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

TL;DR

This paper computes the thermal stress tensor of a massless scalar field on a 2D black hole, analyzing its properties near the horizon for different states and observers, revealing finiteness and specific initial conditions.

Contribution

It provides a detailed calculation of the thermal stress tensor and examines its near-horizon behavior for various quantum states and observer trajectories in a 2D black hole.

Findings

Energy density and flux are finite at the horizon in thermal equilibrium states.

In Hartle-Hawking state, both static and free-falling observers see finite quantities at the horizon.

A free-falling observer starting at a critical radius finds zero energy density at that point.

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.