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

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.