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 ds² = (1+α² U(r))²(dt²- dr²) - r²dθ²-r²sin²θ dΦ², where α² is fine structural constant, U(r) is an scalar potential and in the presence of the electromagnetic field A_μ = (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 applying a unitary transformation and defining the vector component of the electromagnetic field A(r) written as a function of V(r) and U(r), so solve the radial spinor for Dirac-Rosen-Morse problem. Graphical analyzes were performed comparing the eigenenergies and the probability densities in curved and flat space-time in order 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.