The Science of Discworld II

information-theoretic entropy. Indeed the entropy of the source remains unchanged, no matter how many messages it generates.
    There is another puzzle associated with entropy in our universe. Astronomical observations do not fit well with the Second Law. On cosmological scales, our universe seems to have become more complex with the passage of time, not less complex. The matter in the universe started out in the Big Bang with a very smooth distribution, and has become more and more clumpy – more and more complex – with the passage of time. The entropy of the universe seems to have decreased considerably, not increased. Matter is now segregated on a huge range of scales: into rocks, asteroids, planets, stars, galaxies, galactic clusters, galactic superclusters and so on. Using the same metaphor as in thermodynamics, the distribution of matter in the universe seems to be becoming increasingly ordered. This is puzzling since the Second Law tells us that a thermodynamic system should become increasingly disordered.
    The cause of this clumping seems to be well established: it is gravity. A second time-reversibility paradox now rears its head. Einstein’s field equations for gravitational systems are time-reversible. This means that if any solution of Einstein’s field equations is time-reversed, it becomes an equally valid solution. Our own universe, run backwards in this manner, becomes a gravitational system that gets less and lessclumpy as time passes – so getting less clumpy is just as valid, physically, as getting more clumpy. Our universe, though, does only one of these things: more clumpy.
    Paul Davies’s view here is that ‘as with all arrows of time, there is a puzzle about where the asymmetry comes in … The asymmetry must therefore be traced to initial conditions’. What he means here is that even with time-reversible laws, you can get different behaviour by starting the system in a different way. If you start with an egg and stir it with a fork, then it scrambles. If you start with the scrambled egg, and very very carefully give each tiny particle of egg exactly the right push along precisely the opposite trajectory, then it will unscramble . The difference lies entirely in the initial state, not in the laws. Notice that ‘stir with a fork’ is a very general kind of initial condition: lots of different ways to stir will scramble the egg. In contrast, the initial condition for unscrambling an egg is extremely delicate and special.
    In a way this is an attractive option. Our clumping universe is like an unscrambling egg: its increasing complexity is a consequence of very special initial conditions. Most ‘ordinary’ initial conditions would lead to a universe that isn’t clumped – just as any reasonable kind of stirring leads to a scrambled egg. And observations strongly suggest that the universe’s initial conditions at the time of the Big Bang were extremely smooth, whereas any ‘ordinary’ state of a gravitational system presumably should be clumped. So, in agreement with the suggestion just outlined, it seems that the initial conditions of the universe must have been very special – an attractive proposition for those who believe that our universe is highly unusual, and ditto for our place within it.
    From the Second Law to God in one easy step.
    Roger Penrose has even quantified how special this initial state is, by comparing the thermodynamic entropy of the initial state with that of a hypothetical but plausible final state in which the universe has become a system of Black Holes. This final state shows an extreme degree of clumpiness – though not the ultimate degree, which would be a single giant Black Hole. The result is that the entropy of the initial state is about 10 -30 times that of the hypothetical final state, making it extremely special. So special, in fact, that Penrose was led tointroduce a new time-asymmetric law that forces the early universe to be exceptionally smooth.
    Oh, how our stories mislead us … There is another, much more reasonable, explanation. The key point is simple: gravitation is very different from thermodynamics. In a gas of buzzing molecules, the uniform state – equal density everywhere – is stable. Confine all the gas into one small part of a room, let it go, and within a split second it’s back to a uniform state. Gravity is exactly the opposite: uniform systems of