## Is the second law of thermodynamics connected to the expansion of the universe?

This is just a bit of idle wondering, another little bit of amateur cosmology from someone who should probably know better. The question I’m asking myself today is, is the second law of thermodynamics connected to the expansion of the universe?

I suspect the gut instinct of most cosmologists would be “no, of course it isn’t, don’t be silly”. But here’s a reason to suspect it might be. An upper bound on the entropy of the universe can be found from the area of the cosmic horizon that surrounds it. This turns out to be a huge number, and it’s quite substantially bigger than the actual entropy of the present-day Universe*.

But the Universe is expanding, and this means that this upper bound on its entropy is increasing**. If we project things back into the past we’ll find that, once upon a time, the universe was tiny, and this upper bound on its entropy was very small – much smaller than the entropy of the present-day Universe. In particular, when the Universe’s size was on the order of the Planck distance, its information capacity would have been of the order one bit, which is a very small amount of entropy indeed.

So perhaps the picture looks like this: the maximum possible entropy of the Universe is determined by the size of the cosmic horizon, which increases rapidly – more rapidly than the actual entropy is able to increase. This means that the universe’s entropy is always smaller than its maximum possible value, giving it room to increase, and the reason the Universe started off in such a low entropy state is simply that it started off so very small.

Here’s a more prosaic way to look at it: imagine, if you will, our sun, shining its hot, low-entropy light down onto our planet’s surface. This light hits the atmosphere and causes weather patterns to form; it hits plants and allows them to grow. But these things require more than just the hot energy from the sun. They also need to let their own, cooler (thus higher entropy) thermal radiation dissipate away into the coldness of space. If outer space were not cold then the Earth would simply equilibriate at 6000˚C (the temperature of the sun’s light) and there would be no weather and no life, just hot, inert gas and molten rock.  The entropy would reach its maximum possible value, making complex far-from-equilibrium structures impossible.

But why is space so cold? It’s cold because it’s expanding. Space is filled with thermal radiation called the cosmic microwave background, which has a temperature of 2.7 K (-270.45˚C), but that same light used to be much hotter, around 3000K (2700˚C). As space expands it increases the wavelength of any radiation passing through it, which cools it down. If we lived in a static Universe – one that was neither expanding nor contracting – then it would be filled with light at the temperature of the stars. There would be no black in the night sky; looking anywhere would be like looking into the sun, because wherever you looked there would be a star somewhere along your line of sight.

I’m no expert on stellar evolution but I’d guess that if stars couldn’t radiate heat into the coldness of space, the energy they produce would blow them apart. So perhaps the stars themselves would disappear in a static or contracting universe, leaving nothing but hot gas everywhere. If all this is right it means that all the complexity we see around us – everything that relies on the second law for its existence, including weather, geology, life and even the stars – all of it is thanks to the fact that we live in an expanding Universe.

And if that expansion is set to continue forever then, well, maybe that’s not such a bad thing after all.

* I estimate the upper bound to be $10^{101}\, JK^{-1}$, or about $10^{124}$ bits.  Estimates of the actual entropy of the present-day Universe seem to vary dramatically depending on the source, from about $10^89$ bits, quoted by Max Tegmark, to of the order $10^{104}$ bits (Egan and Lineweaver), up to about $10^{100}\, JK^{-1}$ ($10^{123}$ bits), as estimated by Roger Penrose.  See this previous post for more background on some of these figures.  The Egan and Lineweaver figure is probably the most up-to-date calculation.

** A quick back-of-the-envelope calculation shows that this is happening at a rate of the order $10^{48}\, JK^{-1}s^{-1}$, or $10^{71}$ bits per second.  This is a big number, but it’s tiny compared to the total value.  I’m not very confident of the calculation because I did it in a naïve, non-relativistic way – I just used the Hubble constant to calculate the rate of expansion of the horizon, and then plugged the rate of increase of its area into the black hole entropy formula.  But the exact value isn’t important – what matters is that the horizon is expanding at some rate or other, and has been since the big bang.

### 4 Comments to “Is the second law of thermodynamics connected to the expansion of the universe?”

1. Here’s another interesting point that relates to this post: there’s an argument in cosmology about whether time would appear to run backwards if the Universe were shrinking instead of expanding. In other words, would reversing the expansion of the Universe also reverse the second law? As far as I know, most cosmologists currently think that it wouldn’t. According to the book ‘Time’s Arrow and Archimedes’ Point’ by Huw Price, Stephen Hawking used to think it would, but he later changed his mind. (Of course, this debate had more practical relevance when we thought the expansion of the Universe might one day reverse, ending in a Big Crunch. This isn’t thought to be the case anymore, because of “dark energy”, but it’s still an interesting thing to think about.)

What does the argument in my post say about this? Well, if the size of the Universe provides an upper bound on its entropy then it’s impossible for the size to shrink unless the entropy is less than the upper bound. If the cosmic horizon is going to shrink down to zero size then, necessarily, the entropy also has to shrink to zero, and it has to do it at least a bit faster than the shrinking of the horizon. So if this upper bound idea is correct then reversing the expansion of the universe would have to involve reversing the arrow of time after all.

2. Have you seen this: http://www.scribd.com/doc/2136072/entropy-in-cosmology
Makes a similar arguement(ish)

• Thanks, I hadn’t seen it. Max Tegmark can always be relied upon to say something interesting. I disagree about the multiverse though. It’s interesting that the figure he quotes for the observed entropy is so much lower than Penrose’s ($10^{89}$ bits versus $10^{123}$) – if they’re both calculating the same thing (which they might not be), I wonder which is correct.

• I’ve updated the post to point out that estimates of the total entropy of the universe vary dramatically; they’re all less than the upper bound, obviously.