The Science of Discworld IV

We’ve mentioned the flat torus, in which we glue together the opposite edges of a square. Squares arenice because they can be fitted with grids of smaller squares, which have a quantum feel to them because they are discrete – they come in tiny lumps. But you can make flat tori from other shapes too, namely parallelograms.
    The shape of the parallelogram can be captured by a number called the modulus, which distinguishes long thin parallelograms from short fat ones. A different modulus gives a different torus. Although the tori obtained in this manner are flat, they have different metrics. They can’t be mapped into each other while keeping all distances the same. The effect of gravity in Torusland is not to create gravitons: it is to change the modulus, the shape of space.
    Steven Carlipp has shown that in Torusland, there is an analogue of the Big Bang. But it doesn’t start with a
point
singularity. Instead, it begins as a
circle
: a torus with modulus zero. As time passes, the modulus increases, and the circle expands into a torus. Initially this looks like a bicycle tyre, and corresponds to a long thin parallelogram; it is heading towards a square, the standard model for a flat torus, which when curled up looks more like a bagel. So the long-term goal of the Flatland Big Bang turns out to be A. Square. Crucially, Carlipp quantised this entire process; that is, he formulated a quantum-mechanical analogue. That let theoretical physicists explore the relationship between quantum theory and gravity in a precise mathematical context.
    Torusland sheds a great deal of light on the process of quantising a gravitational theory. One apparent casualty of this process, however, is time. The quantum wave function of Torusland does not involve time at all.
    In
The Science of Discworld III
Chapter 6 , we discussed Julian Barbour’s
The End of Time
, which proposes that time does not exist in a quantum world because there is only one universal wavefunction, not involving time. The book was widely interpreted as telling us that time is an illusion. ‘There can only be once-and-for-all probabilities,’ Barbour wrote. We argued that alongside the universalwavefunction, our universe has another basic quantum-theoretic feature, which describes how likely transitions between different states are. These transition probabilities show that some states are closer together than others, and that lets us arrange the events in a natural order, restoring a sensible notion of time.
    Torusland supports this idea, because it has several sensible notions of time, even though its quantum wavefunction is timeless. Time can be measured using Torusland’s equivalent of GPS satellites, by using the lengths of curves between its version of the Big Bang and ‘now’, or by the current size of the universe. Torusland is not timeless at all. You just have to look at it in the right way. In fact, Torusland time leads to an intriguing thought: perhaps time is a consequence of gravity.
    Another idea that Torusland casts doubt on is the holographic principle. This says that the quantum state of the entire observable universe can be ‘projected’ onto any black hole’s event horizon – the point of no return from which nothing can escape – so the universe’s three spatial dimensions can be reduced to just two. It’s like taking a photograph, with the startling property that the photograph faithfully represents
all
aspects of reality. In Roundworld, if someone shows you a photo of a field with a dozen sheep lying down, you can’t tell whether there are lambs hiding behind some of the sheep. But in this event-horizon photo of the universe, nothing can be hidden. The behaviour in two dimensions corresponds perfectly to that in three. The laws of physics change, but everything matches up.
    This is a bit like the way a two-dimensional hologram creates a three-dimensional image, which is why this idea is called the holographic principle. It suggests that not only is the dimension of the universe an open question: it may not be well defined – the answers ‘two’ and ‘three’ may both be true at the same time. This idea has led to some advances in the way string theory represents gravity, and also to articles in the press stating ‘
You are a hologram!
’
    Physicists began to suspect that a similar principle works in any number of dimensions. But it turns out that in Torusland, there isno holographic principle. A. Square may be flat, but