# The Dirac equation across the horizons of the 5D Myers–Perry geometry: separation of variables, radial asymptotic behaviour and Hamiltonian formalism

## TL;DR

This paper explores the behavior of the Dirac equation in a 5D rotating black hole geometry, focusing on variable separation and spectral representation of the Dirac propagator.

## Contribution

The paper introduces a novel Hamiltonian formalism and spectral representation for the Dirac propagator in a 5D Myers–Perry geometry.

## Key findings

- The Dirac equation is separated using an orthonormal frame formalism in the extended 5D Myers–Perry geometry.
- Asymptotic behavior of radial solutions is analyzed at horizons and infinity.
- An integral spectral representation of the Dirac propagator is derived using Green’s functions of the radial ODE.

## Abstract

We analytically extend the 5D Myers–Perry metric through the event and Cauchy horizons by defining Eddington–Finkelstein-type coordinates. Then, we use the orthonormal frame formalism to formulate and perform separation of variables on the massive Dirac equation, and analyse the asymptotic behaviour at the horizons and at infinity of the solutions to the radial ordinary differential equation (ODE) thus obtained. Using the essential self-adjointness result of Finster–Röken and Stone’s formula, we obtain an integral spectral representation of the Dirac propagator for spinors with low masses and suitably bounded frequency spectra in terms of resolvents of the Dirac Hamiltonian, which can in turn be expressed in terms of Green’s functions of the radial ODE.