TY - JOUR

T1 - Geometric post-Newtonian description of massive spin-half particles in curved spacetime

AU - Alibabaei, Ashkan

AU - Schwartz, Philip K.

AU - Giulini, Domenico

N1 - This work was supported by the Deutsche Forschungsgemeinschaft via the Collaborative Research Centre 1227 'DQ-mat'—Project Number 274200144, Project A05. A A acknowledges funding from Trinity College, Cambridge via a Rouse Ball Travelling Studentship.

PY - 2023/11/7

Y1 - 2023/11/7

N2 - 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.

AB - 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.

KW - quantum matter in gravity

KW - post-Newtonian expansion

KW - Dirac equation

KW - generalised Fermi normal coordinates

KW - formal WKB-like expansion

UR - http://www.scopus.com/inward/record.url?scp=85177493671&partnerID=8YFLogxK

U2 - 10.1088/1361-6382/ad079c

DO - 10.1088/1361-6382/ad079c

M3 - Article

VL - 40

JO - Classical and Quantum Gravity

JF - Classical and Quantum Gravity

SN - 0264-9381

IS - 23

M1 - 235014

ER -