# The Fermionic Signature Operator in the Exterior Schwarzschild Geometry

## TL;DR

This paper analyzes the Dirac equation's solution space in Schwarzschild spacetime, deriving a mass decomposition involving the fermionic signature operator, and computes its spectrum to understand quantum states near black holes.

## Contribution

It introduces a novel mass decomposition framework for solutions of the Dirac equation in Schwarzschild geometry, incorporating the fermionic signature operator and flux across the horizon.

## Key findings

- Spectrum of the fermionic signature operator is explicitly computed.
- Generalized fermionic projector states are characterized.
- A new mass decomposition formula involving flux terms is derived.

## Abstract

The structure of the solution space of the Dirac equation in the exterior Schwarzschild geometry is analyzed. Representing the space-time inner product for families of solutions with variable mass parameter in terms of the respective scalar products, a so-called mass decomposition is derived. This mass decomposition consists of a single mass integral involving the fermionic signature operator as well as a double integral which takes into account the flux of Dirac currents across the event horizon. The spectrum of the fermionic signature operator is computed. The corresponding generalized fermionic projector states are analyzed.