BVG Theorem

The 2003 Borde-Vilenkin-Guth Theorem (the BVG Theorem):

Borde, Vilenkin, and Guth joined together to formulate an elegant and vastly applicable demonstration of a beginning of expanding universes (in a famous article in Physical Review Letters). Alexander Vilenkin explains it as follows:

“Suppose, for example, that [a] space traveler has just zoomed by the earth at the speed of 100,000 kilometers per second and is now headed toward a distant galaxy, about a billion light years away. [because of the expansion of the universe as a whole], that galaxy is moving away from us at a speed of 20,000 kilometers per second, so when the space traveler catches up with it, the observers there will see him moving at 80,000 kilometers per second. [As the universe continues to expand, the relative velocity of the space traveler will get smaller and smaller into the future]. If the velocity of the space traveler relative to the spectators gets smaller and smaller into the future, then it follows that his velocity should get larger and larger as we follow his history into the past. In the limit, his velocity should get arbitrarily close to the speed of light [the maximum velocity attainable by mass energy in the universe].”^[4]

The point where relative velocities become arbitrarily close to the speed of light constitutes a boundary to past time in any expanding universe or multiverse. Though the conclusion of Borde, Vilenkin, and Guth is somewhat technical for non physicists, its importance makes their precise words worth mentioning:

Our argument shows that null and time like geodesics are, in general, past-incomplete [requiring a boundary to past time] in inflationary models, whether or not energy conditions hold, provided only that the averaged expansion condition Hav > 0 hold 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 space-time in a finite proper time.^[5]

This proof is vastly applicable to just about any model universe or multiverse that could be connected with our universe. Alexander Vilenkin put it this way in 2006:

We made no assumptions about the material content of the universe. 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, no matter how small. This assumption should certainly be satisfied in the inflating false vacuum. The conclusion is that past-eternal inflation without a beginning is impossible.^[6]

Physicists do not use the word “impossible” very often. So, Vilenkin’s claim here is quite strong. The reason he is able to make it is that there is only one condition that must be fulfilled – an expansion rate of the universe greater than zero (no matter how small).

It is important to note that Borde, Vilenkin, and Guth applied their theorem to the string multiverse as well as to higher dimensional oscillating universes. I present their own words here (which might be quite difficult for non-physicists) because it gives a sense of their own appreciation of the vast applicability of their theorem:

Our argument can be straightforwardly extended to cosmology in higher dimensions [arising out of string theory/M Theory]. For example, [1] in [some models of a string multiverse], brane worlds are created in collisions of bubbles nucleating in an inflating higher-dimensional bulk space-time. Our analysis implies that the inflating bulk cannot be past-complete [i.e. must have a boundary to past time]. ¶ [2] We finally comment on the cyclic Universe model [in the higher dimensional space of string theory] in which a bulk of four spatial dimensions is sandwiched between two three-dimensional branes…In some versions of the cyclic model the brane space-times’ are everywhere expanding, so our theorem immediately implies the existence of a past boundary at which boundary conditions must be imposed. In other versions, there are brief periods of contraction, but the net result of each cycle is an expansion.…Thus, as long as Hav > 0 for a null geodesic when averaged over one cycle, then Hav > 0 for any number of cycles, and our theorem would imply that the geodesic is incomplete [i.e. must have a boundary to past time].^[7]

The boundary to past time (required in the BVG theorem) could indicate an absolute beginning of the universe or a pre-pre-big bang era with a completely different physics. If it is the latter, then the pre-big-bang period would also have to have had a boundary to its past time (because it would have a rate of expansion greater than zero). Eventually, one will reach an absolute beginning when there are no more pre-pre-pre-big-bang eras.

This is an extraordinary conclusion, because it shows that a beginning is required in virtually every conceivable pre-big-bang scenario – including the string multiverse and oscillating universes in higher dimensional space. By implication, then, even if there were multiple pre-big-bang eras, it is likely that these eras would have to have an expansion rate greater than zero, which means that they too would have to have a beginning, which would make an absolute beginning virtually unavoidable. This absolute beginning would be the point at which the universe came into existence. Prior to that point the universe (and its physical time) would have been nothing, which as we saw above, implies a Creator.

Exceptions to this theorem are very difficult to formulate and are quite tenuous because they require either a universe with an average Hubble expansion less than or equal to zero (which is difficult to connect to our inflationary universe) or a deconstruction of time which is physically unrealistic. (For an extended discussion of these exceptions, you may consult Chapter One, Section III.D-E of NPEG). For this reason all attempts to get around the BVG Theorem to date have been unsuccessful. Even if physicists in the future are able to formulate a hypothetical model which could get around the BVG Theorem, it would not mean that this hypothetical model is true for our universe. It is likely to be only a testimony to human ingenuity. Therefore, it is probable that our universe (or any multiverse in which it might be situated) had an absolute beginning. This implies a creation of the universe by a Power transcending our universe.

There is another impressive set of data which corroborates the above three space-time geometry proofs, namely, the Second Law of Thermodynamics (i.e. entropy). The constraints of time will not permit me to address this topic, however, those interested in explication of it may consult Chapter One (Section III A-C) of NPEG). In conclusion, the evidence from physics (from both space-time geometry proofs and the second law of thermodynamics) indicates the probability of a beginning of our universe. In as much as a beginning indicates a point at which our universe came into existence, and prior to that point, the universe was nothing, then it is probable that the universe (and any hypothetical multiverse in which it might be situated) was created by a transcendent power outside of physical space and time.