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

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.