The Arrows of Time

‘This is from your work?’ Ramiro asked. ‘Pelagia found a way to complete the calculations?’

‘Not so much complete them as see into my blind spot,’ Agata explained. ‘Listen, it’s simple. A luxagen is described by a wave that changes sign when you rotate it by a full turn. That has no effect on any probability you calculate – so long as you apply the same rotation to everything in sight – because the probability comes from squaring the value of the wave. Minus one squared is one, so the change of sign makes no difference. It only shows up in more complicated experiments where you rotate some things and leave others unaltered.’

Tarquinia said, ‘I follow that much.’

‘Pelagia’s idea just replaces rotations with trips around the cosmos. Suppose the cosmos were a four-dimensional torus, and you carried a luxagen all the way around it in a giant loop. What happens to the sign of its wave? Does it come back unchanged, or does it come back reversed?’

Ramiro frowned. ‘I can tell that you want us to say “reversed” by analogy, but I thought there had to be perfect agreement around a loop.’

Agata buzzed. ‘I didn’t want you to give either answer. There could be perfect agreement, or the sign could be reversed: nothing rules out either possibility. If the sign’s reversed, that will be undetectable: everything you can measure locally will still be in perfect agreement.’

Tarquinia said, ‘Hang on, if the sign changes… where exactly does it change? What’s this special place on the torus where it flips over?’

‘There is no special place,’ Agata insisted. ‘It’s like cutting open a band and rejoining the ends with a twist: once you’ve glued them together, there’s really nothing special going on at the join. The twist isn’t located there – or anywhere. It’s a property of the whole band.’

Agata began to form a sketch, but Ramiro saw that she was having trouble so he drew what she’d described on his own chest.

‘So you’re talking about the cosmos being some kind of… twisted torus?’ Tarquinia asked.

‘No, not the cosmos,’ Agata replied. ‘The two bands, twisted or not, both have identical circles as their midlines—’

Ramiro added the midlines to his diagram.

‘—and you should think of those circles as the cosmos. What happens with the bands is an additional structure that the topology of the cosmos doesn’t fix, one way or the other. It’s all about the luxagens, not space itself.’

Tarquinia said, ‘All right. I think I’ve got it.’

‘Then the next step is to remember that we’re talking about a four-dimensional torus,’ Agata continued. ‘So there are four completely different ways you can travel in a loop. There’s nothing that compels those four routes to have the same effect – it would be perfectly consistent to have a luxagen whose sign changed around some of those loops but not the others. So there are sixteen possibilities altogether: for each loop, the sign might change or it might not.’

Ramiro understood the counting argument, but he couldn’t see where it was leading. ‘Aren’t these distinctions all invisible, though? They have no effect on any probabilities.’

‘They have no effect on probabilities,’ Agata agreed. ‘But if there were sixteen times more choices for the state of every luxagen, that would multiply their contribution to the vacuum energy by a factor of sixteen. Photons give a positive vacuum energy, but luxagens make the vacuum energy negative, and a factor of sixteen would be enough to guarantee that the total energy density in the cosmos was negative, absolutely everywhere.’

Ramiro struggled to recall the implications of this, but Tarquinia beat him to it.

‘A negative energy density means positive curvature,’ she said tentatively. ‘But you can’t have a torus that’s positively curved everywhere.’

Agata chirped. ‘Exactly! You end up with a contradiction. So the cosmos can’t be a torus. But in a four-sphere, every route you might travel can be shrunk down gradually to a tiny circle, and then to a point: a path that goes nowhere. The sign of the wave can’t change along a path that goes nowhere, so there are no extra modes for the luxagens. The vacuum energy stays positive, which means the curvature will mostly be negative – but it also has to change from place to place, because you can’t have uniform negative curvature on a sphere. And because the curvature depends on the entropy of matter, that has to change too. That’s why the cosmos isn’t in a state of equally high entropy everywhere. That’s why there’s a gradient. That’s why we exist at all: with a history, with memories, with an arrow of time.’

Watching her as she spoke, Ramiro couldn’t help sharing her joy. Perhaps the discovery changed nothing tangible, but it vindicated all her years of effort – and it proved that the Peerless was back on course. New ideas were possible again. The paralysis was over.

‘And that settles everything?’ he asked. ‘Cosmology is complete now?’