In my last post I mentioned that I have been reading Alex Vilenkin's book, Many Worlds in One: The Search for Other Universes (New York: Hill and Wang, 2006). I also mentioned that I was trying to understand Vilenkin's explanation of the theorem he developed with Arvind Borde and Alan Guth. It's actually pretty simple, but it involves Einstein's special theory of relativity, which always short-circuits my brain a little. But I think I understand what Vilenkin is saying now, and it's really interesting.
The explanation is found on pp. 174–6 of his book. Basically, the claim is that an expanding universe cannot be past eternal, because it would involve a contradiction. Now the word "eternal" can mean different things, so it's important to be clear that here it means being infinitely extended in time rather than timeless. An expanding universe that is past eternal is one in which a series of temporal events exists but has no beginning.
Quick digression: You might wonder why the theorem is necessary, given the expansion of the universe. If the universe is expanding, it means that as you go further and further into the past, the universe contracts and becomes more and more dense, approaching a point of infinite density. This point would seem to represent the beginning of the universe, at least (if I have understood correctly) according to the standard Big Bang model. So, if the expansion of the universe implies a beginning, then what use is there for a theorem that proves that an expanding universe must have a beginning? Isn't the Borde-Guth-Vilenkin theorem redundant?
The answer is that ever since the expansion of the universe was discovered, physicists have proposed numerous models in order to avoid an absolute beginning. For instance, some models have involved the idea that the expansion eventually stops and then the universe contracts, only to expand again. In that case, what we call the Big Bang was only the beginning of the present cycle of expansion, but not the absolute beginning of the universe (or of physical reality). Alternatively, if the expansion of the universe was preceded by a period of eternal inflation (which I won't try to explain because I'm not sure if I even understand it), then there is no beginning. So this is what the theorem ultimately rules out.
In order to understand the BGV theorem, one has to understand special relativity. To the best of my understanding here are the points that need to be kept in mind. If you are moving with uniform velocity relative to me, then you and I will measure the duration of the same event differently. That is, our clocks will give us different answers. From my perspective, your clock will run slower than mine does (while from your perspective, your clock will run normally). This is what is meant when we say that moving clocks run slower. The speed of light plays a crucial role in how time is measured in special relativity, because it is the same for all observers (an observer in this case is anyone who is using a clock to measure the duration of some event). Furthermore, the speed of light is the fastest speed there is. It's the universal speed limit.
Another thing that is helpful to keep in mind is that the expansion of the universe is not the expansion of matter into existing space; rather, it is the expansion of space itself. One analogy that often gets used is that of a balloon with buttons taped on its surface. If you inflate the balloon, the buttons will grow further apart from each other, even though they are at rest with respect to the balloon. In the same way, it's not as if the galaxies are flying through space away from each other. Rather, galaxies are at rest with respect to space, but as the universe expands, they grow further apart from each other.
The BGV theorem claims that the idea of an expanding universe with an infinite past involves a contradiction. To see why, imagine an observer in place of each galaxy. As the universe expands, the observers grow further apart from each other. Vilenkin calls these observers "spectators." He says that they are moving under the action of gravity and inertia.
Now imagine that there is another observer traveling in a spaceship who is moving relative to the spectators. Vilenkin calls him "the space traveler." The ship's engines are turned off and he is moving by inertia. (I believe this means that the space traveler is moving at a uniform velocity.)
We are to imagine that the space traveler and the spectators have been moving like this for all eternity. But once you do this, you end up with a contradiction, and it stems directly from the way that the clocks of the traveler and the spectators measure time. To begin with, note that, as the traveler moves through space, the spectators continue to move further apart from each other, due to the expansion of the universe. This means that each successive spectator will measure the traveler moving at a lower velocity than the previous spectator did. The further into the future we go, the lower the traveler's velocity will be measured. This also means that the further into the past we go, the higher the traveler's velocity will be measured. "In the limit," Vilenkin writes, "his velocity should get arbitrarily close to the speed of light."
Since moving clocks run slowly, this means that, relative to the spectators, the traveler's clock should be running slower and slower as we go backward in time. Here I will simply quote Vilenkin:
As we go backward in time, the speed of the space traveler approaches the speed of light and his clock essentially comes to a halt. This is from the spectator's point of view. But the space traveler himself does not notice anything unusual. For him, what spectators perceive as a frozen moment, stretched into eternity, is a moment like any other, which has to be preceded by earlier moments. Like the histories of the spectators, the space traveler's history should extend into the infinite past.
This is the point that I was initially having a hard time understanding, but I see what it means now. This is why it's so important to understand special relativity and to take note of the fact that the spectators and the space traveler are both observing different things with regard to the space traveler's clock, and this is where the contradiction lies.
If the spectators are moving further away from each other (because of the expansion of the universe), then it follows that the further you go into the past, you will eventually reach a point where the traveler's clock isn't ticking at all, from the spectators' perspective. Because it is an eternally frozen moment, it cannot be preceded by another moment, because you can't have a moment that comes "before" eternity. But from the traveler's perspective this is all irrelevant. He observes his clock to still be ticking at a normal rate, and so as far as he is concerned, it is perfectly possible for this very same moment to be preceded by another one. What this means is that the moment in question both can and cannot be preceded by another moment, which is a contradiction.
The contradiction reveals that the scenario we are considering is impossible. A universe that is (on average) expanding cannot have an infinite past. This applies both to inflationary and cyclic models of the universe (Vilenkin explains that in cyclic models, the volume of the universe increases with each expansion, which means that on average the universe is expanding as it goes through each cycle of expansion and contraction). Since we can't deny the expansion of the universe, we are led to the conclusion that the universe has an absolute beginning—which (to tie this all back to my last post) is just to affirm the second premise of the Kalam cosmological argument.