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(5) will come from again.
Captain’s log: stardate Spacehog Day. The Time-Traveller from the era of H.G.Wells now has a soild grasp of Special Relativity. The Cat still wants to know ‘Flat along what?’ Outa-Data has gone on to discuss the Twin Paradox, gravitational lensing, and the definition of ‘time machine’ as ‘closed timelike curve’ or CTC. When I last saw them they were discussing the topology of the universe. The invisible nanofleet has been eaten by a mutant star-goat, which has inexplicably become infatuated with the Forward Photon Torpedo Launcher. Deflector Shields are operating at half power. Situation normal.
"There are many mathematically possible topologies for the universe, but you cannot get to all of them from here," Outa-Data kindly recapped. "However, you can get to some remarkably interesting ones."
"Yes, I remember one episode where we " Kert began.
"Me too," said Pickup.
"Yes, sir. And you, sir. I am sure they will be very interesting stories when you tell them tonight in the holodeck quarter. Now, in classical Newtonian mechanics there is no limit to the speed of a moving object. Particles can escape from an attracting mass, however strong its gravitational field, by moving faster than the appropriate escape velocity. In an article presented to the Royal Society in 1783, John Michell observed that this idea, combined with that of a finite velocity for light, implies that sufficiently massive objects cannot emit light at all because the speed of light will be lower than the escape velocity. In 1796 Pierre Simon de Laplace repeated these observations in his Exposition of the System of the World. Both of them imagined that the universe might be littered with huge bodies "
"That’s my kinda universe," said Kert.
"Pardon, captain?"
"Littered with bodies."
"Huge bodies, Captain Emeritus bigger than stars, but totally dark."
"That is a remarkable idea," said the Time Traveller.
"Indeed. They were both a century ahead of their time. In 1915 Karl Schwarzschild took the first step towards answering the analogous question within the context of General Relativity, when he solved the Einstein equations for the gravitational field around a massive sphere in a vacuum. His solution behaved very strangely at a critical distance from the centre of the sphere, now called the Schwarzschild radius. It is equal to 2GM/c2 where G is the gravitational constant, M the mass of the sphere, and c the speed of light. When it was discovered, its mathematical significance seemed to be that space and time lost their identity in Schwarzschild’s solution, and became meaningless. However, the Schwarzschild radius for the Sun’s mass is 2km, and for the Earth 1 cm buried inaccessibly deep. What would happen to a star that was so dense that it lay inside its own Schwarzschild radius?"
"If it tried anything funny I’d clobber it with a dilithium bomb," said Kert.
"In 1939 Robert Oppenheimer and Hartland Snyder showed that it would collapse under its own gravitational attraction. Indeed a whole portion of spacetime would collapse to form a region from which no matter, not even light, could escape. This was the birth of an exciting new physical concept. In 1967 John Archibald Wheeler coined the term black hole, and the new concept was christened."
The development over time of a static non-rotating black hole is shown in Fig.6, in which space is represented as two-dimensional and time runs vertically from bottom to top. An initial radially symmetric distribution of matter (the shaded circle) shrinks to the Schwarzschild radius, and then continues to shrink until, after a finite time, all the mass has collapsed to a single point, the singularity. From outside, all that can be detected is the event horizon at the Schwarzschild radius, which separates the region from which light can escape from the region that is forever unobservable from outside. Inside the event horizon lurks the black hole.
Figure 6. Formation of a black hole as seen by (a) an observer at the surface of the collapsing mass and (b) an external observer.
Fig.6a is the sequence of events seen by a hypothetical observer on the surface of the star, and the time coordinate t is the one experienced by such an observer. If you were to watch the collapse from outside you would see the star shrinking, towards the Schwarzschild radius, but you’d never see it get there.
"As it shrinks," said Outa-Data, "its speed of collapse as seen from outside approaches that of light, and relativistic time-dilation implies that the entire collapse takes infinitely long when seen by an outside observer, as in Fig.6b. However, you would see the light emitted by the star shifting deeper and deeper into the red end of the spectrum."
"It all depends on your frame of reference," added Pox.
"You mean," said Pickup, "that if an astronaut fell into a black hole he’d be squashed flat in a finite time in his frame of "
"Flat along what?" asked the Cat.
" but if you watched an astronaut falling into a black hole he’d turn redder and redder and seem to go more and more slowly, because the light emitted from him would be red-shifted further and further."
"So a Black Hole is really a Red Fraud?" asked the Cat. "I mean, it looks red even though it’s black?"
"Yes. "
"So why do they call it a black hole? I mean, black goes OK with red, but "
"They call it black because no light gets out," Pickup explained quietly.
"Right. That’s why it looks red? You guys nuts?"
"It is frame-dependent, Cat. In fact, inside a black hole the roles of space and time are reversed. Just as time inexorably increases in the outside world, so space inexorably decreases inside a black hole."
"You mean it collapses down to nothing? That’s crazy," said the Cat. "I don’t believe it."
"What we need," said Renault Pickup, "is a volunteer".
"Hi guys," said Harold Strimmer. "Time for your dandruff check. Don’t brush the lapels. Now, I just wanted to say that I don’t hold that jape with the asteroid against you. All good fun, only sat there on my own for forty years "
* * * * * * *
"Hey, he does turn red," said the Cat. "Neat."
"Because a black hole warps space and time, there is plenty of scope for engineering," Outa-Data said. "Federation engineers have developed a whole battery of techniques, from quantum foam enlargement to improbability calculus. Because the spacetime topology of a black hole is asymptotically flat like Minkowski spacetime at large distances it can be cut-and-pasted into the spacetime of any universe that has reasonably large asymptotically flat regions such as our own. This makes black hole topology physically plausible in our universe. Indeed, the scenario of gravitational collapse makes it even more plausible: you just have to start with a big enough concentration of matter, such as a neutron star or the centre of a galaxy. That is what I meant by heavy engineering. The technology of 2366 can build black holes. We use matter-processors modified neutron stars mostly, with gravitational traps and heavy-duty laser-compressors.
"However, a static black hole does not possess CTCs. The next step is to notice that Einstein’s equations are time-reversible: to every solution there corresponds another that is just the same, except that time runs backwards. The time-reversal of a black hole is a white hole, and it looks like Fig.6 turned upside down. An ordinary event horizon is a barrier from which no particle can escape; a time-reversed horizon is one into which no particle can fall, but from which particles may from time to time be emitted. So, seen from the outside, a white hole would appear as the sudden explosion of a star’s worth of matter, coming from a time-reversed event horizon."
"Why should the singularity inside a white hole suddenly decide to spew forth a star, having remained unchanged since the dawn of time?" protested Sweetly Coi. "Oh, of course, because "
"You have an excellent point, Counsellor. It makes sense for an initial concentration of matter to collapse, if it is dense enough, and thus to form a black hole; but the reverse seems to violate causality. It does not, of course but the cause lies outside our own universe, so we do not see the result coming. Let us just agree that white holes are a mathematical possibility, and notice that they too are asymptotically flat."
"Flat along what?"
"So if you knew how to make one, you could glue it neatly into your own universe. The Federation has just developed an effective method for doing that, based on the uncertainty principle. They use a Heisenberg amplifier to make the position of matter so uncertain that it may well be outside the normal universe altogether. Not only that: they can glue a black hole and a white hole together. They cut them along their event horizons with a cosmotome and sew the edges together with cold dark matter."
Figure 7 A wormhole.
Figure 8. Using a wormhole as a matter-transmitter. (The length of the wormhole is exaggerated in the picture because the picture is drawn in normal spacetime. It can actually be very short, even if the ends are far apart in ‘normal’ spacetime, because distance is intrinsic to the spacetime in the wormhole.)
The result (more accurately, a fixed spacelike section of it) is shown in Fig.7: a sort of tube. Matter can pass through the tube in one direction only: into the black hole and out of the white. It’s a kind of matter-valve. The passage through the valve is achieved by following a timelike curve, because material particles can indeed traverse it.
Because the topology of Fig.7 is asymptotically flat at both ends of the tube, both ends can be glued into any asymptotically flat region of any spacetime. You could glue one end into our universe, and the other end into somebody else’s; or you could glue both ends into ours anywhere you like (except near a concentration of matter). Now you have a wormhole. Federation engineers make the best wormholes in the universe. They are called wormholes because they look like the holes that a maggot bores in an apple. Only here the apple is well, not so much spacetime as everything that is not spacetime.
A wormhole is shown schematically in Fig.8; but you have to remember that the distance through the wormhole is very short, whereas that between the two openings, across normal spacetime, can be as big as you like.
"I see," said Sweetly Coi. "A wormhole is a short cut through the universe."
"But that’s matter-transmission, not time travel," Pox pointed out.
"But it has some connection with time travel?" the time Traveller asked, his fingers shaking.
"Well," Outa-Data said, "that’d be telling as I believe you humans say..."
To Be Continued in Episode 5: Back to the Past
John Gribbin, In Search of the Edge of Time, Bantam Press, New York 1992.
R.Penrose, Singularities and time-asymmetry, in General Relativity: an Einstein Centenary Survey (editors S.W.Hawking and W.Israel), Cambridge University Press, Cambridge 1979, 581-638
Ian Stewart, The real physics of time travel, Analog 114 (January 1994) 106-130.
(5)It has been pointed out to us that this is singularist. We'll get it right next time.