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Borde-Guth-Vilenkin singularity theorem

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The Borde-Guth-Vilenkin singularity theorem (or BGV theorem) was developed in 2003 by three leading cosmologists; Arvind Borde, Alan Guth and Alex Vilenkin. Subsequently in recent years since, the BGV theorem has become widely respected and accepted within the physics community.[1] The theorem is based on, what Alan Guth calls a "well-known fact", something traveling on a geodesic through an expanding universe becomes redshifted.[2] Geodesics within general relativity are what describe the motion of what are called "point particles". A meteor, a satellite or anything traveling in space has geodesics within space-time, and are also considered point particles.[3] Words like expansion and contraction of the universe have to do with "congruences of timelike geodesics (the potential trajectories of test particles)."[4] Important to the 2003 finding was the assumption of a single congruence "with a positive average expansion rate throughout some specified region."[4] Discovered by Edwin Hubble (1889 – 1953) expansion of the universe has become a law. According to Hubbles law; "The apparent recession velocity of a galaxy v is proportional to its distance d from the observer: v=H0d, where the constant of proportionality H0 is known as the Hubble constant." [5] Borde, Guth and Vilenkin follow an imagined observer back into time by way of a "timelike or null geodesic", which according to Guth will be blueshifted within a universe obeying Hubbles law. A timelike or null geodesic have a tangent vector with a norm of negative and zero, respectively. A spacelike geodesic has a tangent vector that is positive.[3] A vector is for example the velocity and acceleration of an object. While a tangent is the point on a curve of a vector. So that a tangent vector is the velocity and acceleration at a particular point on a curve. The world line of a geodesic is the sequence of events that relate to the point particle in question. Under some circumstances the blueshift will reach "infinite rapidity" or the speed of light within a "finite amount of proper time (or affine parameter)".[2] Along this trajectory with an affine parameter shows that such trajectory is "geodesically incomplete."[2]

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 H_{{a,v}} > 0 holds along these past-directed geodesics. This is a stronger conclusion than the one arrived at in previous work 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).[6]

The 2003 paper outlining the BGV theorem, Inflationary spacetimes are not past-complete highlights the finding of a space-time boundary at the inflation event. The paper asks if it is even possible if inflationary universes could be past-eternal as opposed to future-eternal only. If this can be so then there is a "viable model of the Universe with no initial singularity. The Universe would never come into existence. It would simply exist."[4] The chief finding of the paper though was that inflation alone is not enough to explain the universe, there needs to be a whole new physics to explain "correct conditions at the boundary."[7] One possibility is that there must have been some sort of quantum creation event according to Alan Guth.[8] The correct conditions fall within what is called Planck time. Planck time is the shortest elapsed amount of time which according to physicists is 10-43 second after the Big bang.[9][10] Prior to this extremely short period of time elapsing fully (zero to approximately 10−43), called the Plank epoch or Planck length, is still to this day poorly understood and has been fertile ground for theorists to appeal to in attempt to skirt a finite past. The BGV theorem though holds independently of any physical description of the very early universe before Planck time.[9] In fact it can support a wide variety of inflationary cosmological models, or completely different physics of a universe within a multiverse. So that even if our universe is just one part of a much grander set of universes called the multiverse, the multiverse itself then would require a beginning, or what has been called by some as a; beginning of beginnings. It maintains sweeping generality making very few assumptions, not even assuming the material content of the universe or that Albert Einstein's general relativity equations actually work.[11] If tweaking needs to be done to Einstein's theory of gravity for the quantum level interactions of particles during the very early universe then so be it. The BGV theorem even remains consistent with "higher dimensional cosmologies based on string theory."[9] This is because "brane worlds" which are posited by string theory create "collisions of bubbles nucleating in an inflating higher-dimensional bulk spacetime." The "higher-dimensional bulk spacetime" cannot be "past-complete" or in other words, they cannot be past eternal.[7] The primary assumption made by Borde, Guth and Vilenkin however is that the cosmic expansion rate will never get to a nonzero value.[11] There can not be past-eternal inflation, there must be a beginning or singularity.[11] The result is that the BGV theorem covers a wide range of cosmogonies (theories on the origin of the universe) because there need only be on average an expansion rate along the geodesic of more than zero for there to be a space-time boundary and therefore cosmic beginning.[1]

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.[12]


There have been numerous cosmological models that attempt to find consistency without appealing to an initial cosmological singularity. These are the; Steady State model, the Oscillating model, Vacuum Fluctuation models. All of these however do not contend with the sweeping BGV theorem in the long run.[13]

According to Alex Vilenkin eternal inflation and cyclic evolution do not yield a full description of the universe.[14]

Steady State Model

Loop Quantum Cosmology

The BGV theorem does not violate the "weak energy conditions" usually assumed in singularity theorems, this is because it does not even require any energy condition in the first place.[7] This "novel feature" according to Abbay Ashtekar enables the theorem to break the strings to general relativity. Assuming an "always positive" expansion rate is simply violated by "bouncing scenarios including LQC" according to Ashtekar.[15] This may seem like it violates and supplants the BGV singularity theorem at first but actually bouncing scenarios have their own special types of problems that have to be dealt with if it is to be favored.


Also called the Big Crunch. The oscillating universe was popular in the 1930's. Under an oscillating model of the universe the current expansion we currently observe is just one part of a cycle, and one cycle of infinite cycles. There have been many cycles of expansion and when traced back in time it contracts then expands out again into a past or future trajectory. This is speculative, and very metaphysical rather than physical and actually scientific it seems. Having to rely upon unobserved past contractions, or at least, less obvious past contractions than the one clearly deduced from the current expansion of the universe does not rely on empirical evidences.

There are also several objections that have been laid against the alternative oscillating model.

  1. There are no known physics that would allow a collapsing universe to bounce back into expansion.[13]
  2. Empirical evidence seems to point to the fact that "the mean mass density of the universe is insufficient to generate enough gravitational attraction to halt and reverse the expansion."[13]
  3. In oscillating cosmological models, although may avoid a cosmic singularity actually have substantial thermodynamics problems. At the beginning of each oscillation cycle, more energy is used and therefore produces longer trajectories of the expansion phase. If followed into the past, the trajectories get smaller and smaller. Exactly what the model attempts to negate is exactly what is implied by the model itself. In other words "each successive oscillation has a larger radius and longer expansion time."[13]
Each one of the hypothetical cycles would exhaust more and more usable energy. This means every cycle would be larger and longer than the previous one, so looking back in time there would be smaller and smaller cycles. So the multicycle model could have an infinite future, but can only have a finite past.[16]

Vacuum Fluctuation

Creationist Implications

The Borde-Guth-Vilenkin theorem does not bring into question anything related to the long ages that are called for by the Big bang theory. According to this theory the universe is approximately 14 billion years old. Although what is actually proved by the theorem is that there must be a beginning to any universe that on average is inflationary. So there seem to be two points of discussion within creationism regarding the BGV theorem. One point is of contention, while the second is one that creationists, either supporting YEC or OEC, can fully support.

  1. The universe and Earth are old as opposed to young.
  2. A universe with inflation has a beginning.

While the former, point 1, is a rich area of debate and disagreement, the latter, point 2, is actually one that creationists can agree with. Creationists can utilize point 2 in order to bridge the gap from the world to natural theology. There can be a two-step approach. Working backwards from point 2, creationists can argue for the existence of God as opposed to trying to prove the age of the universe and Earth. This challenges the popular arguments of atheists. The kalam cosmological argument is particular way to philosophically argue for a space-less, timeless, beginning-less, metaphysically necessary personal being, and all-powerful cause to the beginning of the universe. When expounded upon the result is what looks awfully familiar to theologians as God. This is an argument brought back into the fold of philosophy of religion in the late 20th and early 21st century, and defended successfully in academic debates, books and papers by William Lane Craig. The kalam cosmological argument specifically demonstrates that what follows logically from the premises is that there must be a transcendent cause to physical space-time reality. The BGV theorem can be used as a powerful empirical evidence of a beginning of any universe which on average has a positive expansion rate, of which the universe presently observed does. In establishing a beginning to the universe, and therefore argument for the existence of God the creationist can be comfortable within the the scientific findings of the BGV theorem specifically, and the Big bang theory more generally. Especially when interacting with atheists or even agnostics upon the existence of something like God.

The beginning of the universe needs some length of history. Usually the emphasis by creationists is the age of the universe and Earth first and then argue about existence of God later. This line of argumentation however assumes God in the process and takes an anachronistic stance by positing in their worldview presentation an already existent God. The philosophical context of the kalam cosmological argument demonstrates the necessary existence of God and from this sophisticated standpoint can the creationist then construct chronological theories regarding the actual history of the cosmos having already dealt with what logically should come first, namely a necessary cause to time itself.


  1. 1.0 1.1 William Lane Craig and J. P. Moreland, The Blackwell Companion to Natural Theology (Blackwell Publishing 2009), pg. 142
  2. 2.0 2.1 2.2 Bruce L. Gordon and William A. Dembski, The Nature of Nature: Examining the Role of Naturalism in Science (Intercollegiate Studies Institute 2011), pg. 498
  3. 3.0 3.1 Geodesic By Wikipedia
  4. 4.0 4.1 4.2 A. Borde, A. Guth and A. Vilenkin, Inflationary space-times are not past-complete, Phys. Rev. Lett. 90 151301 (2003), pg. 1[1]
  5. Peter Coles, Routledge Companion to the New Cosmology (Routledge 2004)[2]
  6. A. Borde, A. Guth and A. Vilenkin, Inflationary space-times are not past-complete, Phys. Rev. Lett. 90 151301 (2003), pg. 3[3]
  7. 7.0 7.1 7.2 A. Borde, A. Guth and A. Vilenkin, Inflationary space-times are not past-complete, Phys. Rev. Lett. 90 151301 (2003), pg. 4[4]
  8. Bruce L. Gordon and William A. Dembski, The Nature of Nature: Examining the Role of Naturalism in Science (Intercollegiate Studies Institute 2011), pg. 500
  9. 9.0 9.1 9.2 Question 74: CERN Probes Big Bang By William Lane Craig
  10. Quentin Smith and L. Nathan Oaklander, Time, Change and Freedom: Introduction to Metaphysics (Routledge University Press 2005), pg. 5
  11. 11.0 11.1 11.2 Alex Vilenkin, Many Worlds In One: The Search for Other Universes (Hill and Wang 2006), pg. 175
  12. Alex Vilenkin, Many Worlds In One: The Search for Other Universes (Hill and Wang 2006), pg. 176
  13. 13.0 13.1 13.2 13.3 The Ultimate Question of Origins: God and the Beginning of the Universe By William Lane Craig (free registration is required for viewing)
  14. Alex Vilenkin, Many Worlds In One: The Search for Other Universes (Hill and Wang 2006), pg. 172
  15. Abbay Ashtekar, Loop Quantum Cosmology: A Status Report (2011), pg. 60[5]
  16. Sarfati, J.D., If God created the universe, then who created God?, Creation ex nihilo Technical Journal, 12(1):20–22, 1998.[6]

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