Hidden Momentum and the Absence of the Gravitational Spin Hall Effect in a Uniform Field

TL;DR

The paper analyzes whether a Dirac particle in a uniform gravitational field exhibits a spin-dependent transverse drift (the gravitational spin Hall effect). It combines a classical hidden-momentum model with a quantum Foldy–Wouthuysen treatment to show that, for uniform fields, spin-dependent motion does not appear at linear order in the gravitational acceleration $g$; such effects only arise at order $O(g^2)$ for a wide range of wave packets. A key finding is that prior claims hinge on inconsistent initial states and misidentifying canonical versus kinetic momentum. The work reinforces the equivalence principle in quantum spin dynamics and clarifies the conditions under which spin–gravity couplings can manifest, informing precision interferometry and polarization-dependent gravitational studies.

Abstract

We re-examine the recent claim that a Dirac particle freely falling in a uniform gravitational field exhibits a spin-dependent transverse deflection (gravitational spin Hall effect). Using a circulating mass model, we show that hidden momentum arises in uniform fields when an object carries angular momentum. On the quantum side, we analyze the Dirac Hamiltonian in a uniform potential, construct its Foldy--Wouthuysen form, and evaluate the Heisenberg evolution of spin-polarized Gaussian packets. The state used previously, with $\langle p\rangle =0$, is not at rest: because canonical and kinetic momenta differ, the packet carries a spin-dependent hidden momentum from $t=0$. Imposing $\langle x(0)\rangle =\langle v(0)\rangle=0$ requires a compensating spin-dependent $\langle p(0)\rangle$; with this preparation $\langle x(t)\rangle =0$ to leading order in the gravitational acceleration $g$. Generalizing, an exact Foldy--Wouthuysen transformation (linear in $g$ but to all orders in $1/c$) shows that spin-dependent transverse motion begins no earlier than at $O(g^2)$ for a broad class of wave packets. We conclude that a uniform field does not produce a gravitational spin Hall effect at linear order; the previously reported drift stems from inconsistent initial states and misinterpreting canonical momentum.

Figures (2)

• Figure S1: Two different types of trajectories for the motion of a particle with spin in a uniform gravitational field. (a) Free-fall motion with an initial horizontal velocity; (b) motion starting from rest, with the horizontal velocity developed through the gravitational spin Hall effect. Based on Equations (22)--(24) in Ref. Wang:2023bmd, the particle should follow a projectile trajectory as in (a); however, the graphic in that work shows trajectory (b), contradicting those equations.
• Figure S2: Masses circulating in a frictionless pipe bent into a rectangle. The rectangle is placed in the constant gravitational field near the Earth.