We consider a Casimir apparatus consisting of two perfectly conducting parallel plates, subject to the weak gravitational field of the Earth. The aim of this paper is the calculation of the energy-momentum tensor of this system for a free, real massless scalar field satisfying Neumann boundary conditions on the plates. The small gravity acceleration (here considered as not varying between the two plates) allows us to perform all calculations to first order in this parameter. Some interesting results are found: a correction, depending on the gravity acceleration, to the well-known Casimir energy and pressure on the plates. Moreover, this scheme predicts a tiny force in the upwards direction acting on the apparatus. These results are supported by two consistency checks: the covariant conservation of the energy-momentum tensor and the vanishing of its regularized trace, when the scalar field is conformally coupled to gravity.
We address a problem of Casimir apparatus in dense medium and weak gravitational field. The falling of the apparatus has to be governed by the equivalence principle, with proper account for contributions to the weight of the apparatus from its material part and from distorted quantum fields. We discuss general expression for the corresponding force in metric with cylindrical symmetry. By way of example we compute explicit expression for Archimedes force, acting on the Casimir apparatus of finite size, immersed into thermal bath of free scalar field. It is shown that besides universal term, pro…
A Casimir--type analysis of the effect of dividing the two--sphere by several lines of latitude is done for conformally invariant Dirichlet and Neumann scalars and for spinors. An effective action combination is shown to have minima for symmetrical arrangements of the circles. For a domain of two caps and a slice, the Dirichlet slice expands to fill the whole sphere while the Neumann one adjusts to an angular separation of 57.92 degrees. The fermion expression is written in terms of the Weber class function, $\gf$, and a connection is noted with an earlier calculation of twisted scalar effecti…
We study the influence of stationary axisymmetric spacetimes on Casimir energy. We consider a massive scalar field and analyze its dependence on the apparatus orientation with respect to the dragging direction associated with such spaces. We show that, for an apparatus orientation not considered before in the literature, the Casimir energy can change its sign, producing a repulsive force. As applications, we analyze two specific metrics: one associated with a linear motion of a cylinder and a circular equatorial motion around a gravitational source described by Kerr geometry.
We calculate the Casimir interaction energy in $d=2$ spatial dimensions between two (zero-width) mirrors, one flat, and the other slightly curved, upon which {\em imperfect\/} conductor boundary conditions are imposed for an Electromagnetic (EM) field. Our main result is a second-order Derivative Expansion (DE) approximation for the Casimir energy, which is studied in different interesting limits. In particular, we focus on the emergence of a non-analyticity beyond the leading-order term in the DE, when approaching the limit of perfectly-conducting mirrors. We also show that the system conside…
In this work, we consider a Casimir apparatus that is put into free fall (e.g., falling into a black hole). Working in 1+1D, we find that two main effects occur: First, the Casimir energy density experiences a tidal effect where negative energy is pushed toward the plates and the resulting force experienced by the plates is increased. Second, the process of falling is inherently nonequilibrium and we treat it as such, demonstrating that the Casimir energy density moves back and forth between the plates after being "dropped,'' with the force modulating in synchrony. In this way, the Casimir ene…
In this paper I study the Casimir effect caused by a charged and massive scalar field that breaks Lorentz invariance in a CPT-even, aether-like manner. I investigate the case of a scalar field that satisfies Dirichlet or mixed (Dirichlet-Neumann) boundary conditions on a pair of very large plane parallel plates. The case of Neumann boundary conditions is straightforward and will not be examined in detail. I use the $ζ$-function regularization technique to study the effect of a constant magnetic field, orthogonal to the plates, on the Casimir energy and pressure. I investigate the cases of a ti…
We calculate the finite vacuum energy density of the scalar and electromagnetic fields inside a Casimir apparatus made up of two conducting parallel plates in a general weak gravitational field. The metric of the weak gravitational field has a small deviation from flat spacetime inside the apparatus and we find it by expanding the metric in terms of small parameters of the weak background. We show that the found metric can be transformed via a gauge transformation to the Fermi metric. We solve the Klein-Gordon equation exactly and find mode frequencies in Fermi spacetime. Using the fact that t…
In this work I study the Casimir effect of a massive complex scalar field in the presence of one large compactified extra dimension. I investigate the case of a scalar field confined between two parallel plates in the macroscopic three dimensions, and examine the cases of Dirichlet and mixed (Dirichlet-Neumann) boundary conditions on the plates. The case of Neumann boundary conditions is uninteresting, since it yields the same result as the case of Dirichlet boundary conditions. The scalar field also permeates a fourth compactified dimension of a size that could be comparable to the distance b…
In the present study, the zero- and first-order radiative correction to the Casimir energy for the massive and massless scalar field confined with mixed (Neumann-Dirichlet) boundary condition between two parallel lines in 2+1 dimensions for the self-interacting $ϕ^4$ theory was computed. The main point in this study is the use of a special program to renormalize the bare parameters of the Lagrangian. The counterterm used in the renormalization program, which was obtained systematically position-dependent, is consistent with the boundary condition imposed on the quantum field. To regularize and…
In this work I investigate the finite temperature Casimir effect due to a massive and charged scalar field that breaks Lorentz invariance in a CPT-even, aether-like way. I study the cases of Dirichlet and mixed (Dirichlet-Neumann) boundary conditions on a pair of parallel plates. I will not examine the case of Neumann boundary conditions since it produces the same results as Dirichlet boundary conditions. The main tool used in this investigation is the $ζ$-function technique that allows me to obtain the Helmoltz free energy and Casimir pressure in the presence of a uniform magnetic field perpe…
In the present study, the first-order radiative correction to the Casimir energy for massive and massless scalar fields confined with mixed boundary conditions (Dirichlet-Neumann) between two points in ϕ^4 theory was computed. Two issues in performing the calculations in this work are essential: to renormalize the bare parameters of the problem, a systematic method was employed, allowing all influences from the boundary conditions to be imported in all elements of the renormalization program. This idea yields our counterterms appeared in the renormalization program to be position-dependent. Us…
Using covariant quantization of the electromagnetic field, the Casimir force per unit area experienced by a long conducting cylindrical shell, under both Dirichlet and Neumann boundary conditions, is calculated. The renormalization procedure is based on the plasma cut-off frequency for real conductors. The real case of a gold (silver) cylindrical shell is considered and the corresponding electromagnetic Casimir pressure is computed. It is discussed that the Dirichlet and Neumann problems should be considered separately without adding their corresponding results.
In the present study, the zeroth- and first-order radiative correction of the Casimir energy for massive and massless scalar fields, confined with mixed boundary conditions (Dirichlet- Neumann) between two parallel plates in $ϕ^4$ theory, were computed. Two issues in performing calculations in this work are essential: first, to renormalize the bare parameters of the problem, a systematic method was used, which allows all influences from the boundary conditions to be imported in all elements of the renormalization program. This idea yields our counterterms appearing in the renormalization progr…
We consider the Casimir interaction between two spheres corresponding to massless Dirac fields with MIT-bag boundary conditions. Using operator approach, we derive the TGTG-formula for the Casimir interaction energy between the two spheres. A byproduct is the explicit formula for the translation matrix that relates the fermionic spherical waves in different coordinate systems. In the large separation limit, it is found that the order of the Casimir interaction energy is $L^{-5}$, where $L$ is the separation between the centers of the spheres. This order is intermediate between that of two Diri…
In the present article, Radiative Correction (RC) to the Casimir energy was computed for the self-interacting massive/massless Lifshitz-like scalar field, confined between a pair of plates with Dirichlet and Mixed boundary conditions in $3+1$ dimensions. Moreover, using the results obtained for the Dirichlet Casimir energy, the RC to the Casimir energy for Periodic and Neumann boundary conditions were also draw outed. To renormalize the bare parameters of the Lagrangian, a systematic perturbation expansion was used in which the counterterms were automatically obtained in a position-dependent m…
I investigate the finite temperature Casimir effect for a charged and massless scalar field satisfying mixed (Dirichlet-Neumann) boundary conditions on a pair of plane parallel plates of infinite size. The effect of a uniform magnetic field, perpendicular to the plates, on the Helmholtz free energy and Casimir pressure is studied. The zeta-function regularization technique is used to obtain finite results. Simple analytic expressions are obtained for the zeta function and the free energy, in the limits of small plate distance, high temperature and strong magnetic field. The Casimir pressure is…
We generalize the derivative expansion (DE) approach to the interaction between almost-flat smooth surfaces, to the case of surfaces which are optimally described in cylindrical coordinates. As in the original form of the DE, the obtained method does not depend on the nature of the interaction. We apply our results to the study of the static, zero-temperature Casimir effect between two cylindrical surfaces, obtaining approximate expressions which are reliable under the assumption that the distance between those surfaces is always much smaller than their local curvature radii. To obtain the zer…
We analyze the Casimir densities and forces associated with a massive scalar quantum field confined between two parallel plates in a D-dimensional cosmic string spacetime. The plates are placed orthogonal to the string and the field obeys the Robin boundary conditions on them. The boundary-induced contributions are explicitly extracted in the vacuum expectation values (VEVs) of the field squared and energy-momentum tensor for both single and two plates. The VEV of the energy-momentum tensor, in additional to the diagonal components, contains an off-diagonal component corresponding to the shear…
One-Loop Correction to the Casimir Energy in Lorentz-Violating $ϕ^4$ Theory with Rough Membrane Boundaries
Department of Physics, Semnan Branch, Islamic Azad University, Semnan, Iran（伊斯兰阿扎德大学塞姆南分校物理系）；Energy and Sustainable Development Research Center, Semnan Branch, Islamic Azad University, Semnan, Iran（伊斯兰阿扎德大学塞姆南分校能源与可持续发展研究中心）
Recent work in the literature has studied rigid Casimir cavities in a weak gravitational field, or in de Sitter spacetime, or yet other spacetime models. The present review paper studies the difficult problem of direct evaluation of scalar Green functions for a Casimir-type apparatus in de Sitter spacetime. Working to first order in the small parameter of the problem, i.e. twice the gravity acceleration times the plates' separation divided by the speed of light in vacuum, suitable coordinates are considered for which the differential equations obeyed by the zeroth- and first-order Green functi…
