Motion of Spinless Particles in Gravitational Fields
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 subseq
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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 quantummechanical 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 weakfield 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.
A.J. Silenko (B) Research Institute for Nuclear Problems, Belarusian State University, 220030 Minsk, Belarus e-mail: [email protected] A.J. Silenko Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, Dubna 141980, Russia © Springer International Publishing Switzerland 2015 D. Puetzfeld et al. (eds.), Equations of Motion in Relativistic Gravity, Fundamental Theories of Physics 179, DOI 10.1007/978-3-319-18335-0_10
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1 Introduction The problem of quantum-mechanical description of a spinless particle in a Riemannian spacetime of general relativity (GR) has a long history. An appropriate form of the initial covariant Klein-Gordon (KG) equation [1–4] has been discovered fifty years ago by Penrose [5]. He has found an appropriate term describing a nonminimal coupling to the scalar curvature and conserving the conformal invariance of the equation for a massless scalar particle. Chernikov and Tagirov [6] have given clear explanations of this wonderful result. Their study involved the case of a nonzero mass and n-dimensional Riemannian spacetime. It is important that the inclusion of the Penrose-Chernikov-Tagirov term has been argued for both massive and massless particles [6]. Next step in investigation of the problem of conf
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