Thermal scalar field stress tensor on a two dimensional black hole and its near horizon properties

TL;DR

This work analyzes a massless scalar field at finite temperature in a $(1+1)$-D static black hole spacetime by computing the renormalized EMT and its trace anomaly, and by evaluating locally defined energy density and flux for static and freely-falling observers across Unruh, Boulware, and Hartle-Hawking thermal states. Using thermal Wightman functions on a conformally flat background, explicit EMT components are derived; horizon regularity requires the horizon temperature $T_H = \kappa/(2\pi)$ for Unruh and Boulware states, while Hartle-Hawking states are finite at the horizon for any $\beta$. In Schwarzschild spacetime, a freely-falling observer beginning at a radius $r_i$ exhibits a horizon energy density that vanishes at a critical radius $r_i^c = (3/2) r_H$, with the flux always nonnegative and vanishing at the horizon under equilibrium. The results illuminate how finite-temperature fields influence near-horizon quantum effects and horizon thermodynamics, and they generalize prior zero-temperature analyses to a broader thermal context.

Abstract

We calculate the thermal renormalized energy-momentum tensor components of a massless scalar field, leading to trace anomaly, on a $(1+1)$ dimensional static black hole spacetime. Using these, the energy density and flux, seen by both static and freely-falling observers, are evaluated. Interestingly for both these observers the aforementioned quantities in the thermal version of Unruh and Boulware states are finite at the horizon when the scalar field is in thermal equilibrium with the horizon temperature (given by the Hawking expression). Whereas in Hartle-Hawking thermal state both the observers see finite energy-density and flux at the horizon, irrespective of the value of field temperature. Particularly in the case of Schwarzschild spacetime a freely falling observer, starts with initial zero velocity, finds its initial critical position $r^c_i = (3/2)r_H$, where $r_H$ is the horizon radius for which energy-density vanishes.