Geometric post-Newtonian description of massive spin-half particles in curved spacetime

Abstract

AbstractWe consider the Dirac equation coupled to an external electromagnetic field in curved four-dimensional spacetime with a given timelike worldlineγrepresenting a classical clock. We use generalised Fermi normal coordinates in a tubular neighbourhood ofγand expand the Dirac equation up to, and including, the second order in the dimensionless parameter given by the ratio of the geodesic distance to the radii defined by spacetime curvature, linear acceleration ofγ, and angular velocity of rotation of the employed spatial reference frame alongγ. With respect to the time measured by the clockγ, we compute the Dirac Hamiltonian to that order. On top of this ‘weak-gravity’ expansion we then perform a post-Newtonian expansion up to, and including, the second order of1/c, corresponding to a ‘slow-velocity’ expansion with respect toγ. As a result of these combined expansions we give the weak-gravity post-Newtonian expression for the Pauli Hamiltonian of a spin-half particle in an external electromagnetic field. This extends and partially corrects recent results from the literature, which we discuss and compare in some detail.