Some exact solutions of the Dirac equation in gravitational fields

Abstract

Some exact solutions of the general covariant Dirac equation in gravitational fields, namely in constant curvature de Sitter spacetime and in Milne's Universe are obtained. The tetrads for both cases are constructed on the basis of global quasi-Cartesian coordinates that allow one to exclude coordinate effects connected with the rotation of the local frame under the transition of the triad from one spacetime point to another. Solutions in the form of a spherical bispinor wave with time amplitude modulation are demonstrated for both cases of gravitational fields and the possibility of analogical solutions in the form of cylindrical and plane spinor waves for the de Sitter spacetime is discussed. All the solutions are obtained by means of the algebraic method of separation of variables.

Polarized Dirac fermions in de Sitter spacetime

“…As suggested in [2], the plane wave solutions of the Dirac equation with m = 0 must be eigenspinors of the momentum operators P i corresponding to the eigenvalues p i , with the same time modulation as the spherical waves. Therefore, we have to look for particular solutions in the chart {t c , x} involving either positive frequency plane waves or negative frequency ones.…”

“…According to the general properties of the Hankel functions [12], we deduce that those used here, H (1,2) ν ± (z), with ν ± = 1 2 ± ik and z ∈ R, are related among themselves through [H (1,2) ν ± (z)] * = H (2,1)…”

“…In what follows we study the Dirac field in the chart {t, x} using the conformal time as a helpful auxiliary ingredient. The form of the line element (21) allows one to choose the simple Cartesian gauge with non-vanishing tetrad components [2] …”

“…The Dirac equation in de Sitter spacetime (of radius R = 1/ω = 3/Λ c , produced by the cosmological constant Λ c ) was studied in moving or static local charts (i.e., natural frames) suitable for separation of variables that should lead to significant analytical solutions [2,3]. In [2] spherical wave solutions were derived in the local chart {t, r, θ, φ} commonly associated to the Cartesian one {t, x} having the line element…”

“…In [2] spherical wave solutions were derived in the local chart {t, r, θ, φ} commonly associated to the Cartesian one {t, x} having the line element…”