Casimir phenomena in bumblebee gravity

Abstract

In this work, we analyze the Casimir effect associated with a massive, non-minimally coupled scalar field in static, spherically symmetric black hole spacetimes arising in bumblebee gravity. Three distinct solutions are considered, corresponding to different vacuum expectation value configurations of the Lorentz-violating vector field, including metric and \textit{metric-affine} scenarios. Finite-size effects are implemented through the Thermo Field Dynamics formalism by compactifying the radial direction, allowing the construction of renormalized vacuum expectation values of the energy-momentum tensor. Closed-form expressions for the Casimir energy and pressure are obtained in the massless limit as functions of the radial position of a spherical capacitor and the plate separation. Both observables depend explicitly on the bumblebee parameters and on the location of the apparatus relative to the horizon $R_0=2M$. In the weak-field regime, $r \gg R_0$, the standard flat-space behavior $E \propto -1/d^4$ is recovered. As $r \to R_0$, the Casimir energy vanishes while the radial pressure diverges. Inside the black hole, the interaction may alternate between attractive and repulsive regimes depending on the plate separation and on the Lorentz-violating couplings. A domain-dependent hierarchy among the three configurations emerges, with \textit{metric-affine} effects amplifying the interior vacuum energy, while configurations with simultaneous temporal and radial deformations dominate in the exterior region. Although all geometries share the same asymptotic Schwarzschild structure, their quantitative deviations become increasingly pronounced as the Lorentz-violating parameters grow.

Figures (5)

• Figure 1: Spherical capacitor placed at radial coordinate $r$ from the center of a bumblebee black hole of radius $R_0$. The two concentric shells are separated by a distance $d$. The Casimir effect manifests as an attractive force between the shells, generated by vacuum fluctuations of the confined scalar field modes within the intervening region.
• Figure 2: Casimir energy $E_i(r)$ as a function of the radial coordinate $r$ for the three bumblebee black hole geometries defined in Eqs. \ref{['eqE1']}--\ref{['eqE3']}. The parameters are fixed to $M=1$, $\xi=1/4$, and $\ell=X=\chi=0.3$. The curves correspond to plate separations $d=0.3$, $0.4$, $0.5$, and $0.6$.
• Figure 3: Casimir energy $E_i(d)$ as a function of the plate separation $d$ for fixed radial positions, with $M=1$, $\xi=1/4$, and $\ell=X=\chi=0.3$
• Figure 4: Casimir pressure $P_i(r)$ as a function of the radial coordinate $r$ for the three bumblebee geometries, with fixed plate separation $d$. The parameters are set to $M=1$, $\xi=1/4$, and $\ell=X=\chi=0.3$. As $r\to R_{0}=2M$, all configurations exhibit a divergence behavior.
• Figure 5: Casimir pressure $P_i(d)$ as a function of the plate separation $d$ for fixed radial distance $r$, considering $M=1$, $\xi=1/4$, and $\ell=X=\chi=0.3$.