Two-point function of a quantum scalar field in the interior region of a Reissner-Nordstrom black hole

Abstract

We derive explicit expressions for the two-point function of a massless scalar field in the interior region of a Reissner-Nordstrom black hole, in both the Unruh and Hartle-Hawking quantum states. The two-point function is expressed in terms of the standard $lm\omega$ modes of the scalar field (those associated with a spherical harmonic $Y_{lm}$ and a temporal mode $e^{-i\omega t}$), which can be conveniently obtained by solving an ordinary differential equation, the radial equation. These explicit expressions are the internal analogs of the well known results in the external region (originally derived by Christensen and Fulling), in which the two-point function outside the black hole is written in terms of the external $lm\omega$ modes of the field. They allow the computation of $<\Phi^{2}>_{ren}$ and the renormalized stress-energy tensor inside the black hole, after the radial equation has been solved (usually numerically). In the second part of the paper, we provide an explicit expression for the trace of the renormalized stress-energy tensor of a minimally-coupled massless scalar field (which is non-conformal), relating it to the d'Alembertian of $<\Phi^{2}>_{ren}$. This expression proves itself useful in various calculations of the renormalized stress-energy tensor.