Geometrization of the theory of electromagnetic and spinor fields on the background of the Schwarzschild spacetime

N G Krylova,V M Red’Kov

doi:10.35596/1729-7648-2021-19-8-26-30

Abstract

The geometrical Kosambi–Cartan–Chern approach has been applied to study the systems of differential equations which arise in quantum-mechanical problems of a particle on the background of non-Euclidean geometry. We calculate the geometrical invariants for the radial system of differential equations arising for electromagnetic and spinor fields on the background of the Schwarzschild spacetime. Because the second invariant is associated with the Jacobi field for geodesics deviation, we analyze its behavior in the vicinity of physically meaningful singular points r = M, ∞. We demonstrate that near the Schwarzschild horizon r = M the Jacobi instability exists and geodesics diverge for both considered problems.

Highlights

The behavior of material fields in the vicinity of cosmological objects such as black holes or neutron stars is of great interest [1, 2]
2L yi y j where L is a Lagrangian function, Gi is called a semispray. The properties of this dynamical system can be described in terms of five KCC geometrical invariants
From the physical point of view, the most interesting invariant is the second one P which is associated with the Jacobi field for geodesics deviation, so it indicates how rapidly different branches of the solution diverge from or converge to the intersection points

Summary

Introduction

The behavior of material fields in the vicinity of cosmological objects such as black holes or neutron stars is of great interest [1, 2]. Krylova N.G, Red’kov V.M. Geometrization of the theory of electromagnetic and spinor fields on the background of the Schwarzschild spacetime. The search for analytical solutions under the background of curved spacetimes remains to be a complicated problem that stipulates the development of other methods to analyze the behavior of the corresponding dynamical systems.

Results

Conclusion