to the past of an arbitrary point, reach the boundary of the inflating region of spacetime in a finite proper time \(\tau\) (finite affine length, in the null case).
- The measure of temporal duration from \(-\infty \to t_{0}\) is a quantity that is
*actually*infinite (\(\aleph_{0}\)) rather than*potentially*infinite (\(\infty\)).

Now, if the velocity of a geodesic observer \(\mathcal{O}\) (relative to commoving observers in an expanding congruence) in an inertial reference frame is measured at an arbitrary time \(t\) to be any finite nonzero value, then she will necessarily reach the speed of light at some time \(t’ < t\) and the interval \(t’ \to t\) will be have a *finite* value. If an infinite contraction preceded the “bounce” (and indeed it must do so *necessarily* if \(H_{av} > 0\) is to be avoided), then the time coordinate \(\tau\) will run monotonically from \(-\infty \to +\infty\) as spacetime contracts during \(\tau < 0\), bounces at \(\tau_{0}\), then expands for all \(\tau > 0\). Thus, if what I have argued above is correct, then the implications are unmistakable: as long as \(\mathcal{O}\):

- is a non-comoving geodesic observer;
- is in an inertial reference frame;
- is moving from \(t = -\infty \to t_{0}\);
- has been tracing a contracting spacetime where \(H_{av} < 0\);

it therefore follows that the relative velocity of \(\mathcal{O}\) will get faster and faster as she approaches the bounce at \(\tau_{0}\). Moreover, since we know that \(\mathcal{O}\) will reach the speed of light in some *finite* proper time \(\tau\), coupled with the fact that the interval \(t = -\infty \to t_{0}\) is *infinite*, we can be sure the she will reach the speed of light well-before ever making it to the bounce—and therefore cannot be geodesically complete.

Jack – I found this site after reading some of your comments (which I really appreciated) on Aron Wall’s page under ‘Did the Universe Begin? III: BGV Theorem’. I didn’t see where Aron directly addressed your contention with respect to the applicability of BGV to the quantum region: “…it seems to me that the BGV theorem does not rely on spacetime being classical…Vilenkin seems to unquestionably affirm the applicability of BGV to the quantum regime—if we include the relevant data for the quantum regime in our calculation of Hav and still find that Hav>0, then we would know that our spacetime (i.e., the combination of classical and quantum) is necessarily past-incomplete.”

Sean Carroll and Aron Wall seem to think on the other hand that BGV is only applicable to classical spacetime and would only imply that inflation has a beginning. If SC and AW are right, it seems BGV is much less of a help in arguing for a cosmic beginning. On the other hand, if your contention (which seems to be backed up by Vilenkin’s own words) is correct then it seems the only way to avoid a beginning is to postulate an emergent universe or pre-Big Bang scenario. Do you have any further insight into this? Just this week, I’ve been having interactions with some atheists on the Kalam and have been reticent to push BGV too far given this concern. Any help you can provide would be much appreciated!

Sincerely,

Calvin Marshall

Calvin,

Welcome to my blog! Thank you for your comment, and for your appreciation of Dr. Wall’s and my discussion on his blog. In response to your question, it seems to me that BGV is without a doubt a piece of forceful evidence when arguing for a cosmic beginning. Here’s why:

First, have a look at page 4 of the paper itself:

The results are clear: (to paraphrase) as long as spacetime has, on average, been in a state of cosmic expansion, then all past-directed geodesics will reach a boundary. In other words, the spacetime cannot be geodesically-complete to the past.

Now, the obvious question arises: Can whatever is on the other side of that boundary be past-eternal? Here is how WLC answers the question in his debate with SC:

As Vilenkin put it in in the footnote, “If indeed all past-directed geodesics encounter a quantum spacetime region where the notions of time and causality no longer apply, I would characterize such a region as the beginning of the universe” (A. Vilenkin to William Lane Craig, personal correspondence, December 8, 2013). Indeed, his answer was no different in our personal correspondence.

Thus, it all boils down to the question of whether or not the quantum regime can exist eternally. This is a question to which SC (or anyone else, for that matter) has been unable to provide a positive answer. WLC further explains the problem:

Indeed, during the Q&A (Questioner #13) I pressed SC for a response to this question in the context of

agent causation:As you can read or watch for yourself, he had no answer. Moreover, I intended the question to be a springboard for WLC—and he didn’t disappoint:

In sum, BGV demonstrates that \(H_{av} \gt 0\) spacetimes cannot be geodesically-complete to the past. What’s more, quantum instability ensures that the quantum regime cannot endure for an indefinite duration of time. Therefore, neither can the quantum regime be past-eternal.

I hope this helps. Please feel free to respond further for any clarification or further questions.

Jack — Thanks so much for this helpful reply. With respect to your comments, “If our universe, our observable universe, had a first moment in time, and naturalism is true then it would seem that we need to explain that with some sort of eternal existing set of conditions, and they would have to be necessary and sufficient to produce the effect, which is our universe coming into being or having a first moment in time. So the difficulty seems to be that if we have an eternally existing set of causal conditions that are sufficient to produce the effect, why isn’t the effect coeternal with the cause?” — doesn’t positing a prior contraction or some kind of pre-Big Bang era get you out of this scenario (granted it would have to be viable)?

Any time, Calvin.

In response to your question: Ignoring BGV for the moment, yes, a model of this sort would evade this problem—if such a model were a viable one. But that is a pretty big “if.” As I’ll bet you are aware, a fatal problem affecting models of this sort is mentioned on p. 1 of BGV.

Thanks again Jack. I hope you don’t mind if I ask you another question related to the Craig-Carroll debate. Craig rebutted baby-universe model in part by appealing to Stephen Hawking in his paper ‘Information Loss in Black Holes’ where he says: “There is no baby universe branching off, as I once thought. The information remains firmly in our universe. I’m sorry to disappoint science fiction fans…” (http://arxiv.org/abs/hep-th/0507171). However, in Sean Carroll’s post debate reflections (http://www.preposterousuniverse.com/blog/2014/02/24/post-debate-reflections/), Sean takes issue with Craig’s use of this quote: “…the Hawking quote was completely out of context; he was talking about the fact that he no longer thought that wormholes would lead to violation of unitarity in black-hole evaporation, nothing to do with cosmology.” What are your thoughts? Did Craig take Hawking out of context?

Of course I don’t mind! I’m a nerd for these discussions; keep em coming!

As I understand it, no, WLC did not take Hawking out of context: whether we are talking baby universes or wormholes, WLC’s point is that both lead to a violation of unitarity; and that (i.e., violation of unitarity) is what Hawking contends is not plausible.

[…] will only mention one—the Borde-Guth-Vilenkin theorem which was proven in 2003. As I previously explained, it proves (in very non-technical terms) that any universe that has been, on average, expanding […]