Stefan-Boltzmann Law and Thermal Casimir Effect in Neutron Star Spacetime via Thermo Field Dynamics

TL;DR

This work analyzes the thermal Casimir effect for a massless scalar field in neutron-star spacetimes using Thermo Field Dynamics, deriving a generalized Stefan–Boltzmann law that incorporates gravitational redshift and TOV-curvature corrections. It provides closed-form expressions for the Casimir energy and pressure with simultaneous temporal and spatial compactifications, covering the NS interior, exterior Schwarzschild region, and flat space limits. The study systematically explores high- and low-temperature regimes, showing that strong gravity and spacetime curvature modify the usual T^4 scaling and suppress thermal Casimir contributions in deep interiors, with the non-minimal coupling ξ controlling the magnitude and sign. A polytropic equation of state is used to obtain NS profiles for numerical analysis, linking quantum vacuum fluctuations to neutron-star thermodynamics and suggesting avenues for future investigations of massive fields and astrophysical implications.

Abstract

We investigate the thermal Casimir effect for a massless scalar field in the curved spacetime of a neutron star within the Thermo Field Dynamics (TFD) formalism. Starting from the renormalized energy-momentum tensor, we generalize the Stefan-Boltzmann law to include gravitational redshift and curvature corrections governed by the Tolman-Oppenheimer-Volkoff (TOV) metric. Finite temperature and spatial compactification are introduced simultaneously, allowing a unified and consistent treatment of both vacuum and thermal contributions inside and outside the star. Analytical expressions are derived for the high- and low-temperature limits, showing explicitly how curvature and redshift modify the characteristic $T^4$ dependence of thermal radiation. The results reveal that strong gravity significantly alters the local energy density and pressure, demonstrating the nontrivial interplay between quantum vacuum fluctuations and compact astrophysical geometries. A polytropic model is considered to perform numerical analyses, highlighting the influence of the spacetime background on vacuum fluctuations.

Figures (4)

• Figure 1: Radial profile of the energy-momentum tensor component $T^0_0(r)$ for a massless scalar field in thermal equilibrium, plotted for both the neutron star interior (solid lines) and the exterior Schwarzschild region (dashed lines) by a fixed temperature $T$, considering the minimal coupling $(\xi=0)$ and conformal coupling $(\xi=1 / 6)$. The dotted lines correspond to the flat spacetime Stefan-Boltzmann limit.
• Figure 2: The left panel shows the renormalized Casimir energy density $\mathcal{E}_0(a;r)$ for a massless scalar field inside and outside a neutron star, with plate separation $a = 0.5$. The right panel shows the behaviour of the Casimir pressure with $a=0.5$ by considering the minimal coupling case ($\xi = 0$), the conformal coupling ($\xi = 1/6$) and the flat spacetime result for each case.
• Figure 3: Thermal behaviour of the renormalized Casimir energy density (left panel) and radial pressure (right panel) inside the neutron star interior at a fixed radius $r = 4.793$. Both quantities are rescaled by $a^{4}$ and plotted as functions of the redshift–corrected combination $2aT\,e^{-\Phi(r)} g_{m}(r)^{-1/2}$, which is the natural local temperature parameter in the TOV background.
• Figure 4: High-temperature behaviour of the renormalized Casimir energy density (left panel) and Casimir radial pressure (right panel) inside the neutron-star interior at the fixed radius $r = 4.793$. Both observables are rescaled by $a^{4}$ and plotted as functions of the redshift-corrected temperature parameter $2aT\,e^{-\Phi(r)} g_{m}(r)^{-1/2}$.