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 extends the 5D Myers-Perry black hole metric across horizons, separates the Dirac equation, analyzes asymptotic behaviors, and develops a spectral representation of the Dirac propagator using Hamiltonian formalism.

Contribution

It provides a detailed separation of variables for the Dirac equation in 5D Myers-Perry spacetime and constructs a spectral representation of the Dirac propagator.

Findings

Extended the metric through horizons using Eddington-Finkelstein coordinates

Separated the Dirac equation and analyzed radial asymptotics

Derived an integral spectral representation of the Dirac propagator

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 and R\"oken 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.