Hidden Variables Read online

  Nessitor (triumphantly): With great precision. We measured the weight shown on the balance at a wide variety of speeds, and from this I have been able to deduce a precise formula between the measured weight, the original weight when the vehicles was at rest, and the speed of movement. It is as follows.

  (Here Professor Nessitor went to the central display screen and sketched on it the controversial formula. It is believed that this was the first time it had ever appeared to public view. In the form that Nessitor used, it reads: (Weight at speed v) = (Rest weight) X (1 - v2/c2) When the formula was exhibited there was a silence, while the others examined its implications.)

  Spottipon (thoughtfully): I think I can follow the significance of most of this. But what is the constant, c, that appears in your equation?

  Nessitor: It is a velocity, a new constant of nature. Since it measures the degree to which an object is lightened when it moves with velocity v, I suggest that the basic constant, c, should be termed the "speed of lightness".

  Spottipon (incredulously): You assert that this holds anywhere on Listwolme? That your formula does not depend on the position where the experiment is conducted? Nessitor: That is indeed my contention. In a series of experiments at many places on the surface, the same result was obtained everywhere, with the same velocity, "c". It is almost four times as fast as our fastest car.

  (There was a long pause, during which Professor Spottipon was seen to be scribbling rapidly on a scribe pad. When he had finished his face bore a look of profound inspiration.)

  Spottipon: Professor Nessitor, the formula you have written has some strange implications. You assert that there is a lightening of weight with speed across the surface. This we might accept, but you have not taken your formula to its logical limit. Do you realize that there must be a speed when the weight vanishes? When v = c, you have a situation where an object does not push at all on the balance! Worse than that, if v exceeds your "speed of lightness" you would calculate a negative weight. If that were true, a car moving at such a speed would fly completely off the surface. You would have created the long-discussed and arguably impossible "flying machine."

  Nessitor (calmly): As Professor Spottipon has observed with his usual profound insight, the speed of lightness is a most fundamental constant. My interpretation is as follows: since it is clearly ridiculous that an object should have negative weight, the formula is trying to tell us something very deep. It is pointing out that there is no way that an object can ever exceed the speed of lightness. The speed that we can deduce from these experiments, c, represents the ultimate limit of speed that can ever be attained.

  (Sensation. The assembled scientists began to talk among themselves, some frankly disbelieving, others pulling forth their scribe pads and writing their own calculations. At last a loud voice was heard above the general hubbub.)

  Voice: Professor Nessitor! Do you have any name for this new theory of yours?

  Nessitor (shouting to be heard): I do. Since the effects depend only on the motion relative to the ground, I suggest the new results should be termed the PRINCIPLE OF RELATIVITY. I think that . . .

  (Professor Nessitor's next comments were unfortunately lost in the general noise of the excited assembly.)

  * * *

  Six months passed before Professor Nessitor appeared again at a meeting of the Academy. In those months, there had been much speculation and heated argument, with calls for more experiments. It was to an expectant but still sceptical audience that the Professor made his second address.

  Nessitor: Distinguished colleagues, last time that I was here there were calls for proof, for some fundamental basis for the formula I presented to you then. It was to answer those calls that I embarked, four months ago, on a new set of experiments with the Tristee Two vehicle. We had installed a new instrument on board our car. It measures distances very accurately, and permits the car's course to be controlled to an absolutely straight line. For it had occurred to me to ask the question, if velocity and weight are so closely linked, could it be that distance itself depends on some unknown factors?

  Spottipon (somewhat irritably): With all due respect, Nessitor, I have no idea what you mean by such a statement. Distance is distance, no matter how fast you traverse it. What could you hope to find? I hoped that you would have repeated the experiments on speed and weight.

  Nessitor: My esteemed colleague, please have patience. Permit me to tell you what happened. We set the Tristee Two to travel a long distance at various speeds. And indeed, we confirmed the speed-weight relation. At the same time, we were measuring the distance travelled. But in performing this experiment we were moving longer linear distances over the surface of Listwolme than any other scientific group had ever done.

  I therefore decided to conduct an experiment. We travelled a long distance in a certain direction, accurately measuring this with our new instrument. Then we made a half turn and proceeded far along this new line, again measuring distance all the way. Finally, we headed straight back to our original starting point, following the hypotenuse of the triangle and measuring this distance also.

  Now, we are all familiar with the Sharog-Paty Theorem that relates the lengths of the sides of a right-angled triangle.

  (Nessitor went to the central display panel and scribed the famous Sharog-Paty relation:

  There was a mutter of comments from behind him.)

  Impatient voice from the audience: Why are you wasting our time with such trivia? This relation is known to every unfledged child!

  Nessitor: Exactly. But it is not what we found from our measurements! On long trips—and we made many such—the Sharog-Paty relation does not hold. The further we went in our movements, the worse the fit between theory and observation.

  After some experiment, I was able to find a formula that expresses the true relation between the distances a, b, and c. It is as follows.

  (Nessitor stepped again to the display panel and wrote the second of his famous relations, in the form: cos(c/R) = cos(a/R) X cos(b/R).

  There was more intense study and excited scribbling in the audience. Professor Spottipon alone did not seem to share in the general stir. His thin face had gone pale, and he seemed to be in the grip of some strong private emotion. At last he rose again to his feet.)

  Spottipon: Professor, old friend and distinguished colleague. What is "R" in your equation?

  Nessitor: It is a new fundamental constant, a distance that I calculate to be about three million paces.

  Spottipon (haltingly): I have trouble saying these words, but they must be said. In some of my own work I have looked at the geometry of other surfaces than the plane. Professor Nessitor, the formula you have written there already occurs in the literature. It is the formula that governs the distance relations for the surface of a sphere. A sphere of radius R.

  Nessitor: I know. I have made a deduction from this—

  Spottipon: I beg you, do not say it!

  Nessitor: I must, although I know its danger. I understand the teachings of our church, that we live on the Great Plain of the World, in God's glorious flatness. At the same time I cannot ignore the evidence of my experiments.

  (The Great Hall had fallen completely silent. One of the recording scribes dropped a scribe pin in his excitement and received quick glares of censure. It was a few seconds before Nessitor felt able to continue. He stood there with head bowed.)

  Nessitor: Colleagues, I must say to you what Professor Spottipon with his great insight realized at once. The distance formula is identical with that for distances on a sphere. My experiments suggest that space is curved. We live not on a plane, but on the surface of an immense sphere.

  (The tension crackled around the hall. The penalty for heresy (smothering in live toads) was known to all. At last Professor Spottipon moved to Nessitor's side and placed one hand on his shoulder.)

  Spottipon: My old friend, you have been overworking. On behalf of all of us, I beg you to take a rest. This "curved space" fancy of yours is
absurd—we would slide down the sides and fall off.

  (The hall rang with relieved laughter.)

  Spottipon: Even if our minds could grasp the concept of a curved space, the teachings of the Church must predominate. Go home, now, and rest until your mind is clearer.

  (Professor Nessitor was helped from the stage by kind hands. He looked dazed.)

  * * *

  For almost a year, the Academy met without Nessitor's presence. There were rumors of new theories, of work conducted at white heat in total seclusion. When news came that he would again attend a meeting, the community buzzed with speculation. Rumors of his heresy had spread. When he again stood before the assembly, representatives of the Church were in the audience.

  Professor Spottipon cast an anxious look at the Churchmen as he made Nessitor's introduction.

  Spottipon: Let me say how pleased we are, Professor Nessitor, to welcome you again to this company. I must add my personal pleasure that you have abandoned the novel but misguided ideas that you presented to us on earlier occasions. Welcome to the Academy!

  Nessitor: (rising to prolonged applause, he looked nervous but determined): Thank you. I am glad to be again before this group, an assembly that has been central to my whole working life. As Professor Spottipon says, I have offered you some new ideas over the past couple of years, ideas without fundamental supporting theory. I am now in a position to offer a new and far more basic approach. Space is curved, and we live on the surface of a sphere! I can now prove it.

  Spottipon: (motioning to other scientists on the stage): Quick, help me to get him out of here before it's too late.

  Nessitor (speaking quickly): The curvature of space is real, and the speed of lightness is real. But the two theories are not independent! The fundamental constants c and R are related to a third one. You know that falling bodies move with a rate of change of speed, g, the "gravitational constant". I can now prove that there is an exact relation, that c2 = g x R. To prove this, consider the motion of a particle around the perimeter of a circle . . .

  (The audience was groaning in dismay. Before Nessitor could speak further, friends were removing him gently but firmly from the stage. But the representatives of the church were already moving forward.)

  * * *

  At his trial, two months later, Professor Nessitor recanted all his heretical views, admitting that the new theories of space and time were deluded and nonsensical. His provisional sentence of toad-smothering was commuted to a revocation of all leaping privileges. He has settled quietly to work at his home, where he is writing a book that will be published only after his death.

  And there were those present at his trial who will tell you that as Nessitor stepped down from the trial box he whispered to himself—so softly that the words may have been imagined rather than heard— "But it is round."

  AFTERWORD: THE NEW PHYSICS.

  Sir Arthur Eddington, who will be quoted elsewhere in this collection, once remarked that one mystery of the universe is the way that everything in it seems to rotate—planets, stars, galaxies, they all do it. Surely Listwolme too would rotate on its axis.

  This is a worrying thought. What would it do to Nessitor's observations? Well, he would have found that his "speed of lightness" was a function of direction; it would be lower if he travelled in the same direction as the planet's spin, higher if he went opposite to it. From this he would have concluded that his universe lacked perfect symmetry—there was a preferred direction to it, the direction of the spin axis of the planet.

  That seems to ruin the analogy between his physics and ours, since our universe appears to exhibit no preferred direction. It is assumed to be isotropic, the same in all directions. Except that . . .

  In 1949, Kurt Gödel published a strange solution of the Einstein field equations of general relativity. It describes a universe in which there is a preferred direction, an "axis of rotation" for the entire cosmos. People usually reject Gödel's model by saying it is "non-physical" and does not represent anything in physical reality. But one other way to look at it is to say that we happen, by accident, to live in a non-rotating universe. Most universes, like most other objects, may rotate—we are the odd man out. Or perhaps, as our observational methods become more and more sensitive, we will conclude that this universe is not perfectly isotropic. Maybe with better instruments we and Nessitor would both find evidence of a slow overall rotation.

  FROM NATURAL CAUSES

  Like droplets of acid, envy had eaten slowly into the soul of John Laker. On the day that the shell of his soul was completely eroded he killed Alan Gifford; and on that day the time of true suffering began.

  The murder occurred in late September but the first taste of acid had come on a perfect June day more than twenty years earlier. Laker saw the world then through a dark haze. Tight bands of pain were closing about his chest as he went round the final bend and on into the straight, clenching his jaws and pushing towards the finishing tape. As he passed it his senses could record only one thing. Alan Gifford, on his right—and half a pace ahead. The friendly arm that held his panting body after the race was Gifford's.

  That scene set the pattern for the school years. Alan Gifford and John Laker, track stars—but always in that order. It didn't help Laker to know his times easily beat those of champions of previous years. Somehow that made it worse. In the single year when Laker swept the board in track events, the sight of Gifford in the crowd, weak from an appendicitis operation but cheering Laker on, turned victory to ashes. Next year Gifford was again an agonizing half stride ahead.

  College brought no relief. John Laker was a good engineering student and he did well, very well. But Gifford was an engineer too, and seemed always to win the top design prize or get extra credit for originality. By the time they left college John Laker was an obsessed and tormented man.

  "What next, John?" asked Gifford on the last day of school.

  "Probably a government job. Aec maybe. That's my strongest suit. How about you?"

  "Still not sure. I've had a good offer from the Bureau of Standards. See you in Washington, by the look of things." He slapped Laker on the back and headed off across the noisy campus.

  In the years that followed the two men ran across each other about every couple of months. Alan Gifford had no idea how closely John Laker followed his career—raises, promotions, new responsibilities. He didn't know there was an unannounced and one-sided competition. Laker resented every success Gifford achieved, was wounded by each sign of his progress. Each year the burn went a little deeper.

  The murder when it came was unplanned, almost an accident. Laker, walking from the bus stop to his house, met Gifford in the street. On an impulse he asked him in for a drink.

  "This your place?" said Gifford in surprise as they turned into the driveway. "I'd no idea you lived in such luxury. How much land do you have, a couple of acres?"

  "A bit more." Laker looked at Gifford, hoping at last to see a sign of envy. There was only simple pleasure at the beauty of the handsome red brick Colonial and well-kept garden.

  They went on into the house. Laker saw their reflection in the full-length mirror by the hall coat stand. Grey-haired, lined with bitterness, he looked the older man by ten years. Envy and grinding work had taken their toll. Gifford went to the living-room window and stood looking out over the smooth lawn and carefully tended flower beds.

  "You do a first-rate job out there, John," he said. "Wish I could get my garden to look like that."

  Laker came over to the window and looked out absently. "That's my gardener. He comes in every day. I don't seem to find the time to do much there myself." He had forgotten the breadth of Gifford's interests. Gifford was an enthusiastic amateur gardener, as he seemed to be an enthusiastic everything else. "What are you doing over this side of town, Alan? I thought you were still living in Arlington."

  Gifford nodded. "I am. Right on the approach path to National Airport, I think. I came over here to see Don Thomson at the Arboret
um. Remember him from the University? He was mad about plants even then. Has the ideal job and spends all day playing about with new varieties of flowers."

  Laker smiled. "Sure, I remember him. All teeth and top-soil." He poured their drinks and shrugged. "Should get over mere myself, but you know how it is—something's only half a mile away, you figure you can see it anytime—so you never do."

  "You should. There's no place like it for plant collections. Don would love to show you round."

  As he spoke, Gifford was strolling slowly round the room, sipping his drink and admiring the furniture and elegant china. He shook his head ruefully. "You've got taste and money. You know, Jean's illness used up most of my reserves."

  His wife had died three years before after a long illness. Gifford had cared for her until the end, giving his days and nights to the struggle. Childless, he lived alone. Laker, driven from within, had never married and also lived by himself. The furnishings of the house had been decided by his married sister, indulging a taste beyond her own means.

  Gifford looked at his watch. "Better begin the trek home. No, I won't have another, I've got to fight the Beltway traffic. But I'd like to take a look at your garden on the way out."

  "Sure. Pity the gardener's not here. He could tell you what everything is a lot better than I can."

  In the big front garden they walked slowly from one flower bed to another in the early evening sunshine. Gifford kept up a running commentary.

  "That's going to be a great show of chrysanthemums in another couple of months. These dahlias are show standard. Ah, now. Here's the sign of a real professional." He stopped before a flower bed set back about fifteen yards from the front double gates. "He's double digging there. Must be two feet deep, that trench. I always swear I'll do that but I never get round to it. It makes all the difference to the soil."

  As they approached the hot-house Laker felt a lump in his chest. When they paused before a tray of seedlings the truth hit him like a hammer blow. He had bought the house to compete with Gifford, to show the world he was richer, more successful. But Gifford was finding more pleasure there in one evening than Laker had ever found.