STM-D-1007Paper1969Settled physics

Vacuum Stress between Conducting Plates: An Image Solution

Lowell S. Brown · G. Jordan Maclay

Abstract and summary · read the original at the source · none found

In one page

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.

Why it matters hereChapter 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.

What it claims

01Symmetry 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
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02The 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
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03The 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
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04At 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
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05In 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
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06The 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
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Read it · abstract

Abstract

The zero-point fluctuations of the electromagnetic field give rise to an attractive force between two perfectly conducting parallel plates, the Casimir force. We discuss the structure of the electromagnetic stress-energy tensor in the region between the plates for finite temperatures as well as for the zero-temperature limit, and we describe the relationship of its components to the thermodynamic variables of the radiation field. The stress-energy tensor is defined so that infinite quantities never appear, and it is explicitly computed with the aid of an image-source construction of the Green’s function. The finite-temperature case involves both an infinite set of spatial images and an infinite sum of temperature-dependent images.

Lowell S. Brown and G. Jordan Maclay, Physics Department, University of Washington, Seattle. Physical Review 184, 1272 (1969). Abstract as published.

(Abstract only — see the rights note above for why the eight pages of derivation are not reproduced here. They are at the source. The paper’s four sections are the introduction; the nature of the stress, which states the results; the zero-temperature Green’s function; and the finite-temperature Green’s function. A closing comparison treats the region outside a perfectly conducting sphere, where the image construction cannot be used at all, because a radiation pulse leaving one point of a curved surface does not reach the others at the same instant — that geometry needs a partial-wave decomposition instead, and Brown and Maclay quote Boyer’s approximate evaluation of the single remaining function rather than a closed form.)

The Casimir line on this site runs through here. Casimir’s own prediction and Sparnaay’s first measurement of it are at /library/stm-2aa45438b3, the paper footnote 6 cites. Maclay’s later work — resolving Robert Forward’s Casimir energy-extraction cycle paradox for rectangular cavities — is at /library/stm-0e25498180, and his review of quantum vacuum energy extraction is at /library/stm-846e4e7fae. What happens to the same stress when the geometry stops being two flat plates is at /library/stm-627c58cd85 for a knife edge and /library/stm-d41136300f for microstructured geometries; the field’s own running survey of the subject is the QFEXT09 proceedings at /library/stm-5b5c581f32.

The way in

https://doi.org/10.1103/physrev.184.1272PUBLICATION. Physical Review volume 184, number 5, pages 1272 to 1279, issue dated 25 August 1969, received 9 April 1969. Both authors wrote from the Physics Department, University of Washington, Seattle, Washington 98105; the work was supported in part by the United States Atomic Energy Commission under contract AT(45-1)-1388. LICENCE. The American Physical Society registers only its default licence for this DOI, Unpaywall and OpenAlex report the record closed with no repository copy, and no Creative Commons statement exists, so nothing beyond the work’s own abstract is reproduced here. WHAT WAS READ. The complete eight-page article was retrieved on 2026-09-08 from the APS harvest full-text endpoint for this DOI, https://harvest.aps.org/v2/journals/articles/10.1103/physrev.184.1272/fulltext, and read in full; every claim below cites the paper’s own section, equation or footnote number. The scan carries optical-character-recognition damage in the mathematics, so equations are described in words here rather than transcribed, and no numerical value is quoted that could not be read cleanly in more than one place in the text. RELATED PAGES. Sparnaay’s 1958 measurement, which this paper cites in its footnote 6 as the experimental test of the parallel-plate force, is at /library/stm-2aa45438b3.

How to cite it

Lowell S. Brown, G. Jordan Maclay (1969) Vacuum Stress between Conducting Plates: An Image Solution. doi:10.1103/physrev.184.1272

Where it sits in the curriculum

What the vacuum is Energy from the vacuum