Regular Quantum States on the Cauchy Horizon of a Charged Black Hole

We consider the quantum stress-energy tensor of a massless scalar field near the Cauchy horizon interior to the Reissner-Nordstr\"om 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.