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

We consider the Dirac equation coupled to an external electromagnetic field in curved four-dimensional spacetime with a given timelike worldline gamma representing a classical clock. We use generalised Fermi normal coordinates in a tubular neighbourhood of gamma and 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 of gamma, and angular velocity of rotation of the employed spatial reference frame along gamma. With respect to the time measured by the clock gamma, 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 of 1/c , corresponding to a 'slow-velocity' expansion with respect to gamma. 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.

Automated summary

This study investigates the Dirac equation coupled to an external electromagnetic field in a four-dimensional curved spacetime with a given timelike worldline representing a classical clock. By using generalised Fermi normal coordinates and performing expansions, a weak-gravity post-Newtonian expression for the Pauli Hamiltonian of a spin-half particle in an external electromagnetic field is derived.