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 $⟨p⟩=0$, 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 $O(g^2)$ 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.

Invited contribution to the Universe Special Issue Geometric Theories of Gravity

References in corpus (3)

• Quantum spin Hall effect of light
• Recent progress on the description of relativistic spin: vector model of spinning particle and rotating body with gravimagnetic moment in General Relativity
• Relativistic effects due to gravimagnetic moment of a rotating body