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

TL;DR

This paper derives a post-Newtonian approximation of the Dirac equation for spin-half particles in curved spacetime, incorporating electromagnetic interactions and corrections due to gravity and acceleration, extending previous results.

Contribution

It provides a detailed weak-gravity, post-Newtonian expansion of the Dirac Hamiltonian in curved spacetime, improving and correcting earlier models.

Findings

Derived the Dirac Hamiltonian up to second order in curvature and acceleration.

Extended the post-Newtonian approximation to include electromagnetic interactions.

Corrected and compared with recent literature on the subject.

Abstract

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.