Greg Egan - The Arrows of Time

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.’

Agata was intrigued. Medoro made cameras for the astronomers from time to time, but he’d never felt the need to consult with her before. ‘What are you building?’ she asked.

‘A new imaging chip,’ he said. ‘One that can visualise the orthogonal cluster.’

‘Visualise it?’ Agata scrutinised his face, half suspecting that she was being set up for a joke, but either way she couldn’t resist the bait. ‘How?’

Medoro said, ‘Instead of polling the array of pixels on the chip and counting how many photons have struck each of them, it will count how many photons each pixel has emitted . Point the camera at the sky… and when it emits light towards the orthogonal stars, you can read off the details.’

Before the turnaround Agata would have been sceptical, but now she could see that the possibility of a camera like this had been implicit in the results of the very first engine tests after the reversal. Just as the engines had happily given off light that the ultimate recipients would consider to be arriving from their future, the orthogonal stars were – presumably – still shining down on the Peerless , despite being rendered invisible by the very same property. People’s eyes had not evolved to know when they were the joint authors of a beam of light, as responsible for creating it as the distant star at the other end. But a camera could be made to catch its own strange radiance in the act.

‘Who commissioned this?’ she asked.

‘Do you know Greta?’

‘No.’ Agata knew all the astronomers, and there was no Greta among them.

‘She’s a technical adviser to the Council,’ Medoro explained. ‘She supervised the turnaround, but now that it’s over she’s been given this new thing.’

‘Which is… ?’

Medoro leant forward as if to share some delicate confidence. ‘I was told that the camera would be part of a general upgrade of the navigation systems. The rationale being that the old maps are fine for most purposes, but if we can find a way to keep getting real-time images of the orthogonal stars, so much the better.’

‘Except that this is better than real-time,’ Agata joked. ‘Instead of seeing where the star was, we’ll know where it will be.’

Medoro said, ‘That, and a great deal more.’

‘I’m sorry?’

He buzzed impatiently. ‘Come on, you’re the physicist! Do I have to spell it out?’

Agata stared at him, bemused. Knowing the future positions of the orthogonal stars would not be a momentous revelation: their trajectories were already predictable over a time-span of eons. And in fact, these stars’ ‘future’ positions would be positions in which they’d already been observed, earlier in the Peerless ’s own twisted history. Telescopes had improved since then, but there were unlikely to be any spectacular, collision-avoiding surprises.

‘You’ve lost me,’ she confessed.

‘Suppose something occults an orthogonal star that I’ve been watching with this camera,’ Medoro said. ‘What happens then?’

‘The occulting object will take the place of the camera as the second source of the light.’

‘So we’ll know about the occultation?’ he pressed her.

Agata said, ‘Of course! If there’s no light passing between camera and star, the “image” of the star will disappear, just as an ordinary image would.’

‘And when will we know about it?’

‘When? The exact time will depend on the geometry: the location of the object that blocks the light, and the speed of light for the part of the star trail that’s obscured.’