Historical research guide · 0868 · 1999
The “friction” of vacuum, and other fluctuation-induced forces
Mehran Kardar · Ramin Golestanian
In one minute
Mehran Kardar and Ramin Golestanian take the Casimir force — the pull between two mirrors that comes from the quantum jitter of the electromagnetic field in the gap — and ask what happens when the mirrors are neither flat nor still. Their answer, built with a path-integral method that handles boundaries of any shape, is that the quantized vacuum behaves essentially as a complex fluid, and that fluid pushes back on things moving through it. A plate picks up an extra effective mass set by its own shape, and that added mass is anisotropic: it differs along different directions. A plate that shakes feels a drag, and the energy lost leaves as real photons radiated from a neutral, uncharged surface. The cavity between two plates supports a continuous band of resonances the plates’ own motion can excite. The same mathematics, the authors show, also delivers the thermal Casimir forces in critical mixtures, superfluid films and membranes, and the van der Waals force — one mechanism wearing many names.
Connection
Chapter 2 needs a source that treats the vacuum as a medium with mechanical properties rather than an empty stage, and this Reviews of Modern Physics colloquium is that source in the authors’ own words. It hands chapter 5 the phrase the site keeps returning to — the vacuum as a complex fluid — chapter 3 a published route from vacuum fluctuations to an object’s effective mass, and chapter 6 the cleanest statement of the exchange every vacuum device is built on: move a neutral boundary the right way and the field pays you back in photons.
From the source
The static Casimir force between two perfectly conducting plates of area A at separation H follows from quantizing the cavity modes and comes out as h-bar times c times the area, times pi squared over two hundred and forty, divided by the separation to the fourth power — so measuring a mechanical force between two macroscopic bodies is in principle a way to read the behaviour of the quantum vacuum.Section II.A, Eq. 1
Surface roughness is not a nuisance correction but a measurable signal: for a self-affine rough surface facing a flat one, the free energy gains a term proportional to the squared roughness width over the squared separation that increases the Casimir attraction, plus a second term of opposite sign that decays with an exponent carrying the roughness exponent itself. For millimetre surfaces one hundred angstroms apart the three terms come to about one point nine times ten to the minus four newtons, four point nine times ten to the minus five newtons and three point seven times ten to the minus six newtons — within reach of the force apparatus of the day.Section IV, Eq. 21 and the closing paragraph
Quantum fluctuations of the electromagnetic field add to the effective mass of a corrugated plate, and the correction is anisotropic: it is finite along the corrugation wavevector and zero across it. The size is tiny — about one part in ten to the thirty-fourth for a macroscopic sample, and only down to one part in ten to the tenth for atomic dimensions — so the authors argue the anisotropy, not the magnitude, is the accessible signature, read by comparing a plate’s oscillation frequencies in two orthogonal directions.Section VI.A, Eq. 30
A moving mirror is damped by the vacuum. Above the light cone the mechanical response is imaginary and defines an anisotropic effective shear viscosity growing as the fourth power of frequency, with the dissipative force proportional to the fifth time derivative of the displacement — so a uniformly accelerating plate loses nothing, while a freely oscillating one decays with a time constant of twice its mass over that viscosity, scaling as the fifth power of the plate’s dimension.Section VI.B, Eqs. 22 and 31
For perfectly reflecting mirrors the dissipated energy can go nowhere except into emitted photons, and the authors calculate the angular distribution and spectrum of that radiation for a single plate undulating at one frequency and one wavevector; the total number of photons radiated per unit time and area matches the independently computed energy dissipation rate exactly, and no radiation appears above the drive frequency, as energy conservation requires.Section VI.D
Two plates form a cavity supporting a continuous spectrum of normal modes, and in that band both parts of the response kernel diverge — resonant dissipation by excitation of cavity photons. Read literally the divergence says components of motion in that frequency range cannot be driven by any finite external force, because exciting them would demand an infinite number of photons; assuming realistic finite reflectivity rounds the divergence off, so in practice the restriction is set by how ideal the mirrors are at the frequency of interest.Section VI.C
The way in
https://doi.org/10.1103/RevModPhys.71.1233How to cite it
Mehran Kardar, Ramin Golestanian (1999) The “friction” of vacuum, and other fluctuation-induced forces doi:10.1103/RevModPhys.71.1233