The Arrows of Time

by Greg Egan

  20

  As the Surveyor drew closer to Esilio’s sun, Agata willed the days to stop slipping away so quickly, robbing her of precious time to work.

  It had taken her four years to reshape the foundations of field theory into a form that made sense to her: a kind of dissection of the behaviour of fundamental particles into a series of simple diagrams. When a photon moved from one place to another, the first diagram of the series showed this happening entirely uneventfully. But in the second diagram, the photon was shown giving up its energy to the luxagen field to create a pair of disturbances with positive and negative source strength, which travelled for a while before recombining into a replacement for the original photon.

  In a sense it was just like the old double-slit experiment that Yalda’s teacher, Giorgio, had used to convince people that light was a wave: light couldn’t be passing through one slit or the other, because the pattern of bright and dark lines it made could only be understood by adding contributions due to the light taking paths through both slits. But in Agata’s version the set of ‘paths’ included not only a variety of routes, but all manner of transmogrifications along the way.

  She had baulked at this, at first: a lone photon couldn’t turn into a pair of luxagens – each with just a third of the photon’s mass – because whatever the velocities of those luxagens, it would be impossible to satisfy the laws of conservation of energy and momentum. But she’d finally understood that each of the diagrams on its own was just a kind of fiction, expressing a narrow sliver of the true history, and any characters that came and went without being present at the start and end of each story were mere flights of fancy, subject to very different rules compared with those that endured. Every part was needed to make up the whole, but only the totality was real.

  With any process the variations were endless, but the more complex the diagram the smaller its contribution, allowing the sum to remain finite. And in this scheme, the vacuum itself was simply the sum of all diagrams that started and ended with no particles at all, its energy due entirely to disturbances that came and went of their own accord, with no connection to anything persistent.

  Agata had been gratified to discover that, in flat space at least, these diagrams rendered the vacuum manageable. But if the vacuum energy curved space, then flat space was actually impossible – and if curvature modified the vacuum energy, the two could only be in harmony at some elusive fixed point that remained beyond the reach of her methods.

  Having come so far, she ached to complete the story. She wanted to return to the Peerless with everything solved: the vacuum energy tied to the curvature of space and the topology of the cosmos – determining once and for all whether the entropy gradient that had enabled life was a stupendously improbable fluke, or simply an unavoidable consequence of a few simple principles.

  When she lifted her gaze up from her desk, the prospect of the Surveyor’s planetfall was glorious and thrilling, the purpose of their journey finally to be fulfilled. But when she looked down at her unfinished calculations, she thought: glorious – but please, not yet.

  Gathered with the rest of the crew around Tarquinia’s console, Agata compared the two images on the screen. One was a grey disc faintly mottled with reds and browns, weakly but uniformly lit, grainy and poorly resolved as the photodetectors struggled at the limits of their sensitivity. The other was a disc of the same size, and two-thirds of it lay in the deepest black night, but the crescent of dayside revealed an impossibly vivid landscape of jagged grey mountains, dusty red plains and twisted brown valleys, sharp enough to touch.

  Esilio by the light of the home cluster’s stars, and Esilio by the light of its sun. Esilio as they’d see it with their own eyes, and Esilio through the time-reversed camera. Esilio as it had been a few chimes ago – and Esilio as it would be a few chimes in the future.

  Tarquinia said, ‘The good news is, the temperature looks tolerable. Hotter than we’re used to, but not by much.’

  Agata was surprised. ‘How did you measure the temperature?’

  ‘The density profile of the atmosphere. A hotter atmosphere will stretch up farther from the surface.’

  ‘Is that reliable?’ Agata had no problem with the general idea, but she suspected that the method would be fraught with uncertainties.

  ‘I’m not sure,’ Tarquinia confessed. ‘I’ve never had a chance to observe a planet before.’

  Ramiro said, ‘If this world’s come all the way around the cosmos, shouldn’t it have had time to grow hotter?’

  ‘No plants, no fires,’ Azelio pointed out. ‘If there’s nothing making light, there’s just slow geochemistry to warm it up.’

  ‘Ah.’ Ramiro turned to Agata. ‘Temperature doesn’t change when you swap the direction of time, does it?’

  ‘Not as such,’ Agata replied cautiously. ‘Imagine reversing the motion of all the particles in a container of gas: it wouldn’t make any difference.’

  ‘But if “temperature as such” is unchanged, what about the implications?’ Ramiro pressed her. ‘Will heat still flow from hot to cold?’

  ‘That depends on exactly what you’re talking about.’ Agata wasn’t trying to be unhelpful, but the worst thing she could do was make a blanket pronouncement that ignored the subtleties of the problem. ‘We ought to be able to find examples on Esilio where two lukewarm objects start out with the same temperature, but then heat flows from one to the other – making one cooler and the other hotter.’

  Ramiro hummed impatiently. ‘That’s obvious: inasmuch as we’re able to act purely as spectators, we can expect to see ordinary things happening in reverse. But when we touch something down there, some rock that’s colder than our hands . . . ?’

  Agata said, ‘Why do you expect there to be a simple answer to that: a rule that will hold true in every case? We’re used to predicting heat flows on the basis that entropy will increase along one direction in time — and the same principle will have held on Esilio for most of its history, for its own notion of the future. But the two arrows point in opposite directions, so each side’s rule flatly contradicts the other. Those rules were never universal laws, and this is the place where we finally have to accept that.’

  ‘But couldn’t the Esilian rock pass some of its heat to us, even if it’s colder?’ Azelio suggested. ‘Its entropy goes down, as we see it, while ours goes up. So both sides get to follow their usual rules.’

  ‘That’s not impossible,’ Agata agreed. ‘But we can’t expect to be able to partition everything as neatly as that. While we’re still far apart we can talk about the two sides and their rules . . . but deep down, matter is just matter, it doesn’t come with allegiances. The real laws of physics treat all directions in time and space identically, and they’re the laws that every photon and luxagen obeys — without knowing or caring about anything called entropy, let alone what side it’s meant to be on in some clash of thermodynamic arrows.

  ‘Suppose we leave a piece of equipment behind on Esilio – say, a small spyglass. Over the eons, from our point of view, we’d expect it to become pitted by dust in the wind, and eventually break up completely and turn to sand. Our spyglass, our rules: that sounds fair, doesn’t it? But if that sand stays on Esilio, what origin will it have from Esilio’s point of view? Most likely, some ordinary Esilian rock will have broken down to make it — which to us, would look like erosion running backwards. But then, in Esilian time the remnants of that rock will eventually form themselves spontaneously into a spyglass, which lies on the ground until we come along to retrieve it. So if you follow the history of the matter that makes up the spyglass far enough in both directions, it’s clear that it’s not committed to either side’s rules.’

  Ramiro said, ‘That’s all very fascinating, but you still haven’t told me whether or not I’d burn myself by touching a cold rock.’

  Tarquinia broke in. ‘No one will be touching anything until we’ve done enough experiments to know what’s safe and what isn’t.’

&n
bsp; Ramiro gave up and dragged himself away, muttering about the uselessness of theoreticians.

  Azelio caught Agata’s eye. ‘Your story about the spyglass was unsettling,’ he said, ‘but I’ll tell you what disturbs me more.’

  ‘What?’

  ‘Swap the roles of Esilio and the Surveyor,’ he replied, ‘then tell the same story again. If something from Esilio takes the place of the spyglass, it must be with us already. We must have been carrying it, or the things that will become it, from the very start. Because according to Esilio’s arrow of time we’ve already visited the planet, and it’s almost certain that something remained with us when we departed.’

  ‘The black sun awaits your pleasure,’ Tarquinia announced from the doorway.

  Agata looked up, startled. ‘Already?’

  ‘It’s now, or wait until we’re on our way back.’

  ‘Of course.’ Agata hesitated. ‘The telescope’s mine, until we switch orbits?’

  ‘Absolutely,’ Tarquinia replied. ‘But if you break it, you can grind a new lens.’

  ‘From what?’

  ‘The other part of your punishment will be hunting down suitable materials on Esilio.’

  Agata could have done everything from her room, but that seemed selfish: the experiment belonged to all of them, and she wanted every member of the crew to feel free to look over her shoulder as she worked. So she dragged herself into the front cabin and strapped herself to her couch there.

  Tarquinia had trained her to use the telescope’s software, but Agata still felt an illicit thrill when she invoked it from her own console and began passing it instructions through her corset. Since they’d shut off the engines the Surveyor had been sweeping in towards Esilio’s sun along a hyperbola, with the home cluster’s stars behind them. But as they swung around the sun in order to help them match velocities with Esilio she’d finally have a chance to juxtapose the two kinds of stars, with the dark mass in the foreground perfectly suited to its role.

  Agata used the navigation system to map out the expected path of the black disc against an ordinary-light image of the sky. Then she chose two dozen points on various star trails that were destined to pass behind the sun, and measured their current positions with as much precision as the instruments allowed. The idea that gravity might distort the appearance of these trails wasn’t all that shocking – if it could bend the path of a planet into an ellipse, why wouldn’t it be able to nudge a beam of light? What was astonishing was the prospect of being able to distinguish between a force tugging on the light and curving its trajectory, and the light merely following the straightest possible history through a space that was itself curved.

  Azelio harnessed himself to the couch beside her. ‘How do you know you won’t just be measuring an optical effect from the sun’s atmosphere?’ he challenged her.

  ‘I’ll need to include that in the final calculations,’ Agata conceded. ‘But there ought to be a point where the gravitational effects are showing up clearly, while the light’s still travelling far above the densest part of the atmosphere.’

  ‘Really? You’ve always talked about starlight “grazing the disc,” ’ Azelio protested.

  ‘I have, haven’t I?’ She’d been trying to stress that the lack of glare from the time-reversed sun would allow her to follow the stars right up to the moment they disappeared behind it. ‘But there’s nothing special about the light passing just above the surface – the effect doesn’t suddenly increase there. It’s the distance from the centre of the sun that counts, not the distance from its surface.’

  Azelio inclined his head, accepting her answer. But he remained sceptical. ‘And this measurement is going to tell you the shape of the cosmos?’

  ‘No – it’s necessary for that, but not sufficient. If I end up disproving Lila’s theory, then I won’t have much hope of working out the shape of anything. All my calculations linking energy to curvature depend on Lila being right.’

  Azelio was confused. ‘Why couldn’t you adapt your work to Vittorio’s theory?’

  Agata said, ‘If the results agree with Vittorio’s theory then I’ll have no choice but to accept that as a fact – but I’d have no idea how to integrate it into modern physics. Lila’s theory makes gravity consistent with the notion that everything should work the same way when we rotate our picture of it in four-space. If gravity doesn’t respect that, it would be the most shocking discovery since Yalda came down from Mount Peerless.’

  ‘Then you should hope for that shock,’ Azelio joked. ‘You’d be as famous as Yalda.’

  ‘And I’d have to throw out half a lifetime’s work and start again.’

  ‘Isn’t that the price of every scientific revolution?’

  ‘Lila’s theory is the revolution!’ Agata countered. ‘It’s just been a quieter one than Yalda’s or Carla’s, because it’s been so hard to test. What the revolution will throw out isn’t my work, it’s Vittorio’s – and he didn’t live long enough to know or care that his beautiful ideas weren’t perfect.’

  ‘I won’t believe that space is curved until I’ve seen it with my own eyes,’ Azelio avowed. He wasn’t usually so invested in any of Agata’s purely theoretical claims, but he seemed to have found this impending empirical affront to his intuition too much to accept without protest.

  Agata gestured at the screen. ‘You’ll see something, soon enough.’

  ‘No, all that will show us is that the light is bent. Which Vittorio’s theory predicts as well.’

  Agata buzzed at his stubbornness. ‘Bent by a different amount – and for some colours, in the opposite direction!’

  Azelio said, ‘Honestly, don’t you think you’re trying to conclude too much from such slender evidence? Even if the bending is exactly what you predicted, couldn’t there be another explanation for it? Maybe the requirement for gravity to fit in with rotational physics implies certain angles of deflection for the light. But that could come from a tiny modification to Vittorio’s force law, couldn’t it? We’ve always known that gravity bends the paths of moving objects. Why not just refine that notion – instead of leaping to the conclusion that it’s actually bending space?’

  Agata didn’t know how to answer him. From the point of view of everyday experience, it probably did sound grandiose to make so much of such a small effect.

  She thought for a while. ‘I’ll tell you why I’m going to believe that space is curved, unless I find overwhelming evidence to the contrary.’

  ‘Go ahead.’ Azelio was probably unswayable, but he was still interested in understanding her position.

  ‘If motion under gravity is due to curvature, rather than a force, it will obey an incredibly simple rule: the history of any object in free fall is just the shortest available path through four-space. In flat space, that’s a straight line. In the curved space around a star, it’s not.’

  ‘That’s simple in itself,’ Azelio allowed. ‘At the cost of making the geometry more complicated.’

  ‘But it’s more than just simple!’ Agata insisted. ‘It also fits perfectly with everything else we know about motion.’

  ‘In what way?’

  ‘When light moves from place to place,’ she said, ‘you need to add up contributions from different paths between its starting point and its destination. Paths where it spends about the same time travelling all add together, because the waves will have stayed more or less in step, with their peaks arriving simultaneously. Paths where the travel time varies rapidly mix up peaks and troughs, so they cancel each other out.

  ‘Imagine a kind of mathematical valley that stretches across the landscape of all paths, where the length of each path determines the height of the landscape. The shortest path becomes the lowest point: the bottom of the valley. If you change the path there slightly, you barely change its length, because the bottom of a valley is horizontal. But if you’re far up on the side of the valley instead, the path isn’t just longer, it’s at a point where the valley slopes much more, so any chan
ge would change the length more – making the waves slip out of phase.’

  Agata sketched an example on her chest, and had the corset display it on her console.

  Azelio frowned, but then he remembered something. ‘We used that principle in our optics class: you can find the law of reflection by looking for the angle that light makes with a mirror that lets it arrive all in phase.’

  ‘Right! So now apply the same logic to starlight moving past Esilio’s sun. Suppose the light does bend. If four-space is flat, then the light won’t be following the shortest path, since in flat space that’s always a straight line. It will be on a path up on the valley’s slopes, where any tiny variation changes the length and throws the light out of phase. There are ways around that: we can postulate some mechanism that messes with the phase in exactly the right manner to favour the bent path – but that’s complicated, because as well as explaining what happens with the light, it needs to explain the force on an orbiting planet.

  ‘If four-space is curved, though, that does the job for everything. Light waves and luxagen waves, it makes no difference: if they’re following the shortest path in four-space, they’ll arrive in phase. That’s enough to bend the beam, and enough to make a planet swing around in its orbit.’

  Azelio pondered this, and found nothing he could object to. ‘It makes more sense than I thought,’ he admitted.

  Agata was delighted. ‘So what’s your conclusion?’

  ‘My own prediction, now,’ he declared, ‘is that the light won’t bend at all. I can see why you think it would make things too complicated if you had to account for the bent paths of both light and matter, in flat space. So the simplest solution would be to keep space flat, but have light unaffected by gravity.’

  Agata was on the verge of embarking on an account of how this would violate conservation of energy, but she stopped herself; she’d reached the point where it would be more economical to let the results speak for themselves. ‘Do you want to put two loaves on it?’ she suggested.