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

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 ⟨p⟩=0, is not at rest: because canonical and kinetic momenta differ, the packet carries a spin-dependent hidden momentum from t=0. Imposing ⟨x(0)⟩=⟨v(0)⟩=0 requires a compensating spin-dependent ⟨p(0)⟩; with this preparation ⟨x(t)⟩=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(g2) 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.