The Fermionic Signature Operator in the Reissner-Nordstr\"om Geometry in Horizon-Penetrating Coordinates

We study the Dirac equation in the Reissner-Nordstr\"om geometry in horizon-penetrating coordinates up to the Cauchy horizon. A mass decomposition theorem is proved, which gives a covariant representation of the spacetime inner product that naturally involves the fermionic signature operator and the fermionic flux operator. We compute their spectra and show that both are bounded symmetric operators on the solution space $\mathcal{H}_m$ of the massive Dirac equation. The corresponding fermionic projector state is constructed and shown to satisfy the Hadamard condition. Lastly, we give some physical interpretations of the fermionic flux operator.