6

On the day that the mountain’s centrifugal gravity returned to full strength, Agata spent the morning tidying her apartment.

Her intention had always been to keep as much of the layout as possible fixed across the changes of vertical, and though the demands of safety and comfort had forced various compromises she’d managed to leave one large, corner-mounted cupboard unopened for the whole three years, in the hope that a strict refusal to meddle with its contents might allow every item to return of its own accord to its original position.

This proved to have been excessively optimistic. In retrospect, she realized that it was probably the brief interludes of weightlessness, rather than the long exposure to sideways gravity, that had wreaked the most havoc, allowing the effects of small bumps and vibrations to accumulate, feeding entropy into the jostling mass of books, papers and knickknacks. If there’d been any prospect of the turnaround being repeated, she would have started by tying a few more items together with string, cutting down the number of degrees of freedom.

Agata had a meeting with Lila in the afternoon, but she’d run out of food so she left the apartment early to give herself time to eat on the way. Striding down the corridor, using a guide rope to help her maintain traction, she ran her free hand over the dusty footprints still clinging to the wall on her left.

“Agata!”

She raised her rear gaze. She hadn’t been mistaken about the voice: the man approaching behind her was her brother.

Pio caught up with her. “I almost missed you.”

“I have an appointment,” Agata said curtly.

“Can I walk with you? I won’t slow you down.”

Agata hummed indifference.

“They let me out yesterday,” Pio explained, moving beside her and taking the same guide rope. “Cira came to meet me, but she said you were still angry.”

“Why would I be angry?”

“I had nothing to do with the gnat at the Station,” Pio declared. “That was a dangerous stunt, and if I’d known anything about it I would have tried to stop it myself.”

Agata didn’t believe him, but she knew she’d only make a fool of herself if she started arguing about the migrationists’ internal power structures with someone who actually knew what they were.

“Well, there’ll never be a chance to repeat it,” she said. The Peerless’s reversal had rendered every cousin of the Object into ordinary matter, and turned the Hurtlers into nothing but slowly drifting sand. “We’re in for six generations of cosmic tranquility.”

“Good” Pio replied.

“And you know there are no restrictions on the engines?” Agata added.

“I heard that at the time,” he said. “They let us watch the news.”

A woman walked past them, looking twice when she recognized Pio then hurrying on. Agata felt herself soften a little. She’d had visions of her brother emerging from prison ranting denials against every unwelcome new fact. “If cooling air escapes from the mountain now,” she said, “it will end up mingling with the orthogonal cluster, violating its arrow of time. And yet—” She stopped and spread her arms. “I don’t feel myself burning up.”

Pio buzzed. “I don’t think you blame me for the gnat; I think you’re still punishing me for my debate with Lila. There might have been a problem for us with the arrows clashing. At the time nobody had proved that there wouldn’t be, and I was right to point that out.”

“So you’d say we’ve been lucky” Agata pressed him, “but you’re satisfied now that there’s no reason not to forge ahead, all the way to the reunion?”

“That’s a weighty demand,” Pio replied lightly. “If you’re asking me whether I’m going to advocate any kind of change in course in the immediate future, the answer is no. There’s nothing we could do at this moment that would make the Peerless any safer, and no risk that we urgently need to avoid.”

He gestured toward the floor—toward the rim, out into the void. “But as Lila said in the debate, the orthogonal worlds are still out there, and they can’t annihilate us any more. So don’t ask me to renounce the possibilities they offer. All I’m calling on people to do right now is to keep an open mind. Is that so terrible?”

Agata said, “You’ve forgotten your own slogan: ‘Let the ancestors burn.’ Why should anyone open their mind to that?”

“Let them burn if necessary,” Pio replied. “If the alternative is even worse.”

Agata stopped walking. “You know, you almost sound convincing sometimes. But you were ready to give up on the home world before on much weaker grounds than necessity.”

Pio raised his hands contritely. “I got carried away in the debate. I know it offended you, and I’m sorry.”

They’d almost reached the turnoff 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 demeanor 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 toward 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 skeptically. “Try me, then. But be warned: if I can’t explain it properly afterward 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 favored 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 hypothesize 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 realized 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 favored—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 toward 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 visualize the orthogonal cluster.”

“Visualize it?” Agata scrutinized 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 toward the orthogonal stars, you can read off the details.”

Before the turnaround Agata would have been skeptical, 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 leaned 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.”

Medoro said, “Now suppose that we arrange a sequence of occultations—of the slowest detectable light, with the blocking taking place as far from the telescope as possible.”

Agata thought she knew where he was headed. “Then the image of the star will blink out before the blocking object is actually in place. But you know, even the slowest detectable infrared is quite hard to outpace. So unless you build some massive engines, these flying shutters of yours would need to be launched long before you see their effect on the star.”

But Medoro wasn’t finished with his thought experiment. “Now add a pair of mirrors and fold up the light path, so we can achieve the same effect while manipulating an object that’s much closer.”

Agata raised a quick sketch of the proposal.

“Depending on the dimensions of the system and the number of bounces before the loss to the mirrors is too great,” she said, “you’ll be able to make observations that reveal the shutter’s position some time into the future. I’m no expert on practical optics, but I’d guess that a realistic time span would be measured in flickers at most.”

Medoro said, “Maybe. But suppose it’s more than twice the response time of an automated signal booster. You might only be able to receive the message from a short way into the future, but so long as you can resend it to a time when the booster will be free to handle it ‘again’—without any overlap with the later boost—the process can go on indefinitely.”

Agata gazed at the picture on his chest. If there was a flaw in his plan, she couldn’t see it.

“Greta didn’t mention anything like this?” she asked.

Medoro scowled. “No—but do you really think anyone could commission a camera that detects light from the future, without this sort of thing crossing their mind?”

Agata was ashamed that she’d failed to see the possibilities herself long ago. This was the most beautiful idea she’d ever encountered.

“You’re right,” she said. “The Council must be working on a messaging system. And if they can boost the signal like this…” She reached over to Medoro, almost touching the diagram. “Then I don’t see why we couldn’t use it to learn about the journey still to come, all the way up to the reunion.”