the previous generation
These are the voyages of the
Starhip Boobyprize,
her continuing mission:
to deplore strange
new worlds,
to shriek at strange new life and new civilizations,
to
cautiously come from where
no organism, herd, flock, swarm, pack, nest,
litter,
brood, gaggle, charm, exaltation, hive, tribe,
collective
consciousness, group mind, or other
single or multiple lifeform(6) will
come from again.
Catpain’s glo, stardate dyslexic.(7) The year is 2366. H.G.Wells’s Time Traveller has unexpectedly found himself on board the Starhip Boobyprize, and heard about Federation heavy engineering. Very heavy: they build black holes. His time machine is irreparably damaged, but the Federation may be able to help. The Time Traveller has been told about Special Relativity, in which the speed of light is constant, and General Relativity, in which gravity is produced by the curvature of space. He knows that in relativity, ‘time machine’ means closed timelike curve’ or CTC, a world-line whose future encounters its own past. An attractive prospect is the twin paradox, in which Rosencrantz stays on earth all his life while Guildenstern travels away at close to the speed of light, and comes back much younger than his twin. ‘The time is out of joint’ but is it unjointed enough to make a CTC? The Federation can make a matter-transmitter by connecting a black hole to a white hole to form a wormhole but matter-transmission isn’t time travel. Is it? The mutant star-goat has been destroyed by tidal stresses, but we have now been trapped between the gas layers of a Lazarus star with all engines out. Situation normal.
"You do realise," Outa-Data said to the Time Traveller, "that people used to think time travel was a theoretical impossibility, a contradiction in terms?"
"You are referring to the hoary old ‘grandfather paradox’?"
"How dare you call my grandfather ‘hoary’ oh, sorry, I forgot I do not have a grandfather."
"I had a hairy grandfather," said the Cat. "Tabby. Looked terrible when he wore blue suede."
"Hoary," said Cleverly Blusher, "means ‘old’. The medical term is ‘geriatric’."
"Hey, my grandaddy’s name was Jerry. Dunno about the hat-trick, but he was old! And hairy! It’s a small univers "
"The idea goes back to Ren Barjavel’s story Le Voyageur Imprudent. You go back in time and kill your grandfather "
"Great," said Kert. "Let’s do it. Get the Rats, give them the weapons, and "
"But because your father is not born, neither are you, so you cannot go back to kill him, so "
"So you don’t, so you are born, so you do, so "
"Quite."
"I only thought of that problem after I made my machine," said the Time Traveller. "I wonder... no, I quite liked the old codger anyway..."
"Don’t even think about it," growled Lieutenant Woof. "Not in my sector of spacetime you don’t."
"If you think about the problem using Quantum Mechanics, you can easily see that the paradox does not exist," said Outa-Data.
"What kind of mechanics?"
"Quantum. New since your day. Quantum Mechanics, the underlying physics of matter, is indeterminate. Many events, such as the decay of a radioactive atom, are random. One way to make this indeterminacy mathematically respectable is the ‘many worlds’ interpretation invented by H.Everett. This view of the universe is familiar to all dedicated trekkies: our world is just one of an infinite family of ‘parallel worlds’ in which every combination of possibilities occurs. In 1991 David Deutsch noted that, thanks to the many worlds interpretation, quantum mechanics involves no obstacles to ‘free will’. Moreover another standard science fiction trope the grandfather paradox ceases to be paradoxical, because grandfather will be (or will have been) killed in a parallel world, not in the original one."
The Time Traveller digested this for a moment. ‘That’s getting me really worried," he said. "If I do get back to my home time, how can I tell whether I’ve moved to a parallel universe?"
"Do not worry," Outa-Data said. "According to the many worlds interpretation, that is what you are doing every time your constituent atoms choose whether or not to change their quantum state."
"And you tell him not to worry?" asked the Cat incredulously.
"You are switching from this universe to a parallel universe one for each possible choice of state." An idea of some sort was brewing in Outa-Data’s brain; he could feel his subconscious trying to tell him something. But the Time Traveller was so eager to find a way home that he couldn’t get enough peace and quiet for it to pop into conscious thought...
"I think we should forget this Quantum Mechanics business," Cleverly said, "and return to a simpler question. Is there a connection between wormholes and time machines?"
Of course! Was that what Outa-Data’s subconscious was trying to tell him? No, he had a strange feeling it involved money...
"Yes, sir, there is," he said. "It was noticed in 1988, when Michael Morris, Kip Thorne, and Ulvi Yurtsever realised that they could combine a wormhole with the twin paradox to get a CTC. I had forgotten it until you asked. The idea is to leave the white end of the wormhole fixed, and to tow the black one away (or zigzag it back and forth) at just below the speed of light."
Figure 9 Turning a wormhole into a Time Machine.
"What a pity Mr Strimmer has left us," said Pickup.
"We’ll just have to take it on trust this time," said Cleverly.
Fig.9 shows how this approach leads to time travel. The white end of the wormhole remains static, and time passes at its normal rate, shown by the numbers. The black end zig-zags to and fro at just less than the speed of light; so time-dilation comes into play, and time passes more slowly for an observer moving with that end. Think about world-lines that join the two wormholes through normal space, so that the time experienced by observers at each end are the same: lines joining dots with the same numbers. At first those lines slope less than 45§, so they are not timelike, and it is not possible for material particles to proceed along them. But at some instant, in this case time 3, the line achieves a 45§ slope. After this ‘time barrier’ is crossed, you can travel from the white end of the wormhole to the black through normal space following a timelike curve. An example of such a world-line runs from point 5 in the white end of the wormhole to point 4 in the black. Once there, you can return through the wormhole, again along a timelike curve; and because this is a short cut you can do so in a very short period of time, effectively travelling instantly from point 4 at the black end to the corresponding point 4 at the white. This is the same place as your starting point, but one year in the past ! You’ve travelled in time. By waiting one year, you can close the CTC and end up at the same place and time that you started from. Notice that the corresponding ‘ends’ of the wormhole are not those with the same t-coordinate in Minkowski space, but those with the same ‘elapsed time’ for an observer that moves with them, as marked by the figures.
"Hey," said the Cat, "I read something about that in Kitten’s Own Young Scientist Magazine. ‘Make your own wormhole in your own home. Take a plastic bin-liner and cut out the bottom. Fix one end, and imagine the other rushing to and fro at just below lightspeed, so that time inside it slows down. When the far end of the bag comes near, walk across to it, arriving at some time in your own past. Climb through it, and you’ll travel back in time. If your imagination is vivid enough, that is.’ Ridiculous. I mean, black plastic, it doesn’t suit my eyes one bit."
"The actual distance you have to travel through ordinary space," said Outa-Data patiently, "need not be huge: it depends on how far the right end of the wormhole has to move on each leg of its zigzag path. In space of more than one dimension it can spiral rather than zigzag, which corresponds to making the black end following a circular orbit at close to lightspeed."
"You could achieve this by setting up a binary pair of black holes, rotating rapidly round a common centre of gravity," Pox pointed out in triumph.
"Naturally. So the further into the future your starting point is, the further back in time you can travel from that point," Outa-Data told the Time Traveller.
"Wonderful! I can wait several years if necessary."
"Unfortunately there is a technical difficulty You can never travel back past the time barrier, and that occurs some time after you build the wormholes. There is no hope of getting back to your home time." The Time Traveller’s face fell. So did Outa-Data’s. He’d finally figured out what his subconscious was trying to tell him, and it did involve money. But it suffered the same fault.
"There is another problem, too," he said. "The Federation’s R&D department is working on it, but they have made only a laboratory prototype. The question is: can one really build such a device? Can one really get through the wormhole? The Federation can build the wormhole all right, and move its ends around around. That is just a matter of creating intense gravitational fields.
"But the problem that bothers me most," Outa-Data continued, "is what I call the ‘catflap effect’. When you move a mass through a wormhole, the hole tends to shut on your tail. See, Dr Blusher, I have developed a sense of humour."
"That is one viewpoint, Mr Outa-Data."
"Oh. Well, it turns out that in order to get through without getting your tail trapped you have to travel faster than light, so that is no good."
"Why?"
"The easiest way to see that is to represent the spacetime geometry using a Penrose map, invented by the twentieth century mathematical physicist Roger Penrose. When a computer draws a map of a planet on a flat sheet of paper it must distort the coordinates for example, lines of longitude may have to become curved. The Penrose map of a spacetime also distorts the coordinates; but it is designed so that light cones do not change they still run at 45§ angles." Fig.10 shows a Penrose map of a wormhole.
Figure 10 Penrose map of a wormhole.
"Any timelike path that starts at the wormhole entrace, such as the wiggly line shown, must run into the future singularity. There is no way to get across to the exit without exceeding the speed of light."
"Which, you have told me, is impossible," said the Time Traveller.
"Federation engineers are hoping to get round round the difficulty by threading the wormhole with exotic matter, exerting enormous negative pressure, like a stretched spring. But in 1991 Matt Visser suggested an alternative geometry for a benign wormhole, and they will test it out once they have located a good source of exotic matter. The idea is to cut two identical cubes in space, and paste their corresponding faces together. Then they will reinforce the edges of the cube with exotic matter."
"It sounds complicated," said the time Traveller.
"That is what engineers do. Make complicated things work. However, there is a more old-fashioned method that avoids the need for exotic matter. And because it does not involve building a wormhole, there is no time-barrier effect. You can go back to any time you want. Depending on what nature has up her sleeve." Lots of money, if we struck lucky, Outa-Data thought.
"I don’t follow you," said the Time Traveller, interrupting the android’s beautiful daydreams.
"I am talking about using a naturally occurring time machine. A rotating black hole. Formed when a rotating star collapses gravitationally. The Schwarzschild solution of Einstein’s equations corresponds to a static black hole, formed by the collapse of a non-rotating star. In 1962 Roy Kerr solved the equations for a rotating black hole, now known as a Kerr black hole."
"Are there not two other kinds of black hole: the Reissner-Nordstro m black hole, which is static but has electric charge, and the Kerr-Newman black hole which rotates and has electric charge?"
"Of course, Pox, everybody knows that. It is almost a miracle that an explicit solution exists and definitely a miracle that Kerr was able to find it. It is extremely complicated and not at all obvious. But it has spectacular consequences.
Figure 11 Cross-sectional structure of a rotating black hole.
"One is that there is no longer a point singularity inside the black hole. Instead, there is a circular ring singularity, in the plane of rotation (Fig.11). In a static black hole, all matter must fall into the singularity; but in a rotating one, it need not. It can either travel above the equatorial plane, or pass through the ring. The event horizon also becomes more complex; in fact it splits into two. Signals or matter than penetrate the outer horizon cannot get back out again; signals or matter emitted by the singularity itself cannot travel past the inner horizon. Further out still, but tangent to the outer horizon at the poles, is the static limit. Outside this, particles can move at will."
"Poor old Will," said the Cat.
Figure 12 Penrose map of a rotating black hole.
"Inside it, they must rotate in the same direction as the black hole, although they can still escape by moving radially. Between the static limit and the outer horizon is the ergosphere. If you fire a projectile into the ergosphere, and split it into two pieces, one being captured by the black hole and one escaping, then you can extract some of the black hole’s rotational energy.
"The most spectacular consequence of all, however, is the Penrose map of a Kerr black hole, shown in Fig.12. The white diamonds represent asymptotically flat regions of spacetime one in our universe, and several others that need not be. The singularity is shown as a system of broken lines, indicating that it is possible to pass through it (going through the ring). Beyond the singularities lie antigravity universes in which distances are negative and matter repels other matter. Any body in this region will be flung away from the singularity to infinite distances. Several legal (that is, not exceeding the speed of light) trajectories are shown as curved paths. They lead through the wormhole to any of its alternative exits. The most spectacular feature of all, however, is that this is only part of the full diagram. This repeats indefinitely in the vertical direction, and provides an infinite number of possible entrances and exits.
"If we used a rotating black hole instead of a wormhole, and towed its entrances and exits around at nearly lightspeed with Federation matter-processing equipment, we would get a much more practical time machine one that you could get through without running into the singularity."
The Time Traveller rubbed his hands together happily. "Then I shall soon be back in my own time. Come, let us prepare the remains of my machine, to accompany me on the return voyage."
"Let me check with the computer," said Pickup. "Blast. There are no rotating black holes within reach."
"But I thought Outa-Data said wormholes could be used as short cuts through spacetime?" said Cleverly.
"Provided they’ve been built. There’s one under construction to the nearest rotating black hole, but the union is on strike and it hasn’t been finished yet."
To Be Continued in Episode 6: Cosmic Sting
Sean M. Carroll, Edward Farhi, and Alan H. Guth, An obstacle to building a time machine, Physical Review Letters 68 (1992) 263-269.
Marcus Chown, Time travel without the paradoxes, New Scientist 28 March 1992, 23.
John Gribbin, In Search of the Edge of Time, Bantam Press, New York 1992.
Michael S. Morris, Kip S. Thorne, and Ulvi Yurtsever, Wormholes, time machines, and the weak energy condition, Physical Review Letters 61 (1988) 1446-1449.
Ian Redmount, Wormholes, time travel, and quantum gravity, New Scientist 28 April 1990, 57-61.
Ian Stewart, The real physics of time travel, Analog 114 (January 1994) 106-130.
(6)It has been pointed out to us that this is organicist. We'll get it right next time.
(7)With apologies to Terry Pratchett.