Greg Egan - The Arrows of Time

‘Was anyone hurt?’

‘Agata lost a lot of flesh. For a couple of stints no one thought she’d survive, but she’s finally recovering.’

Tarquinia hummed softly. ‘Can we visit her?’

‘Of course.’

As they dragged themselves towards the nearest stairwell, Tarquinia recounted her own misadventure. ‘They let me have the observing time, and everything was looking perfect, but then the most officious busybody among my colleagues decided to check in on me just when I was inspecting the jetpack from the emergency kit. She decided that was suspicious enough to execute a citizen’s arrest; most of the other staff thought she was an idiot, but she had an ally. I was afraid that if I was released, the two of them would be disgruntled enough to make a real effort to get the attention of someone with the power to mention the incident in an official message.’

‘And then the Council would have known from the start.’ The whole crew would have been under close surveillance from the moment the Surveyor arrived.

‘Exactly.’ Tarquinia buzzed. ‘So I ended up having to let myself be detained in a room at the observatory, with the idiots kept busy watching over me and arguing with my colleagues who wanted me released. It was only after the disruption that they managed to get the security department involved.’

They’d reached the level of the hospital; it was just a short walk up-axis now. Ramiro said, ‘Before we see Agata, there’s something I need to tell you.’ He explained his debunking of the hoax. ‘Don’t be angry with me,’ he pleaded. ‘The whole thing about the ancestors was making her crazy.’

Tarquinia said, ‘I’m not angry, but you shouldn’t have told her that.’

‘Why not?’

‘I didn’t make the inscription,’ Tarquinia declared. ‘I went out there to try, but nothing happened: no shards of stone rose from the ground to meet the chisel. I tried different tools, different movements… but I couldn’t unwrite those symbols. If anything, when I left they were sharper than I’d found them – as if all I’d done was make the message less clear for Agata and Azelio than if I’d stayed away completely. I wasn’t the author of those words. Someone else must be responsible for them.’

Ramiro didn’t know if she was telling the truth or just trying to hold on to the benefits of the hoax, but he wasn’t going to start questioning her version of events now. If this was her story, there was nothing he’d seen with his own eyes that contradicted it.

When they entered the hospital ward, Agata caught sight of Tarquinia and called out excitedly, ‘Ah, you’re free! Congratulations! Come and hear some great news!’

As they approached the sand bed, Ramiro could see that Agata had gained some flesh since his last visit, but she was still limbless. The doctors had told him that she would need every scrag of tissue to support her recovering digestive tract.

‘What’s the news?’ Tarquinia asked her.

‘I just had a visit from Lila and her student Pelagia,’ Agata replied. ‘The innovation block is well and truly over!’

‘Yeah?’ Tarquinia had probably been expecting to spend the whole visit trying to put the record straight about the inscription, but Agata’s mind was on something else entirely.

‘Pelagia’s settled the topology question,’ Agata proclaimed. ‘The cosmos is a four-dimensional sphere. It’s not a torus – it can’t be!’

‘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.’