In truth, Yalda longed for a distraction, and she wished she shared Tullia’s discipline and resolve. But the problem itself seemed as intractable as their incarceration. “Giorgio was right,” she said. “The equation I found has exponential solutions. And if I try to damp them down—if I try to get rid of them by adding new terms to the equation—I just lose the original solutions.”
“It’s a strange equation for a wave,” Tullia conceded. “The nice thing about the wave equation on a string is that you can set the initial conditions and just watch them unfold: you can pluck the string into any shape you like, and give it any kind of movement, and from that the equation lets you find the shape of the string at any time in the future. What’s more, if you make a small mistake measuring the initial setup, it’s not a calamity; the errors in your prediction are equally small.
“But your light equation is more like the equation for the temperature distribution in a solid. If you have, say… a thin slab of stone and you want to know its temperature at every point, to get reliable solutions you need to specify the temperature all the way around the border of the slab. If you tried to start with the temperature along a single edge and its inward gradient, any tiny error in the data there would blow up exponentially as you moved across the slab. Your equation acts the same way.”
Yalda pondered this in the darkness. “So by analogy, to calculate the behavior of light in a certain place, over a period of time… I’d really need to know what it does on the entire border of that four-dimensional region? Not only what it’s doing at the start, but everything that happens at the boundary, and how it all ends up?”
“Precisely,” Tullia said. “The equation you’ve come up with might be said to govern the behavior of light, but practically speaking it’s no good for making predictions. Everything it tells you could be tested in retrospect, but you’d always need to know how the story ends before you can start reliably ‘predicting’ the middle.”
Yalda said, “Waves on a string can only have one velocity. We know that violet light can travel much faster than red light—and it’s not implausible that there are even faster hues, beyond our ability to detect. So why should we expect that knowing the state of the light in just one place should ever be enough to say what happens next? Some other wave we haven’t accounted for—just beyond the edge of the region we know about—could always be on the verge of crashing in and spoiling our prediction.”
“Good point,” Tullia replied. “So let’s accept the possibility of light that travels as fast as you wish… but then to compensate, I let you know about every wave that presently exists as far away as you wish. That way, you get as much warning as you could hope for; you can’t complain that some fast wave came hurtling in from a place beyond the reach of your data. And if the cosmos goes on forever, you get to know about the entire, infinite present: for one moment in time, there are no secrets from you anywhere.”
“Go ahead and grant me that!” Yalda urged her. “Wouldn’t that eliminate the problem?”
“No!” Tullia sounded both exasperated and amused that Yalda had taken the bait so readily, without thinking it through. “Your equation still has solutions that can blow up exponentially out of the tiniest mismeasurement. You still wouldn’t be able to predict what happens right in front of your eyes over the next few pauses. Does that really accord with your instincts about the way the physics of light should work?”
“No,” Yalda admitted. She adjusted her position, then cursed softly and braced herself for the sound of tearing flesh. She’d been trying to keep her arms a few scants apart within their shared sleeve of skin, hoping to make her eventual liberation less traumatic. But her body thought it knew best. Every time she dozed, or her attention wandered, she had to rip apart a fresh bundle of muscle fibers that had formed between the ends of her limbs.
When she was done, she thought back over the steps in Tullia’s argument. “What if the cosmos didn’t go on forever?” she said. “In space, or in time?”
“Then you’d still need to know what happens at its boundary,” Tullia replied. “Like the slab of stone: you need to know what’s going on at all the edges.”
Yalda considered the possibilities this offered. By declaring that the boundary of the cosmos was subject to some special rule—perhaps that the wave simply had to be zero there—she could probably keep it from blowing up in the interior. But that was an ugly resolution, an arbitrary constraint that came from nowhere and offered no deeper understanding.
“What if there are no edges?” she suggested. “What if the cosmos is like the surface of the world—finite, but borderless?”
Tullia lapsed into silence for so long that Yalda grew worried. She extruded a new, free arm and thumped the wall. “Are you all right?”
“Yes! I’m thinking!” Tullia almost sounded happy, as if Yalda had finally suggested something novel enough to be truly diverting.
Eventually Tullia declared, “I’m fairly sure that that would solve the exponential blow-up. You can wrap an oscillation around a sphere so that it joins up with itself smoothly—but you can’t do that with an exponential growth curve, which never revisits its earlier values.”
Yalda chirped with delight. “So if the cosmos is a four-dimensional version of the surface of a sphere—”
“Things would still be very strange,” Tullia warned her. “The prediction problem jumps from one extreme to the other.”
“What do you mean?”
“Think about the two-dimensional version,” Tullia said. “If you draw a circle around Zeugma, the data lying on that circle—combined with your equation—tells you everything that happens in the city. Information about the border gives you information about the interior.”
“But that’s nothing new. Where’s the problem?”
“A circle around Zeugma,” Tullia replied, “is also a circle around everything else in the world. The border of the city is also the border of everything that lies beyond. So from the data on that one circle, you can find the solution to your equation on the entire sphere.”
“Oh.”
Tullia drove the point home. “In the four-dimensional version, that’s like claiming that you can measure the light in a patch a few scants across, for a couple of pauses… and learn everything there is to know about light throughout the history of the cosmos. Because the border of your tiny region is also the border of everything else.”
Yalda buzzed wryly. “I can’t say that that accords with my instincts about the physics of light.”
“Nor mine.” Tullia’s surge of enthusiasm was gone, but she was doing her best not to sound despondent.
“We tried,” Yalda said. “And it was worth trying.”
They had managed to leave the prison for a while, but nothing was easy, even in freedom.
When it was quiet in the cells, Yalda could hear the bells from one of the city’s clock towers; she missed a few from noise, or sleep, or inattentiveness, but never so many as to lose track of time. So she knew it was the middle of the morning on their third day when the guards came and took Tullia.
This had to be her hearing with the sergeant. Yalda waited, trying to be patient. Tullia had told her that there was usually a large group of prisoners to be dealt with in each session, and the whole thing could last a bell or two.
By evening, Tullia had not returned. Either they’d set her free, or they’d moved her to a different cell while she arranged the payment of her fine.
Yalda chose to believe that she’d been freed. Tullia hadn’t resisted arrest, and she knew the system well enough to say the right things at her hearing. If the fine had been small enough she might have been released on a promissory note, rather than forced to wait until actual coins had been delivered to the sergeant. Tullia would be at the Solo Club, celebrating her freedom and looking for ways to help her friend.
Yalda blocked out the sad humming of her neighbors; she felt for them, but she didn’t have the strength to involve herself in their plights. She would have her turn before the sergeant soon; she needed to decide what she would say.
When the guards came the next morning, the light of their lamp almost blinded her. She’d planned to sneak a look at the tool they used to disconnect her chain from its place on the wall, but everything was veiled in painful brightness. As they tugged on the chain to draw her out of the cell, she quickly lengthened one of her arms and shortened the other, allowing the force to be borne by solid flesh instead of the loose tube of skin between them that she’d fought to keep empty.
She stumbled up the broad staircase into a corridor filled with searing daylight, then hurried along with slitted eyes, not wishing to drag the chain and provoke her captors. In a room full of prisoners, she was secured to the wall again. Yalda raised her gaze cautiously; there were more than a dozen men and women chained up beside her, most of them with melded limbs. Everyone looked as wretched and afraid as she was.
She felt herself shivering. No friends were permitted here to offer support. Nobody could advise her now, or speak in her defense. All she had to guide her was Tullia’s counsel, which she’d been so vehement in opposing.
The sergeant entered the room—wearing a belt much like his subordinates’, but adorned with no less than four knives—and took his place behind an impressive calmstone desk. An assistant brought in a stack of paper, smelling of fresh dye; for an instant the scent was almost comforting.
As the first case was heard, Yalda tried to concentrate on the procedure and learn what she could. A young man had stolen a loaf from the markets, and then fled from the police. He did not deny the charge.
The sergeant fined him a dozen pieces. “How will you pay this?” he asked.
“My brother might help me,” the man said, his voice soft with shame.
“Give his details to the messenger; you can wait in your cell.” A guard took the man’s chain and led him away.
The next prisoner, another young man, had trespassed in a private garden; he was not accused of stealing anything, but his fine was three times as much as the thief’s.
Everything about the process was humiliating, but Yalda prepared herself to swallow her pride. Tullia had offered to help her find the money for her fine; Daria would probably be willing to loan her a few dozen pieces. She could be out of this place by nightfall if she was suitably humble and penitent. And whatever blame fell upon her for Antonia’s fate, there was nothing to be gained by making trouble for herself. No one was going to rise up against the Council to overthrow the law on runaways because one fat solo argued with the police about an unrelated assault.
When her turn came, a guard unclamped her chain from the wall and ushered her in front of the sergeant’s desk.
“Are you Yalda, daughter of Vito?”
“Yes, sir.” The sound of her father’s name stung; Yalda had no wish to imagine him witnessing any of this.
The sergeant scanned the paper in front of him. “You are charged, firstly, with possession of a substance contrary to the order of nature and the public good. Do you dispute the charge?”
“No, sir.” In the darkness of her cell she’d rehearsed speeches on the insanity of banning a drug that spared the world fatherless children, fantasizing about the power of her impeccable logic to sway even the most hostile audience.