The Science of Discworld IV

support, stated that the universe must be either a hypersphere, an infinite Euclidean space, or an infinite hyperbolic space. Other topologies were largely forgotten.’
    In fact, more than one topology is possible in each of the three cases. Friedmann had said as much in his 1924 paper, for negative curvature, but this remark somehow became forgotten. Finite spaces of zero curvature had already been discovered, the most obvious being the flat torus. Elliptic space was finite anyway. But even that space was not the only possibility with constant positive curvature, a fact known to Poincaré in 1904. Unfortunately, once the misconception had set in, it was very hard to root it out again, and it obscured the question of the shape of the universe for decades.
    However, at that time cosmologists were after bigger game: the origin of the universe. According to the Big Bang solution of the field equations, both space
and time
sprang into existence from nothing, and then evolved into today’s universe. Physicists were ready for this radical theory because quantum mechanics had already softened them up for the idea that particles can arise spontaneously from nothing. If a particle can do it, why not a universe? If space can do it, why not time?
    Looking back at Einstein again. He could even have
predicted
an expanding spherical universe, but he got it into his head that the static one was the right one. To obtain a static solution, he modified his field equations to include an extra term depending on a ‘cosmological constant’. By choosing this constant suitably, the universe could be rendered static. Precisely
why
the cosmological constant would have that value was less clear, but the new term in the equations obeyed all of the deep symmetry principles that drove Einstein’s philosophy of how the universe ought to behave. It would actually take a lot of special pleading to eliminate that term. When telescopic observations of the spectra of galaxies revealed an expanding universe, Einstein decided that including the cosmological constant had been his ‘biggest blunder’. If he had left it out, he could have predicted the expansion.
    Well … that’s the standard story, but it requires an unstated assumption. In order to derive a formula for how the shape and size of theuniverse changes over time, the mathematical physicists of the early twentieth century looked only for spherically symmetric solutions of Einstein’s field equations. This assumption reduces the spatial variables from three to one: the distance from the centre. As a bonus, it simplifies the Einstein field equations, which can now be solved by an explicit formula. Although there is a hand-waving justification of spherical symmetry, ‘the universe should be the same everywhere’, it doesn’t have a solid basis. Einstein had insisted that the
laws
should be the same everywhere, but that doesn’t imply the same
behaviour
everywhere. Both planets and the vacuum obey the same laws, for example.
    With the advent of computers, it turned out that the Einstein field equations have zillions of solutions – infinitely many, depending on the choice of initial conditions – almost all of which are not spherically symmetric. Space might expand in some regions, contract in others, or swirl round and round. It could change its behaviour as time passed. So although an expanding universe is one possible solution of the Einstein field equations, it no more constitutes a prediction of an expanding universe than the
possibility
of rain tomorrow, as a solution of the weather equations, constitutes a firm forecast of rain.
    A few years ago, all was sweetness and light. The Big Bang fitted all the important observations. In particular, it predicted that the cosmic microwave background radiation should have a temperature of about 3 degrees absolute. Score one to the Big Bang.
    As research continued, however, difficulties emerged. Today’s universe has a lot of large-scale structure – vast skeins of galaxies surrounding even vaster voids, like the foam in a glass of beer, with galaxies forming on the surfaces of beer bubbles, and voids corresponding to the air inside them. Backtracking from its present state and using current theories, the universe should be about 13.5 billion years old. On the one hand, that’s too short a time to explain the current clumpiness of matter. On the other hand, it’s not long enough to explain the current flatness of space.
    A second