Vacuum energy density and pressure inside a soft wall

Abstract

In the study of quantum vacuum energy and the Casimir effect, it is desirable to model the conductor by a potential of the form [Formula: see text]. This “soft wall” model was proposed so as to avoid the violation of the principle of virtual work under ultraviolet regularization that occurs for the standard Dirichlet wall. The model was formalized for a massless scalar field, and the expectation value of the stress tensor has been expressed in terms of the reduced Green function of the equation of motion. In the limit of interest, [Formula: see text], which approximates a Dirichlet wall, a closed-form expression for the reduced Green function cannot be found, so piecewise approximations incorporating the perturbative and WKB expansions of the Green function, along with interpolating splines in the region where neither expansion is valid, have been developed. After reviewing this program, in this paper, we apply the scheme to the wall with [Formula: see text] and use it to compute the renormalized energy density and pressure inside the cavity for various values of the conformal parameter. The consistency of the results is verified by comparison to their numerical counterparts and verification of the trace anomaly and the conservation law. Finally, we use the approximation scheme to reproduce the energy density inside the quadratic wall, which was previously calculated exactly but with some uncertainty.

Vacuum Energy, the Casimir Effect, and Newton’s Non-Constant

“…However, lacking a detailed first-principles approach for the material at frequencies > c/ with being the typical length scale of interatomic spacings, quantitative predictions will strongly depend on the assumptions taken. The same is true for the region inside the walls, for which no analytic form has been found and only numerical methods on the basis of ad hoc assumptions for a smooth potential lead to finite values for ρ C [49]. Independent of these details, however, ρ C can be expected to be large near the boundaries, for which it could potentially contribute to gravitational interactions [50].…”

“…Since, further, the numerical value of Λ(k) is negligibly small in all regions, this means that in the regions outside the plate, the usual vacuum Einstein equations apply, and thus Equation ( 21) can be used with ∇ 2 ∆G = 0 there. Since the above Equation ( 35) is a bold simplification of a still unknown, but likely more complicated, functional form of ρ C (z) [41][42][43]48,49], we defer a detailed numerical integration of Equation (21) and leave this topic for future work. Instead, we revisit the WG-WSD assumption of Equation ( 14) to obtain a rough estimate of what we could expect to find for the phenomenological parameters.…”