Timescape

CHAPTER THIRTY ONE

MARKHAM SPREAD HIS WORKING PAPERS OUT ON THE narrow little drop tray the airline provided. He had hours of boring Atlantic crossing ahead, jammed in next to the window. Cathy Wickham’s equations swam before him, tensor indices beckoning to be turned this way and that, a dense notation compacted with promise.

“Lunch, sir,” the professionally blank-faced steward muttered. This echo of politeness was to put a gloss on his casual dumping of a cardboard package on the drop table. Markham pried open the wedged boards. A rain of packets thumped down among his papers. They were the now-universal, easy-for-them modular units of food. He unwrapped one and found it to be the obligatory rubber chicken. He bit in reluctantly. Pasty, sour stuff. The only saving grace in this was the absence of plastic packaging, he thought. The bombing of the Saudi fields several years ago had brought an abrupt end to that, and a return to humble cardboard. The pulpy gray surface of the package recalled his boyhood, before hydrocarbons ruled the world. The humanizing side of paper containers was the simple fact that they accepted the touch of a pen, would carry a message; plastic’s sheen rejected the imprint of its temporary users. Idly, he jotted the new quantum field equations on the lunchbox. The elegant epsilons and deltas made their stately march across the UNITED AIRLINES block letters. He chewed absently. Time passed. Markham saw a way to separate the tensor elements into several reduced equations. With descending strokes he paired off field components. He jotted side calculations to check himself. Other passengers moved in the distance. In a while the five new equations lay aslant on the ribbing of cardboard. Three he suspected were old friends: the Einstein equations, with modifications for quantum effects when the length scale became small enough. These were well known. The other two seemed to imply more. A deeper sense of the quantum effects added a fresh term here, a tangle of tensors there. There seemed no way to reduce the system further. Markham tapped idly at them with his pen, frowning.

“Say, give-a!” the man next to him cried out. Markham peered out his window. An immense cloud, sulfurous yellow and veined with orange, hung ahead of them. “First I’ve seen,” the man said excitedly. Markham wondered if the pilot would fly through it. In seconds the window was hazed by strands of cloud and Markham realized that they were already passing through a lower segment of the yellow mass ahead. Fresh weight tugged him down; the plane tilted to rise.

“Right ahead of us, folks, is one of those clouds we’ve been hearing about. I’m taking us over it, for a better look.”

This explanation seemed transparently false to Markham. Pilots didn’t change altitude for a lark. The cloud seemed gravid and somehow more solid than the fluffy white cumulus around it. Threads of a coiling, dark blue wound through its cap, giving it a domed profile.

Markham murmured something and went back to his papers. He copied the new equations off the cardboard and studied them, trying to screen out the thin, high wail of the engines. An engineer had once told him the new generation of superfast engines screamed at unbearable levels. Rockwell International had had to go into cost overrun to blunt the jagged spikes of sound. Six months had been spent to bury the screech in a reassuring bass blanket, so the warm, ticketed bodies carried inside would remain woozy and complacent in the metallic grasp. Well, it didn’t work for him. He had always been noise-sensitive. He found the earplugs in the flap compartment in front of him and inserted them. A blanket sealed him in. The only remnant of the engine scream was an acoustic tremor that came up through his legs and settled in his teeth.

He spent an hour testing the new equations. They gave sensible solutions to limiting-case problems he knew. Taking the scale length small and neglecting the gravitational effects, he got the standard equations of relativistic particle theory. Einstein’s work emerged easily, with a few fluid strokes of the pen. But when Wickham’s equations were taken face on, with no side-stepping onto familiar ground, they became opaque.

He squinted at the short, stubby notation. If he sliced through this knot of terms here, just dropped them—but no, that wasn’t right. There was more than remorseless crank-turning to be done here. The work had to have the right deft feel, to glide forward on its own momentum. Beyond the logical standards, there were aesthetic questions. New developments in physics always gave you, first, a logical structure that was more elegant. Second, once you understood it, the structure was not only elegant, it was simpler. Third, from the structure came consequences that were more complex than before. The ever-present trap in seeking a new path was to invert the steps. It was hard to express this to a philosopher; there was something in the art of mathematics that eluded you, unless you watched for it. Plato had been a great philosopher, and he had decided he wanted the planets to move in sets of circles, all compounded together to give the observed orbits. But as Ptolemy found out, the laws that were needed to give those stacked circles were horrendously complex. That would mean complex laws leading to simple consequences, the wrong way round. So all of Ptolemy’s labors gave forth a theory that clanked and groaned, crystalline spheres grinding around, sprockets and wheels and rachets whirring in a doomed machine.

On the other hand, Einstein’s theory was logically more elegant than Newton’s. Subtle, but simple. Its consequences were much harder to work out, which was the right way round. Markham scratched his beard absently. If you kept that in mind, you could discard many approaches before you began, knowing they would ultimately fail. There was no choice between beauty and truth, really. You had to wind up with both. In art, elegance was a whore of a word, bent a different way by each generation of critics. In physics, though, there was some fragile lesson to be learned from past millennia. Theories were more elegant if they could be transformed mathematically to other frames, other observers. A theory that remained invariant under the most general transformation was the most deft, the nearest to a universal form. Gell-Mann’s SU(3) symmetry had arrayed particles into universal ranks. The Lorentz group; isospin; the catalogue of properties labeled Strangeness and Color and Charm—they all cooked gauzy Number into concrete Thing. So to proceed beyond Einstein, one should follow the symmetries.

Markham scratched equations across a yellow pad, searching. He had intended to spend this time plotting his tactics with the NSF, but politics was dross compared with the actual doing of science. He tried different approaches, twisting the compacted tensor notation, peering into the maze of mathematics. He had a guiding principle: nature seemed to like equations stated in covariant differential forms. To find the right expressions-He worked out the equations governing tachyons in a flat space-time, doing the exercise as a limited case. He nodded. Here were the familiar quantum-mechanical wave equations, yes. He knew where they led. The tachyons could cause a probability wave to reflect back and forth in time: The equations told how this wave function would shuttle, past to future, future to past, a befuddled commuter. Making a paradox meant the wave had no ending, but instead formed some sort of standing wave pattern, like the rippling patterns around an ocean jetty, shifting their troughs and peaks but always returning, an ordering imposed on the blank face of the churning sea. The only way to resolve the paradox was to step in, break up the pattern, like a ship cutting across the troughs, leaving a swirl of sea behind. The ship was the classical observer. But now Markham added the Wickham terms, making the equations symmetric under interchange of tachyons. He rummaged through his briefcase for the paper by Gott that Cathy had given him. Here: A Time-Symmetric, Matter and Anti-Matter Tachyon Cosmology. Quite a piece of territory to bite off, indeed. But Gott’s solutions were there, luminous on the page. The Wheeler-Feynman forces were there, mixing advanced and retarded tachyon solutions together with non-Euclidean sums. Markham blinked. In his cottony silence he sat very still, eyes racing, imagination leaping ahead to see where the equations would fold and part to yield up fresh effects.

The waves still stood, mutely confused. But there was no role left for the ship, for the classical observer. The old idea in conventional quantum mechanics had been to let the rest of the universe to be the observer, let it force the waves to collapse. In these new tensor terms, though, there was no way to regress, no way to let the universe as a whole be a stable spot from which all things were measured. No, the universe was coupled in firmly. The tachyon field wired each fragment of matter to every other. Hooking more particles into the network only worsened things. The old quantum theorists, from Heisenberg and Bohr on, had let in some metaphysics at this point, Markham remembered. The wave function collapsed and that was the irreducible fact. The probability of getting a certain solution was proportional to the amplitude of that solution inside the total wave, so in the end you got only a statistical weighting of what would come out of an experiment. But with tachyons that dab of metaphysics had to go. The Wickham terms—

Sudden motion caught his eye. A passenger in the next row was clutching at a steward, eyes glassy. His face was laced with pain. A stretched mouth, pale lips, brown teeth. Mottled pink splotched his cheeks. Markham pulled his earplugs. A brittle scream startled him. The steward got the man down on the floor in the middle of the aisle and pinned his frantically clawing hands. “I can’t—can’t—breathe!” A steward murmured something comforting. The man shook with a seizure, eyes rolling. Two stewards carried him past Markham. He noticed an acrid smell coming from the sick man and wrinkled his nose, forcing his glasses upward. The man panted in the enameled light. Markham replaced his plugs.

He settled again into the embalming quiet, conscious only of the reassuring hum of the engines. Without peaks and valleys of sound the world had a stuffed, spongy feel, as though Maxwell’s classical ether were a reality, could be sensed at the fingertips. Markham relaxed for a moment, reflecting on how much he loved this state. Concentration on an intricate problem could loft you into an insulated, fine-grained perspective. There were many things you could see only from a distance. Since childhood he had sought that feeling of slipping free, of being smoothly remote from the compromised churn of the world. He had used his oblique humor to distance people, yes, keep them safely away from the center of where he lived. Even Jan, sometimes. You had to form for yourself a lucid language for the world, to overcome the battering of experience, to replace everyday life’s pain and harshness and wretched dreariness with—no, not with certainty, but with an ignorance you could live with. Deep ignorance, but still a kind that knew its limits. The limits were crucial. Galileo’s blocks gliding across the marble Italian foyers, their slick slide obeying inertia’s steady hand—they were cartoons of the world, really. Aristotle had understood in his gut the awful fact that friction ruled, all things groaned to a stop. That was the world of man. Only the childlike game of infinite planes and smooth bodies, reality unwrinkled, cast a web of consoling order, infinite trajectories, harmonic life. From that cartoon world it was always necessary to slip back, cloaking exhilarating flights in a respectable, deductive style. But that did not mean, when the papers appeared in their disguise of abstracts and Germanic mannerisms, that you had not been to that other place, the place you seldom spoke of.

He paused in the impacted hush, and then went on.

He wondered distantly if his first guess was right: these new Wickham equations allowed no way out of the paradox, because the whole universe was swept into the experiment. The consequence of setting up the standing wave was to send tachyons forward and backward in time, yes, but also to spray them at superlight speeds throughout the entire universe. Within an instant, every piece of matter in the universe learned of the paradox. The whole structure of space-time became woven into one piece, instantly. That was the new element with tachyons; until their discovery, physics assumed that disturbances in the space-time metric had to propagate outward at light speed.

Markham realized he had been hunched forward, scribbling mathematical statements of these ideas. His back stabbed him with small, hot knives. His writing hand protested with a sweet ache. He leaned backward, reclining the seat. Below he saw the slate-gray plain of the sea like a giant blackboard for God’s idle equations. A freighter plowed a wake that curved with the currents, silver in the sun. They were descending toward Dulles International on a gentle long parabola.

Markham smiled with serene fatigue. The problems caught you up and carried you along, unminding currents. Was there any way to resolve the paradox? He knew intuitively that here lay the heart of the physics, the way of showing whether you could reach the past in a rigorous way. Peterson’s laconic bank vault note proved something had happened, but what?

Markham twisted uncomfortably, irked by the narrow, cramped seat. Air travel was getting to be a rich man’s route again, only this time without the perks. Then he fetched his mind back from these passing reminders of the relentlessly real world. The problem was not solved, and time remained.

But is the paradox decidable at all? he thought. The German mathematician Gödel had shown that even simple systems of arithmetic contained things which were true, but unprovable. In fact, you couldn’t even show that arithmetic itself was consistent—that is, didn’t contain paradoxes. Gödel had forced arithmetic into describing itself in its own language. He had trapped it into its own box, deprived it of ever proving itself by reference to things outside itself. And that was for arithmetic, the simplest logical system known! What of the universe itself, with tachyons launching through it, threading the cloth of space-time? How could all the squiggles on all the yellow pads in the world ever trap that vast weave into the old boxes of yes/no, true/false, past/future? Markham relaxed in his brimming warmth. The plane went clunk and tilted earthward.

The point that continued to puzzle him was whether Renfrew needed to send a message at all, to make a paradox. Tachyons were constantly being produced by natural collisions of high-energy particles—that’s how they had been discovered. Why didn’t those natural tachyons produce a paradox somehow? He frowned. The plane nosed further, giving the illusion of hanging over the lip of a pit, legs dangling. Natural tachyons… The answer had to be that it took some minimum impulse to trigger a paradox. Some critical volume of space-time had to be tweaked, and then the disturbance would propagate outward instantaneously, with enough amplitude to matter. You could change the past at will, yes, so long as you didn’t make paradoxes that had large amplitude. Once you exceeded the threshold, the tachyon wave would have a significant impact on the whole universe. But if so, how could you tell that had happened? What was the signature? How did the universe pick a way to resolve the paradox? They knew they had reached the past—Peterson proved that. But what more could happen?

Markham felt a sudden stab of perception. If the universe was a wholly linked system with no mythical classical observer to collapse the wave function, then the wave function did not have to collapse at all. It—

A wrenching thump. Markham looked out in surprise and saw the ground veer suddenly. Ahead were the patient green fields of Maryland. A clump of forest swarmed beneath the wings. In the cabin, a babble of voices. Shouts. A rasping buzz. The forest went whipping by. The trees were sharp, precise, with the clarity of good ideas. He watched them flick past as the airplane became light, airy, a gossamer webbing of metal that fell with him, mute matter tugged by gravity’s curved geometry. Skreeeeeee. The trees were pale rods in the slanting light, each with a ball of green exploding at the top. They rushed by faster and faster and Markham thought of a universe with one wave function, scattering into the new states of being as a paradox formed inside it like the kernel of an idea.—If the wave function did not collapse… Worlds lay ahead of him, and worlds lay behind. There was a sharp crack and he saw suddenly what should have been.