The goal in this effort is twofold: (1) to develop an understanding of Casimir forces in geometries more complicated than the usual parallel-plate geometry and (2) to provide extensive numerical computations to elucidate quantitative and qualitative aspects of the vacuum fluctuation energy and Casimir forces for the rectangular cavity. We review geometries for which Casimir forces and vacuum energy have been computed, and point out some of the difficulties with the ideal-conductor boundary conditions and ideal-shape boundary conditions, e.g., infinitely sharp edges. We investigate the vacuum electromagnetic stress-energy tensor at 0 K for a perfectly conducting three-dimensional rectangular cavity with sides ${a}_{1}\ifmmode\times\else\texttimes\fi{}{a}_{2}\ifmmode\times\else\texttimes\fi{}{a}_{3}.$ The elements of the tensor are averaged over the appropriate spatial coordinates of the cavity. We first consider the average energy density ${T}^{00}=e(\mathit{a})/V$ from the viewpoint of symmetry, where ${e(a}_{1}{,a}_{2}{,a}_{3})=e(\mathit{a})$ is the finite change in the zero-point energy from the free-field case. The vacuum energy $e(\mathit{a})$ and the total vacuum force on the wall normal to the i direction, ${F}_{i}=\ensuremath{-}\ensuremath{\partial}e/\ensuremath{\partial}{a}_{i},$ are both homogeneous functions of the cavity dimensions. Because of this symmetry, the energy and forces are related by the equation $e(\mathit{a})=\mathit{a}\ensuremath{\cdot}\mathit{F}(\mathit{a}).$ We compute the vacuum forces and energy numerically for cavities with a broad range of dimensions. The implications of the perfect-conductor boundary conditions and the effects of the edges of the cavity are both considered. The ${C}_{3v}$ symmetry of the constant-energy surfaces is apparent. The zero-energy surface, which is invariant under dilations and therefore extends to infinity, separates the nested, concave, positive-energy surfaces from the open, negative-energy surfaces. The positive- (negative-) energy surfaces are mapped into each other by scale changes. The force $\mathit{F}(\mathit{a})$ is normal to the constant-energy surface at $\mathit{a}.$ The surfaces corresponding to zero forces, ${\mathit{F}}_{i}(\mathit{a})=0,$ are invariant under dilations and are therefore infinite. The zero-energy surface and the zero-force surfaces delineate the different geometries for which there are zero, one, or two negative (inward or attractive) forces on the cavity walls, along with the sign of the corresponding energy. There is no rectangular cavity geometry for which all forces are negative or zero; conversely, only geometries that are not too different from a cube have all positive (outward or repulsive) forces. Only for the last case is the energy $e(\mathit{a})$ necessarily positive. To provide an intuitive feeling for these vacuum energies, comparisons are made to other forms of energy in small cavities. We consider the energy balance for changes in cavity dimensions.
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Jan 1, 2008
Ryota Kusama + 3
Open Access
