The Arrow of Time

So I’ve been busy, but with good reason. Long ago I mentioned some research I have been doing with Jennie Chen, a grad student here, on the arrow of time. And finally the paper is done! We have just submitted the manuscript to the online e-print server at, where it should appear Thursday evening. (Update: here it is!) Some day we’ll even submit it to a journal, although that seems so twentieth-century to me.

As a reward to all our faithful readers at Preposterous Universe, here is the paper itself, revealed a full twenty-four hours before ordinary physicists get to see it! Getting in on the cosmological ground floor, as it were. The paper is available as a postscript file or as a pdf file; unfortunately the pdf version will look crappy on your screen, but it should print okay.

We have a grandiose-sounding goal: to explain the century-old puzzle of why the entropy of the universe was low in the past. The entropy, of course, is a measure of how “disorderly” a system is; more properly, how “generic” or “random” it is. Low-entropy states are suspiciously orderly; the classic example is a box of gas in which all of the gas just happens to be located in one corner. This is a perfectly acceptable configuration, but a statistically very unlikely one; if you let it go, the gas will quickly evolve to fill the room. This is the celebrated Second Law of Thermodynamics — in closed systems, entropy tends to increase (or stay constant). Long ago Boltzmann developed a mathematical understanding of this phenomenon, by showing how entropy measured the number of equivalent ways we could re-arrange the elements of the system to give a state that was macroscopically indistinguishable. For the box of gas, there aren’t that many ways we could re-arrange the molecules to keep them in one corner, but there are many ways we could re-arrange them smoothly throughout the box. It is therefore very natural to evolve from a low-entropy state to a high-entropy state, simply because there are so many more high-entropy states to evolve to.

The fact that entropy increases defines the arrow of time. It is a statistical phenomenon, valid for large systems, and the Second Law looks very different from the “microscopic” laws of physics, which are generally don’t care which direction time is running in. Of course, the only reason we see that entropy increases is because it used to be small in the past. Once the gas fills the box, it essentially stays there forever (apart from rare fluctuations).

So, as cosmologists, we have an issue to address — why was the entropy of our early universe so small? If high-entropy states are “natural,” why don’t we live in one? You might think to appeal to the dreaded anthropic principle, and argue that life couldn’t exist in a state with really high entropy. But that turns out not to be good enough; the entropy of our universe is much much lower than it needs to be to support the existence of life. So we are faced with the “arrow of time problem.”

Although there isn’t a consensus view of the solution to this problem, most cosmologists would guess that it has something to do with inflation. The idea of inflation states that the very early universe went through a period of incredible acceleration, which smoothed out the bumps and wiggles and gave us the big smooth universe we observe today. If it works, inflation tends to wipe out any pre-existing features and leave us with a universe similar to what we observe today.

But there’s a problem — even though cosmologists think that it’s quite natural for inflation to start and leave us with the universe we see, very few would think it was quite natural for a collapsing universe to smooth itself out and anti-inflate; that is, to undergo a process that is the time-reversed version of inflation. So secretly, the very idea of inflation has some time-asymmetry built into it; it makes sense forward but not backward. Therefore it doesn’t really count as a solution to the arrow of time problem. We need to explain how the conditions for inflation to start in the first place could naturally arise.

Jennie and I do the following thought experiment — if it weren’t for inflation, what would be a “natural” state for the universe to be in? Different people have addressed this question, with different answers; Roger Penrose, for example, has suggested that it would be a lumpy universe full of black holes. Our answer is almost exactly the opposite — the only natural state is empty space. This is basically because gravity makes everything unstable, and the entropy of any given configuration can always be increased by just expanding the universe by a huge factor. Sure, black holes will form, but they will ultimately evaporate away. If you let the universe evolve forever, it will ultimately get emptier and emptier (generically).

But we now know that even empty space has energy — vacuum energy. (Or at least some sort of dark energy.) So when we evolve to “empty space,” there is still some energy pushing the universe around; the resulting spacetime is called “de Sitter space.” Along with this energy comes a small nonzero temperature, which keeps all the fields in the universe gently fluctuating. Gentle or not, however, if we wait long enough we will find a really big fluctuation — one that is large enough to make inflation spontaneously begin. In other words, we are suggesting (although it’s not original with us) that de Sitter space is unstable; it doesn’t last forever, but eventually starts inflating here and there. These little inflationary patches will ultimately convert into ordinary matter and radiation, leaving behind universes just like our own.

And here is the fun part: this story can be told either forward or backward in time. In other words, you give me some state of the universe, chosen however you like. (Maybe you calculated the wavefunction of the universe, who knows.) I evolve it using the laws of physics. If Jennie and I are correct, it first empties out into a cold de Sitter space, dominated by a tiny shred of dark energy. But eventually we get lucky, and a small patch of inflating universe is born within this de Sitter background. This will happen at different places and times, give rise to a fractal distribution of spacetime geometry in the far future. And I can do the same thing going backwards in time from the initial state you gave me; the generic evolution is the same. It will empty out, and eventually begin to spontaneously inflate. So in the super-far past of our universe, before our “Big Bang” (which is nothing special in this picture), we will find other Big Bangs for which the arrow of time is running in the opposite direction. On the very largest scales, the entire universe is symmetric with respect to time.

Is this scenario correct? Interesting? Important? We’re not sure yet. This is certainly not one of those brilliant flashes of insight, like the original discovery of inflation was. Rather, it’s a concatenation of several intriguing ideas, most of which had already been suggested by someone or another. And there are a million questions remaining to be answered, especially about the onset of inflation in an empty background spacetime. But I like how the whole picture hangs together, and wouldn’t be surprised if something like it eventually came to be accepted as a reasonable picture of the universe on the very largest scales.

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