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

## TL;DR

This paper constructs a quantum state for a scalar field near the Cauchy horizon of a Reissner-Nordström black hole, ensuring it satisfies the Hadamard condition and providing a closed-form expression for the stress-energy tensor.

## Contribution

It introduces a novel quantum state construction on the Cauchy horizon by complexifying coordinates and enforcing periodicity, satisfying the Hadamard condition and explicitly calculating the stress-energy tensor.

## Key findings

- Quantum state satisfies the Hadamard condition on the Cauchy horizon.
- Explicit closed-form expression for the quantum stress-energy tensor.
- Construction method involves double analytic continuation and Euclidean periodicity.

## Abstract

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.