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

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 \( {r}_i^c \) = (3/2)r_H, where r_H is the horizon radius for which energy-density vanishes.