Spin Vector Symmetry

"Aren't you successful now?"

"Yes, but Oakshott Engineering is a small firm and I'm thinking I could do more with a bigger enterprise. It's moot now, under the circumstances, but I was planning to leave Oakshott Engineering."

"That sounds like heresy to me," Danielle said, "but tell me anyway."

"I've been five years with Oakshott and, though I love my job, I want a greater challenge. I've applied for a position with the Beltway consortium, the company that's going to link up all the hyperspace beacons in a thirty to fifty light-year radius of Earth."

"It's an impressive project," Danielle agreed, "but it's more technology than science. It's just a big database and an algorithm to line up the hyperspace beacons and calculate optimum transit times between them. The biggest part will be the computing."

"That was the original project. The new version hasn't been published on the Physics Web yet, but it's much more ambitious. There's cutting-edge astrophysics to be done. That's if the proposed technology can be invented. You'll see. Or you would have. Anyway, I might have become a spokesman for the Beltway consortium. I could have been the face of the Beltway Hyperspace System."

"No despairing, Ellen. You'll yet be the face - and the gorgeous body - of the Beltway Hyperspace System. And if you marry Malcolm and get pregnant, then it'll be even better for you. The public love a pregnant woman."

******

After more heaving and moving the dead mining machine inch-by-tortuous-inch from the wall, the riggers made a big-enough gap for a man to push through.

Again they shone their torches and shouted into the black tunnel.

It was an anxious time. Gil went first, scanning his torch left and right, stopping at any likely-looking pile of rubble, hoping not to find a body. They stopped every few yards to shout: "Hello! Lads! Can you hear us?" and listened a minute for an answer.

Ten yards into the tunnel, at last, they heard something. It was a voice, croaky and weak, as if speaking with a mouthful of dust. But it was definitely a man's voice.

Rigging team B cheered. The trapped men were alive! They were about 100 feet away at the other end of the shaft, where they had tried for hours to shift the fallen rubble by hand. But they were weak, having used up all the water and oxygen they carried. Lying down, they gasped for breath in the depleted atmosphere of the shaft.

The whole team ran to them, bringing oxygen canisters and water.

"Good boyos," Geraint said to the grateful men. "We'll get you out now."

They told the men to sip the water and not glug it straight down. The trapped men rested a few minutes, sharing the oxygen, recovering their strength. Then, exhausted and dehydrated, but very much alive, they were helped out of the tunnel.

"You boyos take these boyos to the ship," Geraint said to his team. "I'm going to check on Rob. Bring us back some more oxygen and another fuel cell. Be snappy, look you."

******

Ellen changed the subject again.

"What do you like to think about?" she asked.

"Physics. I'm a geek and proud of it."

"Who are your heroes?"

"The usual ones: Galileo, Newton, Maxwell, Einstein, Feynman (the Caltech connection, you see). Plus those who formulated quantum gravitation; and those who made hyperspace travel possible, though the theory's a big fudge that we use only because it works."

"They're all men. Tell me about a woman who inspires you."

"Have you heard of Emmy Noether?"

"I remember learning about the Noether symmetries when I did physics at university, but I've forgotten the details. Remind me who she was."

"Emmy Noether was one of the best mathematicians of about five centuries ago, but she worked without a salary and wasn't allowed to take up the academic positions she earned in her native country. She was a victim of horrible discrimination as a woman, which harmed her career and might have harmed the progress of science as well. Even worse, because of her religion, she had to flee Germany to save her life.

"Noether's Theorem says that conservation laws are associated with symmetries. For instance, if a physical system behaves the same way when it's rotated about an axis, then its angular momentum is conserved. There are other symmetries for conservation of parity, entropy and energy, and so on."

"I remember now. I learned about Noether's Theorem in relation to the spin quantum number, but I never understood quantum spin."

"It's confusing because spin in quantum theory has both a real part (the mechanical spin of a beam of electrons in a vacuum tube delivers a measurable torque to a blade) and a peculiar quantum part, in which spin is intrinsic and neither speeds up nor slows down. It's easier to think of it mathematically. If you measure the direction of an electron's spin in a magnetic field, then the vector component of the angular momentum is quantised and can have only values relating to the distribution of energy in the field."

"Yes, that makes it completely clear. Thanks for enlightening me."

"Sorry, Ellen. I could do better if I had a computer tab to show you the maths."

"Don't worry. I wouldn't understand it even if you did what some of those crazy Chinese kids do and implanted a chip in my brain to download directly from the Physics Web."

Danielle laughed at that image, then she stopped dead.

"Oh, my God! How could I have forgotten? Spin vector symmetry! We can separate mechanical spin from quantum spin. But we can also separate spin from other properties in classical mechanics ..."

Danielle stopped talking and stared straight ahead without blinking.

Surprised, Ellen waited a few seconds but Danielle made no sound. Ellen waved her hand in front of Danielle's face. There was no response. Her eyes did not flicker.

Ellen checked her pulse. It was strong and regular. There seemed nothing physically the matter with her, so Ellen sat back in the chair and waited patiently for Danielle to snap out of her trance.

In the silence, just at the edge of her senses, Ellen heard a sort of hissing bubbling sound. The noise seemed to be coming from the ceiling, but it was too muffled and quiet for her to be sure. It was reassuring none the less because it sounded man-made.

Five minutes later, Danielle continued talking as if there had been no hiatus.

"... which means we can turn over the mass without turning over the angular momentum."

"Do you often do that?"

"Do what?"

"Go silent for five minutes."

"When I'm concentrating .... Give me your lipstick."

"Is this really the time for make up?"

"I need to write something down and there are no pens here or working computer tabs. Come on, it's in the belt pocket where your laser penknife should be."

"That's where I put my mascara," Ellen said with a smile. She took her lipstick from the spanner pocket.

"I use the spanner pocket for chocolate," Danielle said taking the lipstick. She pulled off the lid and looked at its deep crimson-red colour.

"What shade do you call this: Courtesan?"

"Temptress," Ellen admitted.

"I should have known," Danielle said with a sad smile and a shake of her head.

"Why, Doctor Goldrick," Ellen exclaimed. "I do believe you're a prude."

"I have good taste is all," Danielle muttered primly as she wrote equations on the wall behind the instrument panel. "Check me as I go, please."

Ellen could follow it quite easily. The equations were all classical mechanics, though with an odd twist. There were calculations of mass, torque, the law of conservation of angular momentum, angles of incidence and calculations of force, mass and time. The final series of equations were unusual.

"What's this?" Ellen asked.

"This is how we're going to save Carina Sunspark."

"How?"

"We're going to turn over the asteroid, like you suggested, so that its magnetic poles are reversed and it should be pushed away from Mu Carinae instead of attracted to it. But your plan was to do it the hard way, and we're going to do it the easy way."

"What do you mean?"

"Your plan of turning MC10 over by pushing it at its poles is the hard way because it turns over both the asteroid's mass and its spin. My easy way will turn over just the mass of the asteroid and leave it spinning in the same direction. That requires much less force."

"I don't get it."

"Do you know what a tippe top is?"

"The child's toy? Yes. It's shaped like a closed-cup mushroom, with a long stalk. You spin it on its mushroom head and when it slows and begins to tip, it turns over and spins on its stalk. What of it?"

"The tippe top spins in the same direction on its stalk as it did on its head."

"I remember learning that. You mean the tippe top turns over its mass but not its angular momentum. How can its mass and its angular momentum be separated?"

"Because they're two different kinds of inertia."

"Go on. Explain from the beginning."

"I was six when my brother gave me a tippe top. He saw I was more interested in playing with his toy trains and aeroplanes than with my dolls, so he encouraged me. That's when I fell in love with physics.

"I played with the tippe top for hours and quickly noticed something interesting. I drew an arrow on the side of the mushroom head, to show which way it was spinning. It confirmed what I thought. If I spun the tippe top clockwise on its head then, when it turned over, it still spun clockwise on its shaft. The arrow was pointing in the backward direction. But there was no point when the tippe top stopped spinning one way and began spinning in the reverse direction.

"Later, when I was ten, after I was bumped up a couple of years at school, I started studying physics properly. I remembered the tippe top and did a demonstration for my class. I vandalised one of my brother's footballs. I put plaster of Paris on its poles, painted it white and drew an arrow on its equator. When I set the ball spinning on an infant's paddling pool full of water, I hit it with a stick to turn it over. Sure enough, it still spun the same way, although it was upside down. You understand what that means?"

"I think so. Hitting the football with a stick is the same kind of impulse the tippe top receives from the friction when it falls over and hits its edge against the table."

"Exactly. The tippe top gets a strong jolt from the table, which makes it rotate about a second axis at a right angle to the horizontal spin, turning it over. We just need to do the same thing with the asteroid. We need to give it a good jolt on the side."

"How?"

"You know the military cruiser that's coming to help the rescue effort?"

"Oh, God! Don't tell me."

"We need it to shoot a missile at us."

******

It took Rob two hours to drill through the rock to the control room with the laser beam, stopping every so often to measure how far he had drilled. When he was very near to 120 feet, he fed a comms device on a lead into the hole. He heard nothing, so he drilled some more, anxious that he might be too late.

Another few inches and another few minutes to wait for the rock to cool down. He fed the comms device in again but still heard no sound. Rob knew he might not be close enough, but he began to worry more seriously.

When Geraint returned to join him, he said:

"We found the boyos, Rob. They're all right. Our lads are taking them to the ship. They'll be back in an hour or less. Are you through yet? How are they?"

"I've drilled 120 feet, less a couple of inches. I'm going slowly because I must be nearly there. I'm worried. It's been quiet every time I've stopped to listen."

"No despairing. That's our Princess down there, look you. Drill the last few inches and make sure."

******

Having explained her theory to Ellen, who checked over the equations and agreed they were right, Danielle said:

"We need to talk to the boys now, otherwise our hard work will be vaporised, along with the asteroid and us."

"We may be in luck," Ellen said. "When you were in your trance, I thought I heard a bubbling sound. It may have been a laser drill."

"That's them," Danielle said confidently, never doubting for a moment that the lads would come to her rescue. "They must be even closer now. Shall we make some noise?"

Still in their underwear, they gathered what solid objects they could find and climbed on the instrument desk to batter on the ceiling, shouting as loudly as they could, not caring if their exertions used up the last of the precious air.

*****

Rob applied the laser drill on a low setting and drilled another two inches. He stopped a last time to let down the comms cable, hoping to see or hear something.

The commotion Ellen and Danielle made was audible even before the microphone reached the final half-inch of rock left to drill.

"I hear them!" Rob exclaimed with joy.

He listened a few seconds more.

"It sounds like they're fighting."

"Aye," Geraint said, meaning either "No comment" or "I'm not surprised."

Rob pulled up the device and turned on the laser to drill through the last half-inch of rock. The molten rock bubbled up on the ceiling above Danielle's equations, raising a mini volcano. The women got out of the way as the laser beam broke through the steel roof and scorched a line on the floor.

Rob turned off the drill. Two minutes later, the thin cable of the comms device, with a camera, light, speaker and microphone packed into its bulbous head, poked through the hole in the control room ceiling.

When there was a foot of cable, Danielle grabbed hold of the end and pointed it at the equations she wrote on the wall with Ellen's lipstick, talking rapidly.

"Geraint, can you hear me? Can you see what I'm pointing at?"

"Yes, Princess. Equations," the miniature speaker replied in scratchy tones.

"Good. Record them and send them with a message to the military cruiser. We want the cruiser to fire an unarmed one-hundred-tonne missile at us at 500,000 metres per second, to hit the asteroid at a tangent dead north at any point 62.5 degrees south of the asteroid's equator. If it works, it'll turn over the asteroid, reversing its poles, pushing us away from the engineering station."

"I've got the recording, Princess. Are you all right?"

He politely did not ask why they were half-naked.

"We're fine," Danielle said. "Pipe us down some oxygen, please, and then get off the asteroid."

"No, Princess. We're staying here."

"You're not. When the missile strikes, we'll be all right down here, where angular velocity is least, but anyone in a tunnel is in danger. Ellen, tell them to go."

"Geraint, we need you boys to be safe so you can come back and dig us out. Please do as Danielle asks."

"Yes, ma'am."

"How are my boys?"

"They're fine, ma'am. We got them out. I sent them and the miners to our ship."

"Thank you," Ellen said quietly.

4Impact

Accelerating with full power on its ion drive, armed missile cruiser, Achilles, was the Anglosphere military's fastest spaceship. With a complement of 80 men and its own hyperdrive motor for free-standing hyperspace jumps, it was armed with two dozen Arrows, the most powerful nuclear missiles in the galaxy.

The Achilles left Earthstation 7 more than a day before and had so far made two leaps through hyperspace beacons and one free-standing hyperspace jump, to cut a corner off its route. Now it was heading at its top acceleration of four-times Earth's gravitation to a hyperspace beacon that would transport it the last fifty light-years to the Earthside beacon at Mu Carinae.

4g acceleration was uncomfortable. It took a toll on the men's muscles and bones, making even the fittest men walk ponderously, like deep-sea divers.

But current speed was already 330,000 metres per second, and there would be relief in about half-an-hour when they were at 400,000 metres per second, which was top speed for the beacon. Then they could cruise weightlessly the last ten minutes or so.

Despite the strain on muscles and lifting equipment, the armoury sergeant prepared an Arrow with an armed nuclear payload sufficient to vaporise asteroid-mine MC10. He waited for the order to launch.

Lieutenant Martin Faris, in charge of telemetry, was at his navicomms console on the ship's bridge when he received the comms from Geraint. He watched the video and set out the equations for himself on his computer tab, checking over the conclusions. He sent instructions to the armoury and took the calculation to Vernon Rhodes, the captain of the Achilles.

There was time. There was still twenty minutes before they had to launch the Arrow.

Lieutenant Faris knocked on the door of the captain's cabin and walked in.

"Sir," he said, "we've received this request via a survey vessel from a Doctor Danielle Goldrick, who is trapped with another woman in the control room of the asteroid-mine, MC10, in the Mu Carinae system."

Captain Rhodes looked at the message, noting the lieutenant's conclusions.

"Our orders are to destroy MC10 when it is more than a million miles from Carina Sunspark," Vernon Rhodes said. "How long until then?"

"Less than an hour, sir. We will have to launch the Arrow through the beacon."

"I see. Give me the figures, Martin."

"An Arrow can accelerate at 12g. It'll reach maximum speed for the beacon in eight minutes, cruise for four more minutes, pass through the beacon, and hit the asteroid-mine another twenty minutes later."

"With what danger?"

"Besides killing anyone on MC10? The danger is that the explosion might not be big enough to vaporise the asteroid but will leave rocks and shards on the same path to smash into the engineering station."

"How serious is the danger?"

"Not very, sir. We're good at damage."

"And the two women on it."

"We can fulfil our orders, sir."

"Indeed. Give me the numbers for the unarmed missile that Doctor Goldrick wants us to launch."

"She wants a 100-tonne payload to hit the asteroid-mine at 500,000 metres per second."

"Can we do it?"

"Almost, sir. An Arrow is 80 tonnes without a nuclear payload. We have enough dense elements (gold, lead, osmium and depleted uranium) in our stores to pack an Arrow out to 110 tonnes. I estimate it will use fifteen tonnes of fuel to hit MC10 at that speed."

"So you need another five tonnes of mass?"

"Yes, sir."

"Is the Arrow waterproof inside?"

"Yes, sir. Good idea. I'm sure there's five cubic metres of space in an Arrow."

"And you've already loaded a missile with all our spare heavy metals, haven't you, Martin?"

"Yes, sir, I gave the order as soon as I read the request, just to be prepared."

"Very good. What complications are there if we launch now?"

"At the top speed requested by Doctor Goldrick, it will take the Arrow forty-two minutes from the beacon to MC10. So we won't know whether or not it has done its job of diverting the asteroid out of the path of Sunspark until it's too late to launch a live missile and blow up MC10."

"Which our current orders require us to do. Hmm. It's a risk. We might fail and allow the men on the engineering station to be killed. If we launch a live missile, we'll certainly kill two women in the asteroid-mine."

"Yes, sir."

"I need a moment to decide. Meanwhile, prepare Doctor Goldrick's Arrow."

"Yes, sir."

Martin gave his orders to the armoury sergeant, who had a hose brought and began to fill the empty space inside the 40-foot missile with five tonnes of water.

The Arrow was a cylinder with a payload in the front quarter of its length and a tube filled with solid rocket fuel for the rest, with flash lasers and a directional exhaust at the rear.

The fuel was a compressed explosive material that was detonated by the flash lasers. The detonation converted the solid fuel, layer by microscopic layer, into an ionised plasma, expanding its volume exponentially, forcing a stream of hot particles out of the exhaust to give a blistering rate of acceleration.