by Tom DeMarco
“Look, solve your equation for a value of time equal to zero.”
Loren did as he was told, working the math in his head. “OK. So what?”
“So what? What does it mean that time is zero? It has no meaning at all.”
“Well, the equation just implies that that is the time at which there was zero energy used up in the universe, that’s all.”
“Right. And since then, the energy used has been increasing, always increasing. So time zero has a significance. It was the beginning of time, whatever that means. There could be no conceivable time before that.”
“Yes. I’m not sure what it implies, but that’s what the equation says.”
“Energy is being used up by the passage of time. I’m not saying that is impossible. Lots of more surprising things have been discovered. Let’s just say it is a bit unlikely. But suppose it is true? Then we could compute what the present value of t is, just by knowing how much energy had been used. We could figure out exactly how old the universe is.”
“I never thought of that. We don’t know the energy used, of course.”
“No, but we could figure out the maximum amount of energy there could ever have been to be used up. Assume that the universe was originally cram-packed with matter, a solid block of stuff.”
“OK.”
“Let’s say it was made of solid…oh, I don’t know, what would it be?”
“Well, nickel perhaps. The cores of the planets appear to be made of nickel.”
“OK, nickel. If the whole universe were pure nickel, it would have the maximum imaginable energy potential. Compute how much energy that would have supplied if all the mass were transformed. Whatever answer you come up with is certainly more energy than there has ever been in the universe.”
“OK,” Loren picked up the thread of Chan’s logic. “The specific weight of nickel is about 9. So a cubic centimeter of nickel weighs 9 grams, more or less. That’s the mass. The m-c-squared conversion says the theoretical energy equivalent is…” Loren worked out the result. “It’s huge.”
“Right. Now, assume that that much energy has been used up since time zero in every cubic centimeter of the universe. Calculate what the present value of t is, consistent with that much used up energy. That will tell you the maximum conceivable age of the universe.” Chan still hadn’t looked back at the equations. Loren decided that the man probably didn’t need to. He looked back down at the equations and solved for the present value of t.
“Um. It comes to a little more than a trillion seconds.”
“Uh huh. So the universe can’t be more than a trillion seconds old. Let’s see, there are about pi times ten times a million seconds in a year. We divide that into a trillion. You have just proved that the universe can’t possibly be more than thirty thousand years old. Nice work. This is Nobel Prize stuff.”
“But we made your stupid assumption. Suppose the universe were not originally cram-packed with matter…”
“Then you have proved the universe is even younger. If the maximum density of the universe were something close to what it is today, then the whole universe may only be a few minutes old.
“What the hell does that mean?”
“It means the equation is wrong. Go back and try again.”
Loren was crestfallen. He looked for some flaw in Chan’s logic, but there wasn’t any. The equation was wrong. It had been wrong all along. There was nothing to do but start over. “I wouldn’t know where to begin.”
Dr. Chan had picked up the spokeshave and was applying it again very lightly to the right side of the bow. “You begin, of course, from where you are. From your six equations. The fact that they are not completely right doesn’t imply that they are completely wrong.”
Loren was dubious. “I could spend the rest of my life on this.”
Chan shaved on for a while. At last he said, “Look, I will tell you a story. Maybe it will help. It has always helped me. The story is about a physicist, named Planck.”
“Max Planck.”
“Yes. You already know all about Planck as a physicist. What I want to tell you is about how the man is viewed as a mathematician, what we mathematicians think of him. Planck was a marvelous physicist, of course, but something of a humbug as a mathematician.”
“He was?”
“He was. But an ingenious humbug. He was studying…this was back in 1899…he was studying the problem of why a metal glows the way it does when it’s heated up. Everybody conceded that it should glow, but they thought it should glow with a blue light, not red. Newtonian physics predicted that a piece of copper, heated to thirteen hundred degrees ought to give off a blue light. But maddeningly, when they did the experiment, it glowed red. A nice bright cherry red. What sense were they to make of this contradiction? Every other physicist kept asking, What could be going on inside the metal to make it behave this way without violating Newton’s law? But Planck asked a totally different question. He asked, what is the simplest change I could make to Newton’s equation to make it predict a cherry red glow? He wrote down the equation, filling in the frequency of the red glow and just kludged it to make it work. The easiest kludge he could think of was to quantify the energy in little indivisible packets, the so-called coins of energy. Once he had hypothecized the coin, he used the cherry red color to figure out how big the coin would have to be.”
“Planck’s quantum of action.”
“Exactly. He had not one single justification for saying that there was a coin of energy, and no rationale for saying that it ought to be that big. He just worked the equation backwards to make it come out. Of course, the scientific and mathematical communities were outraged. They were all furious that he had backed into this result. Well, not quite all. There was one person who took the result seriously, in spite of its poor credentials.”
“Einstein.”
“Yes. And so, in his second paper of 1905, Einstein used Planck’s coin of energy and the value of Planck’s derived constant to show what was going on in Brownian motion. Planck had found an important truth. He had used a less than pure method to do it, but he found the truth. And that’s why today, Planck, the humbug mathematician, is known as a great physicist.”
“So what does that say to me?”
“Try a little kludge, my friend. Find the second stable state…”
“But I tried that!”
“You were looking for it where you expected it to be. You’ve got to look for it where it is.” Chan gave him a look usually reserved for children and dumb animals.
“Oh. I see. You mean an entirely empirical experiment. I just keep changing the energy of the beam until I finally find some value that is stable.”
“Exactly. And then when you have the value of that energy, plug it back into the equation, and kludge it to make it come out.”
“It’s going to take a long time. The second state could be anywhere.”
“So try a second piece of fraud. Imagine that you have already done the experiment and found that the actual energy value is not 0.00013, but, oh, let’s say, 0.000022. Convince yourself that you have just done that. Convince yourself that 0.000022 is the empirically observed value. Plug that back into the equation, and see what you get, see what change you will have to make to account for the difference.”
“But, that’s too much fraud. Since the actual value when I find it is almost certainly not going to be 0.000022, my explanation of why it had to be 0.000022 will be useless.”
“Inaccurate, but not useless. You might have to change the values of some of your correction factors, but not the concept.”
“I see.” Loren shrugged. It was going to be a lot of work. And all the time spent on it was time taken away from what was most important to him now, planning for the defense against the second attack. He doubted he’d ever find the time to follow through on the suggestion. “Thank you for the advice, Dr. Chan. I’ll go away and think about it.”
“Ah, my dear friend, Dr. Martine. That is the recipe f
or a five-year project. Don’t go away. Sit right down on this platform and plug in the value 0.000022. Kludge up the equation. I will help you, in between shaping the bow. Together we will solve the problem, now, before supper.”
Loren nodded. He sat himself down with his back against Columbia’s topsides. There was a piece of waste planking there that he picked up for a writing surface. Dr. Chan handed him a pencil. Loren turned over the daybook page and wrote down the third equation with t-prime-two equal to 0.000022. Then he sat back to consider what that might mean. It was not a five year project, not even a five-minute project, in fact. After a scant ninety seconds he looked up. “Well, the very same artifice that Planck came up with could be used to explain this.”
“Aha.”
“If we can conceive of time being packetized into little indivisible coins, just like energy is, then the value of t-prime-two just depends on the size of the coin.”
“Aha.”
“That’s why you told me the story, I guess.”
“Maybe.”
“But this is dumb. Coins of time: I have no right to speculate on such a thing. I have no theory, no explanation, no reason to believe it’s true.”
“You will never be a very good humbug, Loren. You are too principled.”
“And the value can’t be 0.000022, anyway. It’s got to be a whole lot smaller. In fact, now that I look at it together with the first equation, I can tell you exactly how big the coin has got to be. Wait…” Loren stared down dumbly at what he had just written, struggling to come to grips with its consequences. “Peter, this is amazing. Look at this!” He jumped to his feet, almost catapulting himself over the scaffold rail. Dr. Chan grabbed him by the back of his shirt to stop him from falling. “Peter, this is stunning. We used pure fraud to come up with the idea of quantum time, but now I can prove to you that it has to be true. I can prove it. It’s not a question of there possibly being coins of time, there have to be coins of time.” He looked up from the paper. “Only I still can’t believe it.”
“Of course you can believe it. There are coins of matter—we call them atoms, coins of energy, of light, of gravity. Why should time be different? Why should time be the only thing that is infinitely divisible?”
Loren stared again at the revised equation. After a moment he was grinning. A short year ago, he would have been thrilled by the stunning nature of the discovery, excited by its effect on the university and the world of physics. He would have been thinking that his reputation was made. Now none of that mattered. Now, all it meant to him was that he knew exactly how to build the ultimate keel.
4
THE ULTIMATE KEEL
Loren returned to the workshop, determined to be “quiet as a mouse,” and not disturb whatever trivial project Barodin and Pease were working on. He would be the ideal tenant laborer. And if they took no note of what he was doing, they would soon have to confront what they had missed. Loren was about to make history. When Baracoa’s boats sailed solidly up to windward with his marvelous keels, he would take all the credit, every bit. So he didn’t say a word to them, just went directly to his bench and started working.
After a few minutes of trying to ignore the furious electric energy he was giving off, Edward and D.D. Pease tossed it in. Edward spoke up in a raised voice: “Well, this has been some productive afternoon, Pease. Who would think we could have finished one hundred and eighty-five projects in so little time? But there’s project number one-eighty-five done and right as rain. Pile it up with the others. What’s next?”
“Let’s see. Oh, yes. There’s project number one-eighty-six. Some sort of a keel we’re going to help Martine with.”
“Oh, I remember that one. Not just a keel, but the ultimate keel. The poor kid is probably sitting over there at his workbench, just waiting for us to pitch in and help.”
“Right,” said Loren. “Pair of dreamers. You guys will be lucky if I let you in on this at all. I’m inclined to keep it for myself.” His unsteady hands spilled hot solder down the edge of the circuit board. “Shit.”
“Let me give you a hand there, young fellow.” Edward nudged him aside and moved in. “Ah, a t-prime-two Effector, if I’m not mistaken, Pease. But with a little twist, this one. It’s got a beefed up input circuit, as though our young apprentice envisions applying a much higher voltage to drive it.”
“Exactly.” Loren let Edward clean up the spilled solder and re-solder the junction. “The second state is not where we predicted it would be. We were miles off. In the second stable state, time doesn’t flow like molasses; it almost doesn’t flow at all. It’s slowed down to less than ten to the minus tenth.”
Edward whistled.
“English only spoken here,” Pease objected. “What, pray tell, is ten to the minus tenth?”
“One ten billionth. Time is slowed to less than a ten billionth of its normal flow. That means a second as perceived inside the beam is equivalent to more than three hundred years, viewed from the outside.”
Barodin was disbelieving. “How could we have been off by so much?”
“Edward, it’s fascinating. Dr. Chan helped me to understand. There are coins of time, just like light or gravity. Think what that means!” He continued to explain as Edward completed the circuit, peppering Loren with questions along the way. In the midst of his discourse, Loren suddenly remembered that they still had no way to supply the voltage required. “We’re going to need something over nine hundred volts to drive this thing. So we can’t use any of our regular power supplies.”
“At nine hundred volts,” Pease observed, “I hope you mean for it to draw next to no current. Otherwise our ultimate keel is going to be a pig for power.”
“A few micro-amps. Maybe less.”
“In that case, we can just charge up a capacitor and let the Effector drain it off. That will give us a few shots, anyway, to test the thing out. For the final units, we can rig a battery powered source with a voltage multiplier.” Pease began laying out a capacitor source on a circuit breadboard. “How much control do you need?”
Loren directed them both as they set up their parts of the system. Pease was done first. “We are going to need a keel to attach this to. How about one of those big pieces of plyboard?”
“Good,” said Loren. “Cut a nine inch circle in the middle of it. The flange of the Effector will clamp right into that.”
“OK.” Pease went to get a drill and a sabre saw. By the time he was done cutting the hole, Loren and Ed were ready to try the beam. Pease crowded up to the workbench to observe. Loren adjusted the input voltage to the level he had calculated from the new equation. He drew a long breath and threw the switch. The beam popped on and stayed on.
“It’s stable. But is it doing anything? Is it locked in its plane?” Loren reached out and pushed tentatively against the unit. It didn’t budge. Then he pushed it perpendicular to the direction of the beam and it moved freely. He lifted the whole unit up on its breadboard. It was free to move in two dimensions, but absolutely locked in the third. “God, this is unearthly,” he said. The device felt as if it were sliding between two vertical planes of glass. He could move it up and down and sideways, but it just would not move at all forward or backward. “Unreal. Let’s attach it to the keel.”
They had to turn the unit off to move it. Or that was the wrong way to think about it, Loren decided. The force they would expect to apply to move it across the room in a few seconds was sufficient to do that. It would move it that distance in a few seconds, but seconds as observed inside the beam. That was the equivalent of several centuries from their perspective. Since they didn’t have centuries to move Effector, it was better to turn it off.
The excitement built up all over again, as they configured the Effector onto the “keel.” Now they knew what was going to happen when they turned it on, but they couldn’t wait to do it. Just as they were nearing completion, Kelly came in.
“Jesus, the adrenaline oozing out of this place is scary. What are you guys u
p to? Dirty movies?”
“Kelly, come see.” Loren was dancing with anticipation. “We built a keel. Not just a keel, but a one hundred percent effective keel. It’s going to make our boats sail like the wind. Rupert Paule will never know what hit him. Turn it on, Edward.”
Ed threw the switch, and Loren demonstrated. “Imagine that this piece of plywood is Columbia’s keel. See, it moves effortlessly through the water in the direction the boat is pointed. But just try to tilt it or to move it sideways. Just try!”
The plywood board was standing unsupported on its edge. Kelly tried to push it over and it didn’t move. Then she dragged it along it length. Of course it moved freely in that direction. She leaned against it with her whole weight, but couldn’t push it sideways at all. Finally she lifted the board in both hands, holding it up about a foot off the floor. It was perfectly stable in one dimension, as though adhering to an invisible wall. She could slide it up and down and along the length of that wall, but simply could not separate it from the wall. Only, there was no wall there, just air. Finally she lifted it up to the level of her face. Holding it in place, she turned her back to it, and leaned her shoulders against it. Then she held her hands out in space. She was standing only on her heels, supported, it seemed, by nothing. “Wow,” she said. “Wow. It’s magic. It’s just magic.” She inched her heels out a little further to accentuate the effect. With a crash, the board slid down its plane and deposited Kelly on her bottom on the floor. She looked up at Loren. “Loren, it’s magic.”
“Not magic at all,” He said peevishly. “It’s a perfectly simple application of physical principle.” He didn’t mention that it was a physical principle that no one had conceived of until a few hours before.
“It’s marvelous though,” Kelly said. She was still leaning against it, seated on the floor. It seemed to delight her that she was supported by a board on its edge and the board was supported by nothing. The delight was all over her face. “Would it work on its side, Loren?” She indicated with her hands an imaginary keel on its side, suspended in the air. “Would it?”
