Casimir Effect and Gravitational Balance: a Search for Stable Configurations

TL;DR

This work investigates whether repulsive Casimir forces can counteract gravitational contraction of a thin, self-gravitating spherical shell in the non-relativistic, weak-field limit. By analyzing massless and massive scalar fields, temperature-dependent Casimir energies, and electromagnetic Casimir effects, the study derives the shell's EOM via Israel-like matching and investigates the existence of a stable rest radius. The results show no robust stable oscillatory configuration (γ) in most cases; metastable stability can occur for certain massive-scalar or low-temperature massless-scalar scenarios, but generally stability requires restrictive conditions and scales beyond typical macroscopic regimes. Overall, within the modeled framework and assumptions, Casimir forces do not provide a generic mechanism for stable gravitational balance in this weak-field setup, highlighting limits of vacuum-energy stabilization in classical, low-curvature scenarios.

Abstract

In this study, we examine the role of the repulsive Casimir force in counteracting the gravitational contraction of a thin spherically symmetric shell. Our main focus is to explore the possibility of achieving a stable balanced configuration within the theoretically reliable weak field limit. To this end, we consider different types of Casimir forces, including those generated by massless scalar fields, massive scalar fields, electromagnetic fields, and temperature-dependent fields.

Figures (6)

• Figure 1: Dimensionless acceleration $\ddot R/m_p$ as a function of dimensionless radius $R m_p$ for (\ref{['eq_MasterM0']}). The parameter $C$ is chosen to be $0.0028$. The blue and orange curves are for values of $m_S=0.034\, m_p,$and $m_S=0.04 \,m_p$ respectively.
• Figure 2: Casimir force in a spherical shell produced by a massive scalar field. The points are the numerical results given in Bordag:2001qi. The continuous curve is the fit given in equation (\ref{['eq_fitREm']}).
• Figure 3: Dimensionless acceleration $\ddot R/m_p$ (\ref{['eq_ddRm']}) as a function of $R m_p$ for $m_S=0.034\, m_p$. The blue curve is for $m_\phi=2.9 \,m_p$. The green curve is for $m_\phi=3.455\, m_p$. The orange curve is for $m_\phi=3.7 \,m_p$. The zeros of these curves are the solutions of $\ddot R=0$. Only the blue curve has a meta-stable configuration. Interestingly, this meta-stable configuration establishes approximately at a radius which is close to the Compton wavelength of the scalar $\lambda\approx 1/m_\phi$, which is for any known particle much smaller than the physical radius of a macroscopic or micro meter sphere.
• Figure 4: Value of the roots $T R_i$ as a function of $m_S/m_p$, where $m_p=1/\sqrt{G_N}$. The black line is the solution $R_1$, the blue line is the stable solution $R_3$. The solution $R_2$ is only plotted as dotted line, because for $m_S^2<G_N$ it is negative and for $m_S^2>G_N$ it is imaginary. The green lines are $1/(m_S 200 ), \, 1/(m_S 20), 1/(m_S2)$. They represent the underlying criterium of large radius expansion, namely that we only should trust scenarios below the green line which means temperatures below the Planck energy.
• Figure 5: Dimensionless acceleration $\ddot R/m_p$ as a function of the dimensionless radius $R m_p$ as given from equation (\ref{['eq_scalarLowTemp']}). Different colors correspond to different temperatures: blue ($T=0.01\, m_p$,), green ($T=0.1\, m_p$), and orange ($T=0.15\, m_p$).