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

TL;DR

This paper critically re-examines claims of a gravitational spin Hall effect for Dirac particles in uniform fields, demonstrating that previous observations were due to initial state inconsistencies and that no such effect occurs at linear order in gravity.

Contribution

The study clarifies the role of hidden momentum in uniform fields and shows that the gravitational spin Hall effect does not manifest at linear order, correcting prior misconceptions.

Findings

Hidden momentum arises in uniform fields with angular momentum.

Spin-dependent transverse motion appears only at second order in gravity.

Previous reports of the effect were due to initial state misinterpretation.

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.