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

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This study efficiently derives a PDE for angular eigenvalues in the Kerr metric, proving ODEs are impossible for certain variables. New perturbative expansions and an asymptotic formula for Kerr naked singularities are presented.

Keywords:
Chandrasekhar-Page equationDirac equationKerr black holelinear hamiltonian systems

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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.