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Matt Visser: Some General Interest Articles
This html document contains a few non-technical discussions of some of the things I have been working on for the last few years. These are written for people who are not experts in physics or science and are hopefully understandable to a more general audience.
What exactly is a wormhole?
© Matt Visser, September 1997
Wormholes are hypothetical entities that show up in theoretical analyses of Einstein's theory of gravity (general relativity). No-one has yet seen a wormhole, nor are we certain that they exist, but they seem to show up so easily when we do calculations that many physicists suspect that they might actually be out there in the real universe.
There are two main types of wormhole of interest to physicists: Lorentzian wormholes (general relativity) and Euclidean wormholes (particle physics).
Lorentzian wormholes are essentially short-cuts through space and time. They are mainly studied by experts in Einstein gravity, and if they exist in real life would be more-or-less similar to the wormhole on Star Trek: Deep Space 9. (Just remember, it's entertainment, so don't try to extract detailed physics from DS9, at best it'll give you a vague general idea of what's going on.)
The good news about Lorentzian wormholes is that after about ten years of hard work we cannot prove that they don't exist. The bad news is that they are very strange objects: If they exist at all they need large amounts of negative mass to hold them open and stop them from collapsing. (Negative mass is not anti-matter, it's a region where the energy of the universe is less than that of ordinary vacuum---definitely wierd stuff.) We can get small amounts of negative energy in the Laboratory (Casimir effect), getting the large amounts needed to hold a decent size Lorentzian wormhole open looks to be hopeless with current technologies (and there may be deep issues of principle preventing us from collecting a lot of negative energy in one place).
If Lorentzian wormholes exist, then it seems classically to be relatively easy to turn them into time machines. This embarrassing feature has led Stephen Hawking to promulgate his Chronology Protection Conjecture. According to this conjecture, quantum effects will conspire to effectively prevent time travel even when it looks like classical physics might allow time travel to occur.
Euclidean wormholes are even stranger: they live in ``imaginary time'' and are intrinsically virtual quantum mechanical processes. These Euclidean wormholes are of interest mainly to the particle physicists (quantum field theorists). You cannot give them a nice classical interpretation in terms of a well-behaved classical gravitational field, and unfortunately have to know a lot of quantum physics to appreciate the basic issues.
What is the 'zero-point energy' (or 'vacuum energy') in quantum physics?
© Matt Visser, September 1997
The Zero Point Energy (ZPE) is an intrinsic and unavoidable part of quantum physics. The ZPE has been studied, both theoretically and experimentally, since the discovery of quantum mechanics in the 1920s and there can be no doubt that the ZPE is a real physical effect. The "vacuum energy" is a specific example of ZPE which has generated considerable doubt and confusion. In a completely empty flat universe, calculations of the vacuum energy yield infinite values of both positive and negative sign--something that obviously does not correspond to the nature of the real world.
Observation indicates that in our universe the grand total vacuum energy is extremely small and quite possibly exactly zero. Many theorists suspect that the total vacuum energy is exactly zero.
It definitely is possible to manipulate the vacuum energy. Any objects that change the vacuum energy (electrical conductors, dielectrics and gravitational fields, for instance) distort the quantum mechanical vacuum state. These changes in the vacuum energy are often easier to calculate than the total vacuum energy itself. Sometimes we can even measure these changes in the vacuum energy in laboratory experiments.
In classical physics, if you have a particle that is acted on by some conservative force, the total energy is
E = (1/2) m v**2 + V(x).
To find the classical ground state, set the velocity to zero to minimize the kinetic energy, (1/2)m v**2 , and put the particle at the point where it has the lowest potential energy V(x) . But this result is only a classical approximation to the real world. Because the classical ground state completely specifies both the particle's speed (zero) and position (at the minimum), it violates the famous Heisenberg Uncertainty Principle (m dv dx > hbar) . Quantum physics, via the Uncertainty Principle, forces the particle to spread out both in position and velocity and so causes it to have an energy somewhat higher than the classical minimum. The ZPE is defined as this shift:
E_(ZPE) = E_(quantum minimum) --- E_(classical minimum) > 0.
Classically, we can calculate the natural oscillation frequency that the particle would have if we were to give it a small push. Quantum mechanically, it is now an undergraduate exercise to use the Heisenberg uncertainty relation (more precisely, Schroedinger's differential equation) to show that
E_(ZPE) approx (1/2) hbar omega_0,
where hbar is Planck's constant and omega_0 is the natural oscillation frequency. The ZPE in this sense shows up almost everywhere: it affects molecular bonds, condensed matter physics, small oscillations of any system.
The next step is to realize that the electromagnetic field can be thought of as an infinite collection of coupled oscillators--one at each point in space. Again, the classical ground state is the case in which the electric and magnetic fields both must be zero. Quantum effects mean that this case does not hold true; there is also a Heisenberg uncertainty principle for electric and magnetic fields (it's a little more complex). The good news is that the potential for electromagnetism is exactly quadratic and so can be solved exactly. The bad news is that there is an infinite number of modes. Formally we can write
(Electromagnetic vacuum energy) = sum over all modes (1/2) hbar omega(mode).
The infinity in this equation is what excites the free lunch crowd (the modern descendants of the perpetual motion crowd), who envision an endless ZPE for humanity to tap into. Not quite. Unfortunately...
The first and most obvious problem is that there are other quantum fields in the universe apart from electromagnetism. Electrons, for starters, plus neutrinos, quarks, gluons, W, Z, Higgs and so on. In particular, if you do the calculation for electrons you will find that what are known as Fermi statistics give rise to an extra minus sign in the calculation.
Adding minus infinity to plus infinity gives mathematicians nightmares and even makes theoretical physicists worry a little. Fortunately, nature does not worry about what the mathematicians or physicists think and does the job for us automatically. Consider the grand total vacuum energy (once we have added in all quantum fields, all particle interactions, kept everything finite by hook or by crook, and taken all the proper limits at the end of the day). This grand total vacuum energy has another name: it is called the "cosmological constant," and it is something that we can measure observationally.
In its original incarnation, the cosmological constant was something that Einstein put into General Relativity (his theory of gravity) by hand. Particle physicists have since taken over this idea and appropriated it for their own by giving it this more physical description in terms of the ZPE and the vacuum energy. Astrophysicists are now busy putting observational limits on the cosmological constant. From the cosmological point of view these limits are still pretty broad: the cosmological constant could potentially provide up to 60 percent to 80 percent of the total mass of the universe.
From a particle physics point of view, however, these limits are extremely stringent: the cosmological constant is more than 10**(123) times smaller than one would naively estimate from particle physics equations. The cosmological constant could quite plausibly be exactly zero. (Physicists are still arguing on this point.) Even if the cosmological constant is not zero it is certainly small on a particle-physics scale, small on a human-engineering scale, and too tiny to be any plausible source of energy for human needs--not that we have any good ideas on how to accomplish large-scale manipulations of the cosmological constant anyway.
Putting the more exotic fantasies of the free lunch crowd aside, is there anything more plausible that we could use the ZPE for? It turns out that small-scale manipulations of the ZPE are indeed possible. By introducing a conductor or a dielectric, one can affect the electromagnetic field and thus induce changes in the quantum mechanical vacuum, leading to changes in the ZPE. This is what underlies a peculiar physical phenomenon called the Casimir effect. In a classical world, perfectly neutral conductors do not attract one another. In a quantum world, however, the neutral conductors disturb the quantum electromagnetic vacuum and produce finite measurable changes in the energy as the conductors move around. Sometimes we can even calculate the change in energy and compare it with experiment. These effects are all undoubtedly real and uncontroversial but tiny.
More controversial is the suggestion, made by the physicist Julian Schwinger, that the ZPE in dielectrics has something to do with sonoluminescence. The jury is still out on this one and there is a lot of polite discussion going on (both among experimentalists, who are unsure of which of the competing mechanisms is the correct one, and among theorists, who still disagree on the precise size and nature of the Casimir effect in dielectrics.) Even more speculative is the suggestion that relates the Casimir effect to "starquakes" on neutron stars and to gamma ray bursts.
In summary, there is no doubt that the ZPE, vacuum energy and Casimir effect are physically real. Our ability to manipulate these quantities is limited but in some cases technologically interesting. But the free-lunch crowd has greatly exaggerated the importance of the ZPE. Notions of mining the ZPE should therefore be treated with extreme skepticism.
© 3 October 1996
This article was originally published in Phlogistin , the magazine of the New Zealand science fiction fandom.
Aficionados of science fiction might have excuses for wondering just what in hell is going on in modern physics. These days, picking up a random copy of The Physical Review 1 one is likely to encounter several papers on time travel, a few more on wormholes, and with luck, something about warp-drives. Likewise, other (presumably saner?) members of the professional physics community might wonder just what is going on at the general relativity/quantum physics interface. What are we doing and why are we doing it?
What we are doing is pushing the limits of knowledge: Twentieth century physics has provided us with two superb models of physical reality-Einstein's theory of gravity (general relativity), and the quantum theory of fields (particle physics). These two theories work very well-they are both logically consistent and are both accurate representations of physical reality in their respective regimes of validity. (And we have not yet found the edges of theses realms of validity.) Unfortunately, when you put the two together you very quickly get a real mess on your hands.
Quantum gravity: right now this is a name in search of a theory to be applied to. Despite some noisy (and mutually incompatible) claims to the contrary, there is currently  no universally accepted theory of quantum gravity. There are a number of ideas that seem as though they might pan out, a larger number of ideas that have failed, and continuing confusion over what exactly it might mean to actually quantize gravity.
Part of the problem is that the discussion is carried out largely without direct experimental input. Our current theories, limited though they are, are quite enough to tell us that things will get almighty interesting at high enough energies. There is something called the Planck scale2 such that we know that quantum gravity will be important in the trans-Planckian regime. The vacuum will try to curdle as vacuum fluctuations try to bootstrap their own little black holes into existence, and the quantum fluctuations will then try to vanish down these self-generated black holes-presumably quantum gravity will prevent this curdling process. At trans-Planckian scales spacetime would seem to become some chunderous froth of topology-changing processes accompanied by distortions in the spacetime manifold that are so violent that the word ``manifold'' starts to loose its meaning-this is what the relativists refer to as ``spacetime foam''. Even though we can be very sure that something interesting happens at the Planck scale, we cannot be sure exactly what goes berserk there.
Our best particle accelerators are no help here-even the late lamented Superconducting Super-Collider would have only provided particle energies up to about 104 GeV; a factor of 1015 = 1,000,000,000,000,000 too small to be interesting to the quantum gravity community.
Confronted with these technological limitations the physics community (or at least the part of the physics community that worries about these issues) has revitalized an old trick of Einstein-the gedanken-experiment. The idea of a gedanken-experiment, or thought-experiment, is to take your theory and test it for internal consistency by pure logical analysis without actually doing a real physics experiment. At times this is the best you can do.
When applied to both special and general relativity the gedanken-experiment program showed quite conclusively that these theories were internally consistent-and also gave the experimentalists some good ideas on where to start looking for actual measurable effects. (And the actual measurable effects have by now been found.)
When applied to the as yet undefined tangle of ideas that currently pass for quantum gravity the gedanken-experiment approach leads to much messier results-it is simply a way of taking what we have, imperfect though it may be, and pushing it to its limits to see what goes wrong. The hope is, of course, that the theory will break down in an interesting way, and so give us some hints as to what a better theory might be. One that might generalize both Einstein gravity and quantum field theory. Hopefully this will lead us to the holy grail-the quantum theory of gravity.
The best stop-gap we currently have is a sort of half-way hybrid between quantum physics and Einstein gravity known as ``semiclassical quantum gravity''. It's hybrid because one actually gives up on quantizing gravity itself, treats gravity classically, and simply quantizes everything else. The quantum average of the matter distribution is then fed into the Einstein field equations to generate the classical gravitational field. This is already complicated enough to keep us quite busy, thank you very much.
It's when we try to push this hybrid model up to the Planck scale that the weirdness starts. It looks like wormholes are almost sensible, or at least are not manifestly nonsensical. People have also started to play with warp-drives as solutions to the Einstein field equations in semiclassical quantum gravity. Just don't hold your breath waiting for the engineers to provide a production model-the technology requirements are beyond difficult, and are more akin to the seriously miraculous. 3
One of the truly wierd side-effects of the wormhole/warp-drive discussion is that it seems to lead, almost inevitably, to the possibility of time travel. (At his stage the more conservative members of the physics community typically shake their heads in befuddlement and leave the room.) There are roughly four broad classes of response suitable for handling the various types of paradox induced by the possibility of time travel:
The radical re-write conjecture: Grit your teeth and proceed to re-write all of modern physics from the ground up. Painful, very painful. (I'll live with this if necessary-but you'd better give me good experimental evidence before I spend too much time worrying about this possibility.)
The consistency conjecture: Since there seems to be only one universe, insist that it must be consistent no matter what. So if you try to change history you cannot succeed no matter how hard you try, because the past is already fixed. You know that you, the reader, are alive right now, so no-one can ever send a time traveller to five minutes ago to kill you as you pick up your copy of Phlogiston. If someone tries, something must go wrong: the gun must misfire, or the time machine malfumction, or the assasin miss the bus, or any of a potenially infinite list of increasingly contrived excuses. (Not my favourite way of dealing with things; it quickly begins to look like a consistency conspiracy.)
The chronology protection conjecture: A much more conservative point of view. Even though time travel seems to be absurdly easy once wormholes/warp-drives are allowed, there are reasons to expect things to go berserk just at the onset of time travel. We know that gravitational fields distort the quantum mechanical vacuum, and that this vacuum distortion heads off to infinity at the onset of time travel. We suspect that this effect destroys the wormhole/warp-bubble just as one is getting round to building a time machine. (This is my personal favourite.)
The boring physics conjecture: Forget all this nonsense. Take a good hard look at the experimental evidence, or rather lack thereof, and move on to greener pastures.
To wrap up then, what we are doing is laying the groundwork for Planck scale physics, and groping our way toward the as yet ill-perceived theory of quantum gravity. We are surveying the lay of the land, and even if we do not yet have definitive answers, we are trying to at least formulate the right questions. In particular, whatever the true theory of quantum gravity is, at energies below the Planck scale it must reduce to semiclassical quantum gravity. So any question you can ask about semiclassical quantum gravity must be askable and answerable in the full-fledged theory of quantum gravity. Whatever the theory of quantum gravity ultimately proves to be, it must (among many other things) be able to give solid answers concerning the existence of, and properties of, wormholes, warp-drives, time machines, and other exotica.
Matt Visser grew up in Lower Hutt and studied undergraduate physics at Victoria University. He obtained his Ph.D. at the University of California at Berkeley. He is presently Research Assistant Professor of Physics at Washington University in Saint Louis.
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These are a few general interest books that do not need any complicated technical background. They are non-mathematical descriptions of some of what is going on in modern physics. Remember that because they are effectively english language translations from the physics, something is unaviodably lost in the translation process.
Kip Thorne, Black holes and time warps: Einstein's outrageous legacy, (Norton, New York, 1994).
A brief history of time: from the big bang
to black holes, (Bantam, London, 1988).
Technical Books (very heavy duty, not for the faint at heart):
If you really want to know what is going on at any deep level you will simply have to buckle down and learn a lot of mathematics and physics. The two basic tools are general relativity, for which you need to know differential geometry, and the quantum theory of fields, for which you need to know essentially all of quantum physics, particle physics, and a good dose of the mathematics of Hilbert spaces. Some technical books for the bold are:
Stephen Hawking and George Ellis, The large scale structure of space-time, (Cambridge University Press, England, 1973).
N. D. Birrell and Paul Davies Quantum fields in curved space, (Cambridge University Press, England, 1982).
Stephen Fulling, Aspects of quantum field theory in curved space-time, (Cambridge University Press, England, 1989).
Matt Visser, Lorentzian Wormholes: from Einstein to Hawking, (American Institute of Physics Press, New York, 1995).
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1 The Physical Review; Section D15; Particles and Fields.
2 Planck scale: Energies above 1019 GeV, that is 1019 Giga-electron-Volts; distances smaller than 10-35 metres; times less than 10-45 seconds.
3 For instance: A one-metre-wide wormhole, or a one-metre-wide warp-bubble, needs about minus one Jupiter mass of very strange ``exotic matter'' to support the gravitational field. This exotic matter has to effectively have a negative gravitational mass-note this is negative mass, energy less than the vacuum, and is not antimatter, which has positive mass.
This article translated from TEX by TTH, version 0.9.