Science of Discworld III

by Terry Pratchett

  Could we really make one of these devices? Could we really get through the wormhole?

  In 1966 Robert Geroch discovered a theoretical way to warp spacetime, smoothly, without tearing it, to create a wormhole. There’s a snag, though: at one stage in the construction, time becomes so twisted that the wormhole turns into a temporary time machine, and equipment from late in the construction gets carried back to the beginning. The builders’ tools might vanish into the past, just as they thought they had finished. Still, with the right work schedule, that might not matter. Perhaps a technologically advanced civilisation could build black and white holes, and move them around, by creating intense gravitational fields.

  But building a wormhole is not the only obstacle. Keeping it open is another. The main trouble is the ‘catflap effect’: when you move a mass through a wormhole, the hole tends to shut on your tail. It turns out that in order to get through without getting your tail trapped, you have to travel faster than light, so that’s no good. Any timelike path that starts at the wormhole entrance must run into the future singularity. There’s no way to get across to the exit without exceeding the speed of light.

  The traditional way round this difficulty is to thread the wormhole with ‘exotic’ matter, which exerts enormous negative pressure like a stretched spring. It is a form of negative energy, and is thus different from antimatter, which has positive energy. In quantum mechanics, a vacuum is not empty – it is a turbulent sea of particles, coming into being and disappearing again. Zero energy includes all these fluctuations, so you can get negative energy if you can calm the waves. One way to do this is the ‘Casimir effect’, discovered in 1948: if two metal plates are held very close together, then in between you find a negative energy state. This effect has been observed in experiments, but it’s very weak. To get enough negative energy, you need galaxy-sized plates. Rigid ones, to maintain the gap.

  Another possibility is a magnetic wormhole. In 1907 the geometer Tullio Levi-Civita proved that in general relativity a magnetic field can warp space. Magnetism has energy, energy is equivalent to mass, and mass is spatial curvature. Moreover, he found an exact mathematical solution to Einstein’s field equations which he called ‘magnetic gravity’. The trouble was, this effect could only be observed using a magnetic field one quintillion times as large as anything that could be obtained in a laboratory. The idea languished until 1995, when Claudio Maccone realised that Levi-Civita had actually come up with a magnetic wormhole. The stronger the magnetic field, the more tightly curved the wormhole mouth is. A wormhole whose magnetic field was the strength you can get in a laboratory would be enormous – about 150 light-years across. And you’d need laboratories everywhere along it. It’s making a small wormhole that needs a gigantic magnetic field. Maccone suggested that the surface of a neutron star, where very strong magnetic fields can occur, might be a good place to look for magnetic wormholes. Why bother? Because a magnetic wormhole can stay open without any need for exotic matter.

  A better solution, though, might be to employ a rotating black hole, which has a ring singularity, not a point one. Passers-by can go through the ring and miss the singularity. The mathematics of Einstein’s equations tells us that a rotating black hole connects to infinitely many different regions of spacetime. One must be in our universe (assuming that we built the rotating black hole in our universe), but the others need not be. Beyond the ring singularity lie antigravity universes in which distances are negative and matter repels other matter. There are legal (slower-than-light) paths through the wormhole to any of its alternative exits. So, if we use a rotating black hole instead of a wormhole, and if we can find a way to tow its entrances and exits around at nearly lightspeed, we’ll get a much more practical time machine – one that we can get through without running into the singularity.

  *

  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!

  Ha-ha, most amusing. Except …

  In the context of general relativity, that’s exactly what Miguel Alcubierre Moya came up with in 1994. He proved that there exist solutions of Einstein’s field equations involving a local ‘warping’ of spacetime to form a mobile bubble. Space contracts ahead of the bubble and expands behind it. Put a starship inside the bubble, and it can ‘surf’ a gravitational wave, cocooned inside a static shell of local spacetime. The speed of the starship relative to the bubble is zero. Only the bubble’s boundary moves, and that’s just empty space.

  The SF writers were right. There is no relativistic limit to the speed with which space can move.

  Warp drives have the same drawback as wormholes. You need exotic matter to create the gravitational repulsion needed to distort spacetime in this unusual way. Other schemes for warp drives have been proposed, which allegedly overcome this obstacle, but they have their own drawbacks. Sergei Krasnikov noticed one awkward feature of Alcubierre’s warp drive: the inside of the bubble becomes causally disconnected from the front edge. The starship’s captain, inside the bubble, can’t steer it, and she can’t even turn it on or off. He proposed a different method, a ‘superluminal highway’. On the outward trip, the starship travels below lightspeed and leaves a tube of distorted spacetime behind it. On the way back, it travels faster than light along the tube. The superluminal highway also needs negative energy; in fact, Ken Olum and others have proved that any type of warp drive does.

  There are limits to the lifetime of any given amount of negative energy. For wormholes and warp drives these limits imply that such structures must either be very small, or else the region of negative energy must be extremely thin. For example, a wormhole whose mouth is three feet (lm) across must confine its negative energy to a band whose thickness is one millionth of the diameter of a proton. The total negative energy required would be equivalent to the total output (in positive energy) of 10 billion stars for one year. If the mouth were one light year across, then the thickness of the negative energy band would still be smaller than a proton, and now the negative energy requirement would be that of 10 quadrillion stars.

  Warp drives, if anything, are worse. To travel at 10 times lightspeed (a mere Star Trek Warp Factor 2) the thickness of the bubble’s wall must be 10-32 metres. If the starship is 200 yards (200m) long, the energy required to make the bubble has to be 10 billion times the mass of the known universe.

  Engage.

  Roundworld narrativium can sometimes be documented. When Ronald Mallett was ten years old his 33-year-old father died of heart failure, brought on by drinking and smoking. ‘It completely devastated me,’ he is reported to have said.2 Soon after, he read Wells’s The Time Machine. And he reasoned that ‘If I could build a time machine, I might be able to warn him about what was going to happen.’

  The childish idea faded, but the interest in time travel did not. As an adult, Mallett invented an entirely new type of time machine, one that uses bent light.

  Morris and Thorne bent space to make a wormhole using matter. Mass is curved space. Levi-Civita bent space using magnetism. Magnetism has energy, energy is (so Einstein tells us) mass. Mallett prefers to bend space using light. Light, too, ha
s energy. So it can act like mass. In 2000, he published a paper on the deformation of space by a circular beam of light. Then it hit him. If you can deform space, you ought to be able to deform time too. And his calculations showed that a ring of light could create a ring of time – a CTC.

  With a Mallett bent-light time machine, you can walk into your past. A time traveller makes his or her way into the closed loop of light, space, and time. Walking round the loop has the same effect as moving backwards in time. The more times he walks round the loop, the further back he goes, tracing out a helical world-line. When he has gone sufficiently far into his past, he exits the loop. Easy.

  Yes, but … we’ve been here before. It takes huge amounts of energy to make a circular beam of light.

  That’s true … unless you can slow the light down. A ring of really slow light, Discworld-speed, like that of sound on Roundworld, is much easier to make. The reason is that as light slows down, it gains inertia. This gives it more energy, and the warping effect is far greater for less effort on the part of the builder.

  Relativity tells us that the speed of light is constant – in a vacuum. In other media, light slows down; this is why glass refracts light, for example. In the right medium, light can be slowed to walking pace, or even stopped altogether. Experiments by Lene Hau demonstrated this effect in 2001, using a medium known as a Bose-Einstein condensate. This is a curious, degenerate form of matter, occurring at temperatures near absolute zero; it consist of lots of atoms in exactly the same quantum state, forming a ‘superfluid’ with zero viscosity.

  So maybe Wells’s time traveller could have included some refrigeration equipment and a laser in his machine. But a Mallett bent-light time machine suffers from the same limitation as a wormhole one. You can’t travel back to any time before the machine was constructed.

  Wells was probably right to eliminate that encounter with a giant hippopotamus.

  These are purely relativistic time machines, but the universe has quantum features too, and these should be taken into account. The search for a unification of relativity and quantum theory – respectably known as ‘quantum gravity’ and often derided as a Theory of Everything – has turned up a beautiful mathematical proposal, string theory. In this theory, instead of fundamental particles being points, they are vibrating multidimensional loops. The best-known version uses six-dimensional loops, so its model of spacetime is really ten-dimensional. Why has no one noticed? Perhaps because the extra six dimensions are curled up so tightly that no one has observed them – very possibly, can observe them. Or perhaps – the Irishman again – we can’t go that way from here.

  Many physicists hope that string theory, as well as unifying relativity and quantum mechanics, will also supply a proof of Hawking’s chronology protection conjecture – that the universe conspires to keep events happening in the same temporal order. In this connection, there is a five-dimensional string-theoretic rotating black hole called a BMPV3 black hole. If this rotates fast enough, it has CTCs outside the black hole region. Theoretically, you can build one from gravitational waves and esoteric string-theoretic gadgets called ‘D-branes’.

  And here we see a hint of Hawking’s cosmological time cops. Lisa Dyson took a careful look at just what happens when you put the gravitational waves and D-branes together. Just as the black hole is within a gnat’s whisker of turning into a time machine, the components stop collecting together in the same place. Instead, they form a shell of gravitons (hypothetical particles of gravity, analogous to photons for light). The D-branes are trapped inside the shell. The gravitons can’t be persuaded to come any closer, and the BMPV can’t be made to spin rapidly enough to create an accessible CTC.

  The laws of physics won’t let you put this kind of time machine together, unless some clever kind of scaffolding can be invented.

  Quantum mechanics adds a new spin to the whole time-travel game. For a start, it may open up a way to create a wormhole. On the very tiny length scale of the quantum world, known as the Planck length (around 10-35 metres), spacetime is thought to be a quantum foam – a perpetually changing mass of tiny wormholes. Quantum foam is a kind of time machine. Time is slopping around inside the foam like spindrift bobbing on the ocean waves. You just have to harness it. An advanced civilisation might be able to use gravitational manipulators to grab a quantum wormhole and enlarge it to macroscopic size.

  Quantum mechanics also sheds light, or possibly dark, on the paradoxes of time travel. Quantum mechanics 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 of Hugh Everett III. This view of the universe is very familiar to readers of SF: our world is just one of an infinite family of ‘parallel worlds’ in which every combination of possibilities occurs. This is a dramatic way to describe quantum superposition of states, in which an electron spin can be both up and down at the same time, and (allegedly) a cat can be both alive and dead.4

  In 1991 David Deutsch argued that, thanks to the many worlds interpretation, quantum mechanical time travel poses no obstacles to free will. The grandfather paradox ceases to be paradoxical, because grandad will be (or will have been) killed in a parallel world, not in the original one.

  We find this a bit of a cheat. Yes, it resolves the paradox, but only by showing that it wasn’t really time travel at all. It was travel to a parallel world. Fun, but not the same. We also agree with a number of physicists, among them Roger Penrose, who accept that the ‘many worlds’ interpretation of quantum theory is an effective mathematical description, but deny that the parallel worlds involved are in any sense real. Here’s an analogy. Using a mathematical technique called Fourier analysis you can resolve any periodic sound, such as the note played by a clarinet, into a superposition of ‘pure’ sounds that involve only one vibrational frequency. In a sense, the pure sounds form a serious of ‘parallel notes’, which together create the real note. But you don’t find anyone asserting that there must therefore exist a corresponding set of parallel clarinets, each producing one of the pure notes. The mathematical decomposition need not have a literal physical analogue.

  What about paradoxes of genuine time travel, no faffing about with parallel worlds? In the relativistic setting, which is where such questions most naturally arise, there is an interesting resolution. If you set up a situation with paradoxical possibilities, it automatically leads to consistent behaviour.

  A typical thought-experiment here is to send a billiard ball through a wormhole, so that it emerges in its own past. With care, you can send it in so that when it comes (came) out it bashes into its earlier incarnation, deflecting it so that it never enters the wormhole in the first place. This is the grandfather paradox in less violent form. The question for a physicist is: can you actually set such paradoxical states up? You have to do so before the time machine is built, then build it, and see what physical behaviour actually occurs.

  It turns out that, at least in the simplest mathematical formulation of this question, the usual physical laws select a unique, logically consistent behaviour. You can’t suddenly plonk a billiard ball down inside a pre-existing system – that act involves human intervention, ‘free will’, and its relation to the laws of physics is moot. If you leave it up to the billiard ball, it follows a path that does not introduce logical inconsistencies. It is not yet known whether similar results hold in more general circumstances, but they may well do.

  This is all very well, but it does beg the ‘free will’ question. It’s a deterministic explanation, valid for idealised physical systems like billiard balls. Now, it is possible that the human mind is actually a deterministic system (ignoring quantum effects to keep the discussion within bounds). What we like to think of as making a free choice may actually be what it feels like when a deterministic brain works its way towards the only decision that it can actually reach. Free will may be the ‘quale’ of decision-making – the vivid feeling we get, lik
e the vivid sense of colour we get when we look at a red flower.5 Physics does not yet explain how these feelings arise. So it is usual to dismiss effects of free will when discussing possible temporal paradoxes.

  This sounds reasonable, but there’s a catch. The whole discussion of time machines, in physics terms, is about the possibility of people constructing the various warped spacetimes that are involved. ‘Get a black hole, join it to a white one …’ Specifically, it is about people choosing or deciding to construct such a device. In a deterministic world, either they are bound to construct it from the beginning, in which case ‘construct’ isn’t a very appropriate word, or the thing just puts itself together, and you find out what sort of universe you are in. It’s just like Gödel’s rotating universe: either you’re in it, or you’re not, and you don’t get to change anything. You can’t bring a time machine into being unless it was already implicit in the unfolding of that universe anyway.

  The standard physics viewpoint really only makes sense in a world where people have free will and can choose to build, or not to build, as they see fit. So physics, not for the first time, has adopted inconsistent viewpoints for different aspects of the same question, and has got its philosophical knickers in a twist as a result.

  For all the clever theorising, the dreadful truth is that we do not yet have the foggiest idea how to make a practical time machine. The clumsy and energy-wasteful devices of real physics are a pale shadow of the elegant machine of Wells’s Time Traveller, whose prototype was described as ‘a glittering metallic framework, scarcely larger than a small clock, very delicately made. There was ivory in it, and some transparent crystalline substance.’

  There’s still some R&D needed.

  Probably this is a Good Thing.

  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.)