The Schwarzschild radius for the Sun is 1.2 miles (2 km), and for the Earth 0.4 inches (1 cm) - both buried inaccessibly deep where they can't cause trouble. So it wasn't entirely clear how significant the strange mathematical behaviour was ... or even what it meant.
What would happen to a star that is so dense that it lies inside its own Schwarzschild radius?
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.
How does a black hole develop as time passes? An initial clump of matter shrinks to the Schwarzschild radius, and then continues to shrink until, after a finite time, all the mass has collapsed to a single point, called a singularity. From outside, though, we can't observe the singularity: it lies beyond the `event horizon' at the Schwarzschild radius, which separates the observable region, from which light can escape, and the unobservable region where the light is trapped.
If you were to watch a black hole collapse from outside, you would see the star shrinking towards the Schwarzschild radius, but you'd never see it get there. As it shrinks, its speed of collapse as seen from outside approaches that of light, and relativistic timedilation implies that the entire collapse takes infinitely long when seen by an outside observer. The light from the star would shift deeper and deeper into the red end of the spectrum. The name should be `red hole'.
Black holes are ideal for spacetime engineering. You can cut-andpaste a black hole into any universe that has asymptotically flat regions, such as our own.[1] 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. A technologically advanced society could build black holes.
A black hole doesn't possess CTCs, though, so we haven't achieved time travel. Yet. However, we're getting close. The next step uses the time-reversibility of Einstein's equations: to every solution there corresponds another that is just the same, except that time runs backwards. The time reversal of a black hole is called a white hole. A black hole's event horizon is a barrier from which no particle can escape; a white hole's event horizon is one into which no particle can fall, but from which particles may emerge at any moment. 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.
White holes may seem rather strange. It makes sense for an initial concentration of matter to collapse, if it is dense enough, and thus to form a black hole; but why should the singularity inside a white hole suddenly decide to spew forth a star, having remained unchanged since the dawn of time? Perhaps because time runs backwards inside a white hole, so causality runs from future to past? Let's just agree that white holes are a mathematical possibility, and notice that they too are asymptotically flat. So if you knew how to make one, you could glue it neatly into your own universe, too.
Not only that: you can glue a black hole and a white hole together. Cut them along their event horizons, and paste along these two horizons. The result is a sort of tube. Matter can pass through the tube
[1] This is a mathematician's way of saying that you can put a black hole anywhere you want. (Or, like a gorilla in a Mini, it can go anywhere it wants.)
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 follows a timelike curve, because material particles can indeed traverse it.
Both ends of the tube 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've got a wormhole. The distance through the wormhole is very short, whereas that between the two openings, across normal spacetime, can be as big as you like.
A wormhole is a short cut through the universe. But that's matter-transmission, not time travel. Never mind: we're nearly there.
The key to wormhole time travel is the notorious twin paradox, pointed out by the physicist Paul Langevin in 1911. Recall that in relativity, time passes more slowly the faster you go, and stops altogether at the speed of light. This effect is known as time dilation. We quote from The Science of Discworld: Suppose that Rosencrantz and Guildenstem are born on Earth on the same day. Rosencrantz stays there all his life, while Guildenstem travels away at nearly lightspeed, and then turns round and comes home again. Because of time dilation, only one year (say) has passed for Guildenstern, whereas 40 years have gone by for Rosencrantz. So Guildenstern is now 39 years younger than his twin brother.
It's called a paradox because there seems to be a puzzle: from Guildenstern's frame of reference, it is Rosencrantz who has whizzed off at near-lightspeed. Surely, by the same token, Rosencrantz should be 39 years younger, not Guildenstern? But the apparent symmetry is fallacious. Guildenstern's frame of reference is subject to acceleration and deceleration, especially when he turns round to head for home; Rosencrantz's isn't. In relativity, accelerations make a big difference.
In 1988 Michael Morris, Kip Thorne, and Ulvi Yurtsever realised that combining a wormhole with the twin paradox yields a CTC. The idea is to leave the white end of the wormhole fixed, and to zigzag the black one back and forth at just below the speed of light. As the black end zigzags, 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. At first those lines are almost horizontal, so they are not timelike, and it is not possible for material particles to proceed along them. But as time passes, the line gets closer to the vertical, and eventually it becomes timelike. Once 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. Because the wormhole is a short cut, you can do so in a very short period of time, effectively travelling instantly from the black end to the corresponding white one. This is the same place as your starting point, but in the past.
You've travelled in time.
By waiting, you can close the path into a CTC and end up at the same place and time that you started from. Not back to the future, but forward to the past. The further into the future your starting point is, the further back in time you can travel from that point. But there's one disadvantage of this method: you can never travel back past the time barrier, and that occurs some time after you build the wormholes. No hope of going back to hunt dinosaurs. Or to tread on Cretaceous butterflies.
Could we really make one of these devices? Could we really get through the wormhole?
There are other time machines based on the twin paradox, but all of them are limited by the speed of light. They would work better, and perhaps be easier to build and operate, if you could follow Star Trek and engage your warp drive, travelling faster than light.
But relativity forbids that, right?
Wrong.
Special relativity forbids that. General relativity, it turns out, permits it. The amazing thing is that the way it permits it turns out to be standard SF gobbledegook, invoked by innumerable writers who knew about relativistic limitations but still wanted their starships to travel faster than light. `Relativity forbids matter travelling faster than light,' they would incant, `but it doesn't forbid space travelling faster than light.' Put your starship in a region of space, and leave it stationary relative to that region. No violation of Einstein there. Now move the entire region of space, starship inside, with superluminal (faster-than-light) speed. Bingo!