Doubt continues to linger over the reality of quantum vacuum energy. There is some question whether fluctuating fields gravitate at all, or do so anomalously. Here we show that for the simple case of parallel conducting plates, the associated Casimir energy gravitates just as required by the equivalence principle, and that therefore the inertial and gravitational masses of a system possessing Casimir energy Ec are both Ec/c 2 . This simple result disproves recent claims in the literature. We clarify some pitfalls in the calculation that can lead to spurious dependences on coordinate system. PACS numbers: 03.70.+k, 04.20.Cv, 04.25.Nx, 03.65.Sq The subject of quantum vacuum energy (the Casimir effect) dates from the same year as the discovery of renormalized quantum electrodynamics, 1948 [1]. It puts the lie to the naive presumption that zero-point energy is not observable. On the other hand, it continues to be surrounded by controversy, in large part because sharp boundaries give rise to divergences in the local energy density near the surface (see Refs. [2,3,4]). The most troubling aspect of these divergences is in the coupling to gravity. Gravity has its source in the local energymomentum tensor, and such surface divergences promise serious difficulties. The gravitational implications of zero-point energy are an outstanding problem in view of our inability to understand the origin of the cosmological constant or dark energy [5,6,7].As a prolegomenon to studying such issues, we here address a simpler question: How does the completely finite Casimir energy of a pair of parallel conducting plates couple to gravity? The question turns out to be surprisingly less straightforward than one might suspect! Previous authors [8,9,10,11,12] have given disparate answers, including gravitational forces, or gravitationally modified Casimir forces, that depend on the orientation of the Casimir apparatus with respect to the gravitational field of the earth. We will here resolve some of this confusion with a convincingly calculated result consistent with the equivalence principle. That is, the renormalized Casimir energy couples to gravity just like any other energy. In our opinion, this fact is evidence that vacuum energy must be taken seriously in gravitational theory and that the problem of boundary divergences must be resolved by a better understanding of the modeling and renormalization processes.We start by recalling the electromagnetic Casimir stress tensor between a pair of parallel perfectly conducting plates separated by a distance a, with transverse dimensions L ≫ a, as given by Brown and Maclay [13]:where the third spatial direction is the direction normal to the plates. This is given in terms of the Casimir energy per unit area, E c = −π 2 c/(720a 3 ). Outside the plates, T µν = 0. Omitted here is a constant divergent term that is present both between and outside the plates, and also in the absence of plates, which cannot have any physical significance. Because the electromagnetic field respects conformal symmetr...

We derive boundary conditions for electromagnetic fields on a -function plate. The optical properties of such a plate are shown to necessarily be anisotropic in that they only depend on the transverse properties of the plate. We unambiguously obtain the boundary conditions for a perfectly conducting -function plate in the limit of infinite dielectric response. We show that a material does not ''optically vanish'' in the thin-plate limit. The thin-plate limit of a plasma slab of thickness d with plasma frequency ! 2 p ¼ p =d reduces to a -function plate for frequencies (( 1. We show that the Casimir interaction energy between two parallel perfectly conducting -function plates is the same as that for parallel perfectly conducting slabs. Similarly, we show that the interaction energy between an atom and a perfect electrically conducting -function plate is the usual Casimir-Polder energy, which is verified by considering the thin-plate limit of dielectric slabs. The ''thick'' and ''thin'' boundary conditions considered by Bordag are found to be identical in the sense that they lead to the same electromagnetic fields.

At air-water interfaces, the Lifshitz interaction by itself does not promote ice growth. On the contrary, we find that the Lifshitz force promotes the growth of an ice film, up to 1-8 nm thickness, near silica-water interfaces at the triple point of water. This is achieved in a system where the combined effect of the retardation and the zero frequency mode influences the short-range interactions at low temperatures, contrary to common understanding. Cancellation between the positive and negative contributions in the Lifshitz spectral function is reversed in silica with high porosity. Our results provide a model for how water freezes on glass and other surfaces. DOI: 10.1103/PhysRevB.95.155422 Although water in its different forms has been studied for a very long time, several properties of water and ice remain uncertain and are currently under intense investigation [1][2][3][4]. The question we want to address in the present paper is to what extent the fluctuation-induced Lifshitz interaction can promote the growth of ice films at water-solid interfaces, at the triple point of water. Particles and surfaces, e.g., quartz, soot, or bacteria, in supercooled water are known experimentally to nucleate ice formation [5][6][7]. Here, we focus on interfaces between water and silica-based materials and examine the roles of several intervening factors in the sum over frequency modes (Matsubara terms) contributing to the Lifshitz free energy.Quantum fluctuations in the electromagnetic field result in van der Waals interactions, which in their unretarded form were explained by London in terms of frequency-dependent responses to the fluctuations in the polarizable atoms constituting the material medium [8]. The understanding of these interactions was revolutionized when Casimir introduced retardation effects [9]. The theory was later generalized by Lifshitz to include dielectric materials [10,11]. The Lifshitz formula in Eq. (1), derived for three-layer planar geometries [11], gives the interaction energy between two semi-infinite dielectric media described by their frequency-dependent dielectric permittivities as well as the dielectric permittivity of the medium separating them (see Fig. 1).The purpose of the present work is twofold. First, we want to show that a finite size ice film, nucleated by a solid-water interface, can be energetically favorable even when only the Lifshitz interaction is accounted for. Second, we want to highlight a relevant contribution from the zero frequency term in the expression for the Lifshitz energy in a region where it is not expected to be important. The temperature dependence * Mathias.A.Bostrom@ntnu.no † prachi.parashar@ntnu.no ‡ iver.h.brevik@ntnu.no of the Casimir force between metal surfaces [11][12][13][14] relies strongly on the exact behavior of the low-frequency dielectric function of metals. These and many other investigations have provided support for the notion that the zero frequency term would only be relevant at high temperatures or large surface separations at a moderate ...

It has been demonstrated that quantum vacuum energy gravitates according to the equivalence principle, at least for the finite Casimir energies associated with perfectly conducting parallel plates. We here add further support to this conclusion by considering parallel semitransparent plates, that is, δ-function potentials, acting on a massless scalar field, in a spacetime defined by Rindler coordinates (τ, x, y, ξ). Fixed ξ in such a spacetime represents uniform acceleration. We calculate the force on systems consisting of one or two such plates at fixed values of ξ. In the limit of large Rindler coordinate ξ (small acceleration), we recover (via the equivalence principle) the situation of weak gravity, and find that the gravitational force on the system is just M g, where g is the gravitational acceleration and M is the total mass of the system, consisting of the mass of the plates renormalized by the Casimir energy of each plate separately, plus the energy of the Casimir interaction between the plates. This reproduces the previous result in the limit as the coupling to the δ-function potential approaches infinity.

We calculate the lateral Casimir force between corrugated parallel plates, described by -function potentials, interacting through a scalar field, using the multiple scattering formalism. The contributions to the Casimir energy due to uncorrugated parallel plates is treated as a background from the outset. We derive the leading-and next-to-leading-order contribution to the lateral Casimir force for the case when the corrugation amplitudes are small in comparison to corrugation wavelengths. We present explicit results in terms of finite integrals for the case of the Dirichlet limit, and exact results for the weak-coupling limit, for the leading-and next-to-leading-orders. The correction due to the next-to-leading contribution is significant. In the weak coupling limit we calculate the lateral Casimir force exactly in terms of a single integral which we evaluate numerically. Exact results for the case of the weak limit allows us to estimate the error in the perturbative results. We show that the error in the lateral Casimir force, in the weak coupling limit, when the next-to-leading order contribution is included is remarkably low when the corrugation amplitudes are small in comparison to corrugation wavelengths. We expect similar conclusions to hold for the Dirichlet case. The analogous calculation for the electromagnetic case should reduce the theoretical error sufficiently for comparison with the experiments.

We investigate long-range forces between atoms with anisotropic electric polarizability interacting with dielectrics having anisotropic permittivity in the weak-coupling approximation. Unstable configurations in which the force between the objects is repulsive are constructed. Such configurations exist for three anisotropic atoms as well as for an anisotropic atom above a dielectric plate with a hole whose permittivity is anisotropic. Apart from the absolute magnitude of the force, the dependence on the configuration is qualitatively the same as for metallic objects for which the anisotropy is a purely geometric effect. In the weak limit closed analytic expressions for rather complicated configurations are obtained. The non-monotonic dependence of the interaction energy on separation is related to the fact that the electromagnetic Green's dyadic is not positive definite. The analysis in the weak limit is found to also semi-quantitatively explain the dependence of Casimir forces on the orientation of anisotropic dielectrics observed experimentally. Contrary to the scalar case, irreducible electromagnetic three-body energies can change sign. We trace this to the fact that the electromagnetic Green's dyadic is not positive definite.Comment: 9 page

The Casimir interaction between two concentric corrugated cylinders provides the mechanism for non-contact gears. To this end, we calculate the Casimir torque between two such cylinders, described by δ-potentials, which interact through a scalar field. We derive analytic expressions for the Casimir torque for the case when the corrugation amplitudes are small in comparison to the corrugation wavelengths. We derive explicit results for the Dirichlet case, and exact results for the weak coupling limit, in the leading order. The results for the corrugated cylinders approach the corresponding expressions for the case of corrugated parallel plates in the limit of large radii of cylinders (relative to the difference in their radii) while keeping the corrugation wavelength fixed. * Electronic address: Ines.Cavero-Pelaez@spectro.jussieu.fr † Electronic address: milton@nhn.ou.edu; URL: http://www.nhn.ou.edu/%7Emilton ‡ Electronic address: prachi@nhn.ou.edu § Electronic address: shajesh@nhn.ou.edu; URL: http://www.nhn.ou.edu/%7Eshajesh

The multiple scattering formalism is used to extract irreducible N -body parts of Green's functions and Casimir energies describing the interaction of N objects that are not necessarily mutually disjoint. The irreducible N -body scattering matrix is expressed in terms of single-body transition matrices. The irreducible N -body Casimir energy is the trace of the corresponding irreducible Nbody part of the Green's function. This formalism requires the solution of a set of linear integral equations. The irreducible three-body Green's function and the corresponding Casimir energy of a massless scalar field interacting with potentials are obtained and evaluated for three parallel semitransparent plates. When Dirichlet boundary conditions are imposed on a plate the Green's function and Casimir energy decouple into contributions from two disjoint regions. We also consider weakly interacting triangular-and parabolic-wedges placed atop a Dirichlet plate. The irreducible three-body Casimir energy of a triangular-and parabolic-wedge is minimal when the shorter side of the wedge is perpendicular to the Dirichlet plate. The irreducible three-body contribution to the vacuum energy is finite and positive in all the cases studied.