Local Dirac energy decay in the 5D Myers-Perry geometry using an integral spectral representation for the Dirac propagator

TL;DR

This paper develops an integral spectral representation for the Dirac equation in a 5D black hole spacetime and proves that Dirac particles disperse over time, with their probability of remaining in any compact region tending to zero.

Contribution

It introduces a novel spectral representation for the Dirac propagator in 5D Myers-Perry geometry and establishes decay properties of Dirac particles in this setting.

Findings

Dirac particle probability decays to zero in any compact region as time approaches infinity.

Constructed an integral spectral representation for the Dirac equation in 5D Myers-Perry spacetime.

Extended decay results known in 4D Kerr-Newman geometry to 5D Myers-Perry case.

Abstract

We consider the massive Dirac equation in the exterior region of the 5-dimensional Myers-Perry black hole. Using the resulting ODEs obtained from the separation of variables of the Dirac equation, we construct an integral spectral representation for the solution of the Cauchy problem with compactly supported smooth initial data. We then prove that the probability of presence of a Dirac particle to be in any compact region of space decays to zero as $t\to\infty$, in analogy with the case of the Dirac operator in the Kerr-Newman geometry.