On the Eigenvalues of the Fermionic Angular Eigenfunctions in the Kerr Metric

Area of Science:

• Theoretical Physics
• General Relativity
• Quantum Field Theory

Background:

• The Dirac equation in the Kerr metric describes relativistic particles in a rotating black hole spacetime.
• Separation of variables in the Kerr metric leads to an angular equation with an eigenvalue problem.
• Deformation theory of linear Hamiltonian systems provides context for the eigenvalue problem.

Purpose of the Study:

• To efficiently derive a quasilinear first-order partial differential equation (PDE) for angular eigenvalues.
• To investigate the possibility of obtaining an ordinary differential equation (ODE) for eigenvalues.
• To construct new perturbative expansions and an asymptotic formula for eigenvalues in the Kerr metric, including Kerr naked singularities.

Main Methods:

• Revisiting the eigenvalue problem associated with the angular equation.
• Applying techniques from the deformation theory of linear Hamiltonian systems.
• Developing new perturbative expansion methods and asymptotic analysis.

Main Results:

• An efficient derivation of a quasilinear first-order PDE for angular eigenvalues is presented.
• It is proven that an ODE for eigenvalues cannot be obtained when particle energy or black hole mass is the independent variable.
• New perturbative expansions for eigenvalues in the Kerr case are constructed.
• An asymptotic formula for eigenvalues in the case of a Kerr naked singularity is obtained.

Conclusions:

• The study provides an efficient method for analyzing angular eigenvalues in the Kerr metric.
• The limitations of obtaining ODEs for these eigenvalues are clearly established.
• The developed perturbative expansions and asymptotic formula offer new tools for studying relativistic quantum phenomena in extreme gravitational environments.