Regular quantum states on the Cauchy horizon of a charged black hole

Abstract

We consider the expectation value of the stress–energy tensor for a massless quantum scalar field near the Cauchy horizon interior to the Reissner–Nordström black hole spacetime. We construct the quantum state by considering the two-point function on a negative definite metric obtained by a double analytic continuation from the Lorentzian manifold, complexifying both the t and polar coordinates. We enforce periodicity in the Euclideanized t coordinate with periodicity equal to the reciprocal of the temperature of the Cauchy horizon, a necessary condition for avoiding a conical singularity at the inner horizon. We show by explicit construction that our quantum state satisfies the Hadamard condition on the Cauchy horizon. The expectation value of the quantum stress–energy tensor on the Cauchy horizon is given in closed form.