Home » Cosmology » Extending the BGV Theorem to Cosmogonic Models Positing an Infinite Contraction

Extending the BGV Theorem to Cosmogonic Models Positing an Infinite Contraction

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 (http://arxiv.org/abs/gr-qc/0110012v2) 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). Well, 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 contraction 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 clip_image002 spacetimes in this post. I want to first lay out some basics:

1.      A spacetime is past-incomplete if there is a null (or timelike) geodesic maximally extended to the past that is finite in length.

2.      As long as the expansion rate averaged over the affine parameter λ along a geodesic is positive (clip_image004), 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 τ (finite affine length, in the null case).

3.      The measure of temporal duration from clip_image006 is a quantity that is actually infinite (clip_image008) rather than potentially infinite (clip_image010).

Now, if the velocity of a geodesic observer clip_image012 (relative to commoving observers in an expanding congruence) in an inertial reference frame is measured at an arbitrary time clip_image014 to be any finite nonzero value, then she will necessarily reach the speed of light at some time clip_image016 and the interval clip_image018 will be have a finite value. If an infinite contraction preceded the “bounce” (and indeed it must do so necessarily if clip_image020 is to be avoided), then the time coordinate clip_image022 will run monotonically from clip_image024 as spacetime contracts duringclip_image026, bounces at clip_image028, then expands for all clip_image030

Thus, if what I have argued above is correct, then the implications are unmistakable: as long as clip_image012:

  1.  is a non-comoving geodesic observer;
  2.  is in an inertial reference frame;
  3.  is moving from clip_image040;
  4.  has been tracing a contracting spacetime where clip_image042;

it therefore follows that the relative velocity of clip_image012 will get faster and faster as she approaches the bounce at clip_image038. Moreover, since we know that clip_image012will reach the speed of light in a finite proper time clip_image022, coupled with the fact that the interval clip_image044 is infinite, we can be sure the she will reach the speed of light well-before ever making it to the bounceand therefore cannot be geodesically complete.

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