GEOMETRIZATION OF THE THEORY OF ELECTROMAGNETIC AND SPINOR FIELDS ON THE BACKGROUND OF THE SCHWARZSCHILD SPACETIME

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.


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]. Relevant spacetime models describe gravitational potentials of these objects. However, 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.
The Kosambi-Cartan-Chern geometrical approach (KCC-theory) was developed in details in numerous mathematical books and papers [3][4][5]. KCC-theory allowed to describe the evolution of a dynamical system in a configuration space of the Lagrange type. At that, the dynamical system is governed by the system of second-order differentials equations.

Results and discussion
where L is a Lagrangian function, G i 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. Explicitly, the second KCC-invariant can be calculated according to the formula In this work we apply the KCC-theory to study systems of differential equations which arise in the theory of electromagnetic and spinor fields on the background of the Schwarzschild spacetime.

Electromagnetic field
In [6] the Maxwell equations were considered on the background of the Schwarzschild spacetime where function 1 M r    , and the differential equation system for the radial components was derived after separating the variables in the initial Maxwell equations both in 3-dimensional Majorana-Oppenheimer and 10-demensional Duffin-Kemmer-Petiau approaches. We start with the second order differential equation for the primary radial function F that was obtained in Majorana-Oppenheimer formalism: The second KCC invariant is found in the form In Duffin-Kemmer formalism the equation for another primary radial function f is ([6]): where a derivative over r is denoted by a prime. Utilizing the notations x r M  , one finds the second KCC-invariant in the form: ( 1) j j   , (8) which coincides with (6).
Near singular points, the eigenvalue of the second invariant P   behaves as follows: It indicates that in the vicinity of x = 0 (this point is nonphysical) the geodesics converge. Vice versa, near the physical points, Schwarzschild horizon x = 1 and at x   , the Jacobi instability exists and the geodesics diverge. The typical behavior of the eigenvalue as a function of the radial coordinate is shown in Fig. 1, а. a b Fig. 1.

Spinor field
We consider the spin 1/2 particle on the background of the Schwarzschild spacetime. We start from a generally covariant form of the Dirac equation: After separating the variables with diagonalization of the total angular momentum [7,8] Each complex function ( ) i f r we resolved into a sum of real and imaginary parts: 3 5 6 f x ix   , 4 7 8 f x ix   .
By substituting expressions (11) into the system (10) one gets the system of 8 connected differential equation systems of the first order. To bring it to the second order differential system of the type (1) we differentiate each equation over the radial variable. After that the second invariant