Quantum kicks near a Cauchy horizon

We analyse a quantum observer who falls geodesically towards the Cauchy horizon of a $(1+1)$-dimensional eternal black hole spacetime with the global structure of the non-extremal Reissner-Nordstr\"om solution. The observer interacts with a massless scalar field, using an Unruh-DeWitt detector coupled linearly to the proper time derivative of the field, and by measuring the local energy density of the field. Taking the field to be initially prepared in the Hartle-Hawking-Israel (HHI) state or the Unruh state, we find that both the detector's transition rate and the local energy density generically diverge on approaching the Cauchy horizon, respectively proportionally to the inverse and the inverse square of the proper time to the horizon, and in the Unruh state the divergences on approaching one of the branches of the Cauchy horizon are independent of the surface gravities. When the outer and inner horizons have equal surface gravities, the divergences disappear altogether in the HHI state and for one of the Cauchy horizon branches in the Unruh state. We conjecture, on grounds of comparison with the Rindler state in $1+1$ and $3+1$ Minkowski spacetimes, that similar properties hold in $3+1$ dimensions for a detector coupled linearly to the quantum field, but with a logarithmic rather than inverse power-law divergence.