Exact solution of Dirac–Rosen–Morse problem in curved space–time

Abstract

In this work, we extend the analysis of the relativistic Dirac–Rosen–Morse problem in curved space–time. For that, we consider the Dirac equation in curved space–time with line element d s2 = (1 + α2 U( r))2(d t2 − d r2) −  r2dθ2 −  r2sin 2θdϕ2, where α is fine structural constant, U( r) is a scalar potential, and in the presence of the electromagnetic field Aμ = ( V( r), cA( r), 0, 0). Because of the spherical symmetry, the angular spinor is given in terms of the spherical harmonics. For the radial spinor, we apply a unitary transformation and define the vector component of the electromagnetic field A( r) written as a function of V( r) and U( r) so as to solve the radial spinor for Dirac–Rosen–Morse problem. Graphical analyses were performed comparing the eigenenergies and the probability densities in curved and flat space–time to visualize the influence of curvature in space–time on the two-component radial spinor, with the upper and lower components representing the particle and antiparticle, respectively.