Historical research guide · 1007 · 1969
Vacuum Stress between Conducting Plates: An Image Solution
Lowell S. Brown · G. Jordan Maclay
In one minute
Twenty-one years after Hendrik Casimir predicted that two metal plates facing each other in empty space would be pushed together by the vacuum, Lowell Brown and Jordan Maclay — both then in the physics department at the University of Washington — worked out what the vacuum is actually doing in the gap. Casimir had computed one number, the force. Brown and Maclay compute the whole stress-energy tensor: the energy density, the pressure on the plates and the sideways stresses, at any temperature rather than only at absolute zero. Their method is the trick every student of electrostatics already knows, mirror images, carried across to the quantum field — the plates are replaced by an endless line of image sources, and a warm cavity by a second endless set displaced in imaginary time. The pay-off is bookkeeping in which no infinite quantity is ever handled, and answers that join straight onto thermodynamics: the stress along the plates turns out to be the radiation field’s free energy per unit volume.
Connection
Chapter 2 needs the vacuum to be a medium with a mechanical state, not merely a force law, and this is the paper that writes that state down component by component. Chapter 6 needs it because anyone proposing to draw work from a Casimir cavity has to keep an honest ledger of the energy, the pressure and the entropy of the field inside it — and this is where that ledger was first set out, by the same Jordan Maclay who spent the following decades on the engineering side of it.
From the source
Symmetry does almost all the work. At zero temperature, the requirements that the electromagnetic stress-energy tensor be traceless and divergence-free, combined with the flat parallel geometry and simple dimensional counting, fix every component of the tensor between the plates in terms of a single pure number — and the explicit image construction then supplies that number as pi squared divided by 180.Section 2, the argument running from equation 5 to equation 7
The stress-energy tensor is defined as a limit of a field product at finite spatial separation, with the homogeneous part that belongs to the infinitely extended vacuum simply discarded. That definition removes the usual vacuum infinity at the start, so no infinite quantity is ever manipulated, and the tensor stays finite even when it is evaluated right at the surface of a plate.Section 2, equations 1 and 2, and the paragraph that follows them
The energy density between the plates and the pressure on a plate come out as minus pi squared over 720 and minus pi squared over 240, each multiplied by Planck’s constant times the speed of light divided by the fourth power of the separation. Both agree with Casimir’s results, the negative pressure corresponds to an attractive force, and the same answer is recovered independently from the principle of virtual work.Section 2, equations 8, 9 and 10
At finite temperature the whole tensor reduces to one function of one dimensionless variable — the temperature times the plate separation, in natural units — and the components of the stress running parallel to the plates are identified with the negative of the Helmholtz free-energy density of the radiation field. The energy-density correction then reproduces the thermodynamic relation between internal energy, free energy and entropy exactly, so every component of the stress carries a thermodynamic meaning.Section 2, equations 16 to 25, with the free-energy function itself given as a double sum in equation 22
In the low-temperature or small-separation limit every term independent of the plate separation cancels out of the answer. Brown and Maclay give the physical reason: at small separation no mode of the radiation field propagating normally to the plates can be excited, and the modes running along the plates carry energy but exert no separation-dependent pressure. Corrections to that limit are exponentially small.Section 2, equations 32 to 35 and the discussion beneath them
The paper hands the experimenter a number. In footnote 6 the authors write the total pressure that would actually be observed at room temperature, once the blackbody radiation outside the plates has cancelled the blackbody contribution inside, as 0.01300 divided by the fourth power of the separation in micrometres, plus two times ten to the minus three, in dynes per square centimetre — and note that the measurements available in 1969, Sparnaay’s included, were consistent with an attractive force but not yet accurate enough to confirm the exponent and the coefficient of the leading term.Footnote 6, attached to the discussion of the measurable limit in Section 2
The way in
https://doi.org/10.1103/physrev.184.1272How to cite it
Lowell S. Brown, G. Jordan Maclay (1969) Vacuum Stress between Conducting Plates: An Image Solution doi:10.1103/physrev.184.1272