The Arrows of Time

They’d almost reached the turn-off to Lila’s office. Agata didn’t want to detour for a meal now in case Pio insisted on joining her.

‘I have to go,’ she said. ‘You can tell Cira that you tried your best, to no avail.’

‘What are you talking about?’ But Pio’s baffled demeanour was a bit too self-conscious to be believable.

‘You should find something useful to do,’ Agata suggested. ‘I’m sure they still need help re-bedding the medicinal gardens.’

‘And your work’s useful?’ he retorted. ‘Try some gardening yourself!’

‘Goodbye, Pio.’ Agata strode towards the intersection, glancing at her brother with her rear gaze in the hope that he’d set off back down the corridor so she could get to the food hall after all. But he must have been hungry too, because he headed for the hall himself.

Agata muttered imprecations against her family and readied herself for a bell or two of higher mathematics through the eyes of a Starver.

‘Are you eating for four now?’ Medoro joked.

Agata looked up. ‘We can share if you want to. I might have ordered too much.’

Medoro sat on the floor, facing her, and helped himself to a loaf. The food hall was quiet, and Agata had been lost in thought.

‘How’s work?’ he asked.

‘I finished proving an interesting result today,’ she said. ‘Lila and I had been fairly sure that it was true, but it took a while to clear up all the technicalities.’

‘Ah. Would I understand it?’

‘Maybe not the proof,’ Agata admitted, ‘but the result itself is simple.’

Medoro buzzed sceptically. ‘Try me, then. But be warned: if I can’t explain it properly afterwards you’ll be hearing from Gineto.’

‘Suppose the topology of the cosmos is that of a four-dimensional sphere,’ Agata began. ‘Not the shape, just the topology: the way it all connects up.’

‘I thought the cosmos was a torus,’ Medoro protested.

‘A torus was Yalda’s preferred model.‘ Agata had nothing but respect for Yalda, but she wished the schools would stop treating this favoured model as an established fact. ‘It makes for a nice, concrete example that’s simple to work with – but the truth is, we don’t know the real topology. It might be a torus, it might be a sphere, it might be something else entirely. The only thing we know for sure is that it has to be finite in all four dimensions.’

Medoro said, ‘All right. So you hypothesise that the cosmos is a sphere. Then what?’

‘Then you ask what kind of curvature it might have.’

‘The curvature of a sphere?’ Medoro ventured.

‘Ha!’ To her amusement, Agata realised that her own intuition now filtered out this eminently sensible guess so rapidly that she hadn’t even thought of mentioning it. ‘Well, you might think so: why shouldn’t the cosmos have the curvature of a perfectly symmetrical four-dimensional sphere? The trouble is, a perfect sphere has equal positive curvature in all dimensions: no direction is different from any other. But in Lila’s theory of gravity, if the disposition of matter is like that – with no direction favoured – what you get is uniform negative curvature. You could only get uniform positive curvature if the energy density were negative, and we have no reason to believe that that’s the case.’

Medoro thought for a while, chewing on a second loaf. ‘So can you have something with the topology of a sphere, but with uniform negative curvature?’

‘You can’t,’ Agata said. ‘In fact that’s what we just proved. A four-sphere with positive curvature is possible geometrically but impossible physically, while a four-sphere with negative curvature would make sense physically, but it’s impossible geometrically.’

‘Hmm.’ Medoro brushed crumbs from his tympanum. ‘Which leaves you with what? That the cosmos can’t really be a four-sphere at all?’

‘No, that doesn’t follow,’ Agata replied. ‘It just means that if the cosmos is a four-sphere, topologically, then it can’t be perfectly uniform: it must differ from place to place.’

‘Aha!’ Medoro chirped appreciatively. ‘So it goes some way towards explaining the entropy gradient?’

‘Some way.’ Agata was pleased with the result, but she didn’t want to oversell it. ‘If we had a reason to believe that the topology had to be a four-sphere, then we could say that the cosmos would need to contain some regions of lower entropy in order to meet the geometrical constraints.’

‘And do you have a reason?’

‘No,’ Agata admitted. ‘As far as anyone knows, the cosmos might just as easily be a torus, in which case our theorem can’t be applied and the entropy gradient is as inexplicable as ever.’

‘Never mind,’ Medoro counselled consolingly. ‘I’m sure someone will work it all out eventually.’

Agata was about to retort that she had every intention of being that ‘someone’, but she caught herself; he was just goading her. ‘That’s enough cosmology,’ she said. ‘How’s the camera business?’

‘Cosmological,’ Medoro replied. ‘Actually, that’s why I came looking for you. I’m starting a new project, and I wanted to hear your thoughts on it.’