Fermionic Casimir densities for a uniformly accelerating mirror in the Fulling-Rindler vacuum

TL;DR

The study addresses fermionic Casimir densities for a Dirac field in the Fulling-Rindler vacuum in the presence of a uniformly accelerated planar boundary obeying bag boundary conditions. By constructing the full set of fermionic modes in the RL and RR regions and applying a generalized Abel-Plana summation, the authors separate boundary-free and boundary-induced contributions to the fermion condensate and the energy-momentum tensor. They show that boundary-induced terms dominate near the mirror, while boundary-free terms control the behavior near the Rindler horizon; for massive fields, the boundary-induced pieces have region-dependent signs, and for massless fields the fermion condensate vanishes in $D eq1$ while the EMT remains nonzero. The results reveal key differences from Minkowski-space scenarios and provide explicit, convergent representations suitable for analytic and numerical analysis in arbitrary $(D+1)$ dimensions.

Abstract

We investigate the local characteristics of the Fulling-Rindler vacuum for a massive Dirac field induced by a planar boundary moving with constant proper acceleration in $(D+1)$-dimensional flat spacetime. On the boundary, the field operator obeys the bag boundary condition. The boundary divides the right Rindler wedge into two separate regions, called RL and RR regions. In both these regions, the fermion condensate and the vacuum expectation value (VEV) of the energy-momentum tensor are decomposed into two contributions. The first one presents the VEVs in the Fulling-Rindler vacuum when the boundary is absent and the second one is the boundary-induced contribution. For points away from the boundary, the renormalization is reduced to the one for the boundary-free geometry. The total VEVs are dominated by the boundary-free parts near the Rindler horizon and by the boundary-induced parts in the region near the boundary. For a massive field the boundary-free contributions in the fermion condensate and the vacuum energy density and effective pressures are negative everywhere. The boundary-induced contributions in the fermion condensate and the energy density are positive in the RL region and negative in the RR region. For a massless field the fermion condensate vanishes in spatial dimensions $D\geq 2$, while the VEV of the energy-momentum tensor is different from zero. This behavior contrasts with that of the VEVs in the Minkowski vacuum for the geometry of a boundary at rest relative to an inertial observer. In the latter case, the fermion condensate for a massless field is nonzero, while the VEV of the energy-momentum tensor becomes zero.

Figures (4)

• Figure 1: The coordinate lines $\tau =\mathrm{const}$ and $\rho =\mathrm{const}$ in the R region of the Rindler spacetime on the plane $(x^{1},t)$. The worldline of a uniformly accelerating mirror is depicted by a thick line. It separates the RL and RR regions.
• Figure 2: The fermion condensate for spatial dimension $D=3$ in the RR and RL regions versus $\rho /\rho _{0}$ (left panel) and $m\rho _{0}$ (right panel). The graphs on the left panel are plotted for $m\rho _{0}=0.5$ and the dashed curve presents the condensate in the boundary-free geometry. The numbers near the curves on the right panel present the values of $\rho /\rho _{0}$.
• Figure 3: The energy density in the Fulling-Rindler vacuum for the $D=3$ Dirac field as a function of $\rho /\rho _{0}$ (left panel) and $m\rho _{0}$ (right panel). For the graphs on the left panel we have taken $m\rho _{0}=0.5$ and the numbers near the curves on the right panel correspond to the values of $\rho /\rho _{0}$. The dashed curve on the left panel presents the energy density in the boundary-free problem.
• Figure 4: The same as in Fig. \ref{['fig3']} for the stresses parallel and perpendicular to the boundary (full and dashed curves, respectively). The dot-dashed curve on the left panel corresponds to the stress in the boundary-free problem.