In spacetime, there were six distinct ways you could rotate an object: you could turn it around any of three perpendicular axes in space, or you could boost its velocity in any of the same three directions. These two kinds of rotation were complementary, or “dual” to each other, with the ordinary rotations only affecting coordinates that were untouched by the corresponding boost, and vice versa. This meant that you could rotate something around, say, the x-axis, and speed it up in the same direction, without the two processes interfering.
When Robert had tried applying the Yang-Mills approach to gravity in the obvious way, he’d floundered. It was only when he’d shifted the algebra of rotations into a new, strangely skewed guise that the mathematics had begun to fall into place. Inspired by a trick that particle physicists used to construct fields with left-or right-handed spin, he’d combined every rotation with its own dual multiplied by i, the square root of minus one. The result was a set of rotations in four complex dimensions, rather than the four real ones of ordinary spacetime, but the relationships between them preserved the original algebra.
Demanding that these “self-dual” rotations satisfy Einstein’s equations turned out to be equivalent to ordinary general relativity, but the process leading to a quantum-mechanical version of the theory became dramatically simpler. Robert still had no idea how to interpret this, but as a purely formal trick it worked spectacularly well — and when the mathematics fell into place like that, it had to mean something.
He spent several hours pondering old results, turning them over in his mind’s eye, rechecking and reimagining everything in the hope of forging some new connection. Making no progress, but there’d always been days like that. It was a triumph merely to spend this much time doing what he would have done back in the real world — however mundane, or even frustrating, the same activity might have been in its original setting.
By evening, though, the victory began to seem hollow. He hadn’t lost his wits entirely, but he was frozen, stunted. He might as well have whiled away the hours reciting the base-32 multiplication table in Baudot code, just to prove that he still remembered it.
As the room filled with shadows, his powers of concentration deserted him completely. His hay fever had abated, but he was too tired to think, and in too much pain to sleep. This wasn’t Russia, they couldn’t hold him forever; he simply had to wear them down with his patience. But when, exactly, would they have to let him go? And how much more patient could Quint be, with no pain, no terror, to erode his determination?
The moon rose, casting a patch of light on the far wall; hunched over, he couldn’t see it directly, but it silvered the grey at his feet, and changed his whole sense of the space around him. The cavernous room mocking his confinement reminded him of nights he’d spent lying awake in the dormitory at Sherborne. A public school education did have one great advantage: however miserable you were afterwards, you could always take comfort in the knowledge that life would never be quite as bad again.
“This room smells of mathematics! Go out and fetch a disinfectant spray!” That had been his form-master’s idea of showing what a civilised man he was: contempt for that loathsome subject, the stuff of engineering and other low trades. And as for Robert’s chemistry experiments, like the beautiful colour-changing iodate reaction he’d learnt from Chris’s brother —
Robert felt a familiar ache in the pit of his stomach. Not now. I can’t afford this now. But the whole thing swept over him, unwanted, unbidden. He’d used to meet Chris in the library on Wednesdays; for months, that had been the only time they could spend together. Robert had been fifteen then, Chris a year older. If Chris had been plain, he still would have shone like a creature from another world. No one else in Sherborne had read Eddington on relativity, Hardy on mathematics. No one else’s horizons stretched beyond rugby, sadism, and the dimly satisfying prospect of reading classics at Oxford then vanishing into the maw of the civil service.
They had never touched, never kissed. While half the school had been indulging in passionless sodomy — as a rather literal-minded substitute for the much too difficult task of imagining women — Robert had been too shy even to declare his feelings. Too shy, and too afraid that they might not be reciprocated. It hadn’t mattered. To have a friend like Chris had been enough.
In December of 1929, they’d both sat the exams for Trinity College, Cambridge. Chris had won a scholarship; Robert hadn’t. He’d reconciled himself to their separation, and prepared for one more year at Sherborne without the one person who’d made it bearable. Chris would be following happily in the footsteps of Newton; just thinking of that would be some consolation.
Chris never made it to Cambridge. In February, after six days in agony, he’d died of bovine tuberculosis.
Robert wept silently, angry with himself because he knew that half his wretchedness was just self-pity, exploiting his grief as a disguise. He had to stay honest; once every source of unhappiness in his life melted together and became indistinguishable, he’d be like a cowed animal, with no sense of the past or the future. Ready to do anything to get out of the cage.
If he hadn’t yet reached that point, he was close. It would only take a few more nights like the last one. Drifting off in the hope of a few minutes’ blankness, to find that sleep itself shone a colder light on everything. Drifting off, then waking with a sense of loss so extreme it was like suffocation.
A woman’s voice spoke from the darkness in front of him. “Get off your knees!”
Robert wondered if he was hallucinating. He’d heard no one approach across the creaky floorboards.
The voice said nothing more. Robert rearranged his body so he could look up from the floor. There was a woman he’d never seen before, standing a few feet away.
She’d sounded angry, but as he studied her face in the moonlight through the slits of his swollen eyes, he realised that her anger was directed, not at him, but at his condition. She gazed at him with an expression of horror and outrage, as if she’d chanced upon him being held like this in some respectable neighbour’s basement, rather than an MI6 facility. Maybe she was one of the staff employed in the upkeep of the house, but had no idea what went on here? Surely those people were vetted and supervised, though, and threatened with life imprisonment if they ever set foot outside their prescribed domains.
For one surreal moment, Robert wondered if Quint had sent her to seduce him. It would not have been the strangest thing they’d tried. But she radiated such fierce self assurance — such a sense of confidence that she could speak with the authority of her convictions, and expect to be heeded — that he knew she could never have been chosen for the role. No one in Her Majesty’s government would consider self assurance an attractive quality in a woman.
He said, “Throw me the key, and I’ll show you my Roger Bannister impression.”
She shook her head. “You don’t need a key. Those days are over.”
Robert started with fright. There were no bars between them. But the cage couldn’t have vanished before his eyes; she must have removed it while he’d been lost in his reverie. He’d gone through the whole painful exercise of turning to face her as if he were still confined, without even noticing.
Removed it how?
He wiped his eyes, shivering at the dizzying prospect of freedom. “Who are you?” An agent for the Russians, sent to liberate him from his own side? She’d have to be a zealot, then, or strangely naive, to view his torture with such wide-eyed innocence.
She stepped forward, then reached down and took his hand. “Do you think you can walk?” Her grip was firm, and her skin was cool and dry. She was completely unafraid; she might have been a good Samaritan in a public street helping an old man to his feet after a fall — not an intruder helping a threat to national security break out of therapeutic detention, at the risk of being shot on sight.
“I’m not even sure I can stand.” Robert steeled himself; maybe this woman was a trained assassin, but it would be too much to presume
that if he cried out in pain and brought guards rushing in, she could still extricate him without raising a sweat. “You haven’t answered my question.”
“My name’s Helen.” She smiled and hoisted him to his feet, looking at once like a compassionate child pulling open the jaws of a hunter’s cruel trap, and a very powerful, very intelligent carnivore contemplating its own strength. “I’ve come to change everything.”
Robert said, “Oh, good.”
Robert found that he could hobble; it was painful and undignified, but at least he didn’t have to be carried. Helen led him through the house; lights showed from some of the rooms, but there were no voices, no footsteps save their own, no signs of life at all. When they reached the tradesmen’s entrance she unbolted the door, revealing a moonlit garden.
“Did you kill everyone?” he whispered. He’d made far too much noise to have come this far unmolested. Much as he had reason to despise his captors, mass murder on his behalf was a lot to take in.
Helen cringed. “What a revolting idea! It’s hard to believe sometimes, how uncivilised you are.”
“You mean the British?”
“All of you!”
“I must say, your accent’s rather good.”
“I watched a lot of cinema,” she explained. “Mostly Ealing comedies. You never know how much that will help, though.”
“Quite.”
They crossed the garden, heading for a wooden gate in the hedge. Since murder was strictly for imperialists, Robert could only assume that she’d managed to drug everyone.
The gate was unlocked. Outside the grounds, a cobbled lane ran past the hedge, leading into forest. Robert was barefoot, but the stones weren’t cold, and the slight unevenness of the path was welcome, restoring circulation to the soles of his feet.
As they walked, he took stock of his situation. He was out of captivity, thanks entirely to this woman. Sooner or later he was going to have to confront her agenda.
He said, “I’m not leaving the country.”
Helen murmured assent, as if he’d passed a casual remark about the weather.
“And I’m not going to discuss my work with you.”
“Fine.”
Robert stopped and stared at her. She said, “Put your arm across my shoulders.”
He complied; she was exactly the right height to support him comfortably. He said, “You’re not a Soviet agent, are you?”
Helen was amused. “Is that really what you thought?”
“I’m not all that quick on my feet tonight.”
“No.” They began walking together. Helen said, “There’s a train station about three kilometres away. You can get cleaned up, rest there until morning, and decide where you want to go.”
“Won’t the station be the first place they’ll look?”
“They won’t be looking anywhere for a while.”
The moon was high above the trees. The two of them could not have made a more conspicuous couple: a sensibly dressed, quite striking young woman, supporting a filthy, ragged tramp. If a villager cycled past, the best they could hope for was being mistaken for an alcoholic father and his martyred daughter.
Martyred all right: she moved so efficiently, despite the burden, that any onlooker would assume she’d been doing this for years. Robert tried altering his gait slightly, subtly changing the timing of his steps to see if he could make her falter, but Helen adapted instantly. If she knew she was being tested, though, she kept it to herself.
Finally he said, “What did you do with the cage?”
“I time-reversed it.”
Hairs stood up on the back of his neck. Even assuming that she could do such a thing, it wasn’t at all clear to him how that could have stopped the bars from scattering light and interacting with his body. It should merely have turned electrons into positrons, and killed them both in a shower of gamma rays.
That conjuring trick wasn’t his most pressing concern, though. “I can only think of three places you might have come from,” he said.
Helen nodded, as if she’d put herself in his shoes and catalogued the possibilities. “Rule out one; the other two are both right.”
She was not from an extrasolar planet. Even if her civilisation possessed some means of viewing Ealing comedies from a distance of light years, she was far too sensitive to his specific human concerns.
She was from the future, but not his own.
She was from the future of another Everett branch.
He turned to her. “No paradoxes.”
She smiled, deciphering his shorthand immediately. “That’s right. It’s physically impossible to travel into your own past, unless you’ve made exacting preparations to ensure compatible boundary conditions. That can be achieved, in a controlled laboratory setting — but in the field it would be like trying to balance ten thousand elephants in an inverted pyramid, while the bottom one rode a unicycle: excruciatingly difficult, and entirely pointless.”
Robert was tongue-tied for several seconds, a horde of questions battling for access to his vocal chords. “But how do you travel into the past at all?”
“It will take a while to bring you up to speed completely, but if you want the short answer: you’ve already stumbled on one of the clues. I read your paper in Physical Review, and it’s correct as far as it goes. Quantum gravity involves four complex dimensions, but the only classical solutions — the only geometries that remain in phase under slight perturbations — have curvature that’s either self-dual, or anti-self-dual. Those are the only stationary points of the action, for the complete Lagrangian. And both solutions appear, from the inside, to contain only four real dimensions.
“It’s meaningless to ask which sector we’re in, but we might as well call it self-dual. In that case, the anti-self-dual solutions have an arrow of time running backwards compared to ours.”
“Why?” As he blurted out the question, Robert wondered if he sounded like an impatient child to her. But if she suddenly vanished back into thin air, he’d have far fewer regrets for making a fool of himself this way than if he’d maintained a façade of sophisticated nonchalance.
Helen said, “Ultimately, that’s related to spin. And it’s down to the mass of the neutrino that we can tunnel between sectors. But I’ll need to draw you some diagrams and equations to explain it all properly.”
Robert didn’t press her for more; he had no choice but to trust that she wouldn’t desert him. He staggered on in silence, a wonderful ache of anticipation building in his chest. If someone had put this situation to him hypothetically, he would have piously insisted that he’d prefer to toil on at his own pace. But despite the satisfaction it had given him on the few occasions when he’d made genuine discoveries himself, what mattered in the end was understanding as much as you could, however you could. Better to ransack the past and the future than go through life in a state of wilful ignorance.
“You said you’ve come to change things?”
She nodded. “I can’t predict the future here, of course, but there are pitfalls in my own past that I can help you avoid. In my twentieth century, people discovered things too slowly. Everything changed much too slowly. Between us, I think we can speed things up.”
Robert was silent for a while, contemplating the magnitude of what she was proposing. Then he said, “It’s a pity you didn’t come sooner. In this branch, about twenty years ago —”
Helen cut him off. “I know. We had the same war. The same Holocaust, the same Soviet death toll. But we’ve yet to be able to avert that, anywhere. You can never do anything in just one history — even the most focused intervention happens across a broad ‘ribbon’ of strands. When we try to reach back to the ’30s and ’40s, the ribbon overlaps with its own past to such a degree that all the worst horrors are faits accompli. We can’t shoot any version of Adolf Hitler, because we can’t shrink the ribbon to the point where none of us would be shooting ourselves in the back. All we’ve ever managed are minor interventions, like sending projecti
les back to the Blitz, saving a few lives by deflecting bombs.”
“What, knocking them into the Thames?”
“No, that would have been too risky. We did some modelling, and the safest thing turned out to be diverting them onto big, empty buildings: Westminster Abbey, Saint Paul’s Cathedral.”
The station came into view ahead of them. Helen said, “What do you think? Do you want to head back to Manchester?”
Robert hadn’t given the question much thought. Quint could track him down anywhere, but the more people he had around him, the less vulnerable he’d be. In his house in Wilmslow he’d be there for the taking.
“I still have rooms at Cambridge,” he said tentatively.
“Good idea.”
“What are your own plans?”
Helen turned to him. “I thought I’d stay with you.” She smiled at the expression on his face. “Don’t worry, I’ll give you plenty of privacy. And if people want to make assumptions, let them. You already have a scandalous reputation; you might as well see it branch out in new directions.”
Robert said wryly, “I’m afraid it doesn’t quite work that way. They’d throw us out immediately.”
Helen snorted. “They could try.”
“You may have defeated MI6, but you haven’t dealt with Cambridge porters.” The reality of the situation washed over him anew at the thought of her in his study, writing out the equations for time travel on the blackboard. “Why me? I can appreciate that you’d want to make contact with someone who could understand how you came here — but why not Everett, or Yang, or Feynman? Compared to Feynman, I’m a dilettante.”
Helen said, “Maybe. But you have an equally practical bent, and you’ll learn fast enough.”
There had to be more to it than that: thousands of people would have been capable of absorbing her lessons just as rapidly. “The physics you’ve hinted at — in your past, did I discover all that?”
“No. Your Physical Review paper helped me track you down here, but in my own history that was never published.” There was a flicker of disquiet in her eyes, as if she had far greater disappointments in store on that subject.
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