In school, we have probably all made experiments with charged objects and the electrodynamical forces. Consider two conducting and charged plates, parallel to each other. If both plates have the same charge, they will repel, if they have opposite charges they will attract. This force acting on the plates can be measured. So far, so good. Now take away the charges, just stick with the two conducting plates. What will happen? Surprisingly, the two plates will still attract each other!
This seemingly weird finding of quantum field theory is a very weak effect that can only be measured at smallest distances of the plates, typically of the order nanometers. Roughly speaking one can understand it as follows: quantum field theory teaches us there is no such thing as a really 'empty' vacuum. Instead, there are constantly particle pairs created. They come into life for a very short time, then annihilate and vanish. They are called 'virtual particles'. As is common for quantum objects, their energy is related to a wavelength. The higher their energy, the smaller the wavelength.
The troublesome thing about these particles is that their energy can actually become arbitrarily large, in fact, it can be infinitely large. This then results in a seemingly enormous 'vacuum energy'. However, as long as we neglect gravity, we can not really measure absolute values of energy anyhow. We can only measure differences between energies. Therefore, one can be bothered by the result of this calculation, but for practical purposes it never actually becomes problematic.
Now come back to the two conducting plates. The relevant fact is that they act like mirrors for the electromagnetic field and define a boundary: There is an area between the plates which has a finite extension into one direction, and there is the part outside the plates which is infinite on both sides. The wavelengths of the virtual particles have to fit between the plates (see figure below). In such a way, the plates select only some of the total of all possible waves in the vacuum.
Now if we ask for the energy differences between the in- and the outside, there is more virtual stuff on the outside than on the inside. The result is an inward directed pressure on the plates, which appears like an attractive force.
Though this pictorial explanation looks very nice it certainly can't replace a sensible calculation. E.g. for a cube the resulting force is confusingly repulsive, and the sign also depends also on whether the field is fermionic or bosonic. If one does the calculation for the case of parallel plates, one finds that the force F depends on the distance d between the plates as F~ 1/d4. The closer the plates are together, the larger the force between them.
Casimir and his Effect
Weird, eh? But experimentally confirmed! Hendrik Casimir and Dirk Polder, who where working at a Philips Research Lab, predicted the existence of such a force in 1948 and proposed an experiment to detect it (Phys. Rev. 73, 360 - 372 (1948)). Though the first experiments were performed already ten years later by Sparnaay, the data had large error-bars and the results were inconclusive. It has only been in the last ten years that these doubts were erased (Lamoreaux '97; Bressi, Carugno, Onofrio & Ruoso 2001), and Casimir's prediction has been confirmed.
Now during the last decade the Casimir force has received an increased and still increasing amount of attention, because more and more devices are 'nano' - and thereby come into the range where they are sensitive to the Casimir effect. The number of citations of Casimir's original paper has been exponentially growing (see Figure below)
[Graph from D. Budker's powerpoint presentation]
Weird and Weirder
But that's actually not the reason why I find the Casimir effect so interesting. No, the reason is that if one computes the energy density between the plates, it turns out to be negative! Please keep in mind, there is nothing in the known universe with negative energy density. The obvious question to ask then is how does that system couple to gravity? Is this really a negative contribution to the total mass? Does it get repelled by the earth?
I find this question intriguing because it addresses the question how manifestations of the quantum vacuum energy gravitate.
Piture credit: Baughman et al., Science 297, 787 (2002)
|Five or six years ago I read an article on superconducting nano-tubes and began to wonder about their Casimir energy. As roughly cylindrically symmetric conducting tubes, would they be subject to the Casimir effect? In a very idealized case, I estimated the contribution to the density to be not too far off the current precision for the equivalence principle in an Eötvös experiment. I can't recall the details though -- maybe some orders of magnitude, not too desperate a situation. I am still not sure whether this makes sense, because I then unfortunately thought about the realistic situation - most notably the tubes are not actually cylindrical but already so small that their molecule structure is relevant.|
I tried to convince an undergrad to look into the matter, but well. It didn't go anywhere. The topic had a brief revival for me some time later when I read a paper that proposed an experiment to test the weak equivalence principle for the gravitational coupling of vacuum energy... HA! I just found the paper. Not sure whether I should be surprised that it's by John Moffat ( Moffat & Gillies, New J. Phys. 4 (2002) 92) It seems whatever I look for, he has written a paper on it.
Anyway, the problem with the gravitational interaction of vacuum energy is if one takes gravity into account, it is no longer true that only energy differences are measurable. So what ought we to make out of an infinite result? If it was really a huge contribution that coupled to gravity, we would already have noticed (worse, we wouldn't even exist then). We can still just define some specific state as 'empty' in the sense that it does not have an energy that couples to gravity. However, the problem is that the definition of this vacuum depends on what an observer calls a particle. Unfortunately, this can differ from one observer to the other*. Either way one turns it, the definition of the vacuum depends on the definition of a particle, and the corresponding annihilation and creation operators. Concepts which in a curved background are not unique.
The actual reason for this post is that I recently came across a paper on the arxiv:
- Gravitational and Inertial Mass of Casimir Energy
By Milton, Fulling, Parashar, Romeo, Shajesh, & Wagner
There's an incredibly large amount of virtual particles popping in and out of existence - right in front of your eyes, in exactly this moment. If if one does it the right way, one can measure them. Thanks to Casimir's ingenuity.
- Kimball A. Milton, The Casimir Effect: Physical Manifestations of Zero Point Energy
- Astrid Lambrecht, The Casimir effect: a force from nothing
- Wikipedia entry on the Casimir Effect
*This is essentially the reason for the Unruh effect - the one's vacuum is a thermal bath of particles for the other. Do these particles gravitate?
TAGS: PHYSICS, CASIMIR EFFECT, VACUUM ENERGY