Motion of Spinless Particles in Gravitational Fields

Abstract

The covariant Klein-Gordon equation describing a scalar particle in a Riemannian spacetime of general relativity is transformed to a Hamiltonian form by the generalized Feshbach-Villars method applicable for both massive and massless particles. The subsequent Foldy-Wouthuysen (FW) transformation allows to derive the relativistic FW Hamiltonian for a wide class of inertial and gravitational fields and find the new manifestation of conformal invariance for a massless scalar particle. Similarity of manifestations of conformal invariance for massless scalar and Dirac particles is proved. New exact FW Hamiltonians are derived for both massive and massless scalar particles in a general static spacetime and in a frame rotating in the Kerr field approximated by a spatially isotropic metric. The latter case covers an observer on the ground of the Earth or on a satellite and takes into account the Lense-Thirring (LT) effect. High-precision formulas are obtained for an arbitrary spacetime metric. General quantum-mechanical equations of motion are derived. Their classical limit coincides with corresponding classical equations. The quantum-mechanical description of the relativistic LT effect is presented. The exact evolution of the angular momentum operator in the Kerr field approximated by a spatially isotropic metric is found. The quantum-mechanical description of the full LT effect based on the Laplace-Runge-Lenz vector is given in the nonrelativistic and weak-field approximation. Relativistic quantum-mechanical equations for the velocity and acceleration operators are obtained. The equation for the acceleration defines the Coriolis-like and centrifugal-like accelerations and presents the quantum-mechanical description of the frame-dragging effect.