Coupling of Linearized Gravity to Nonrelativistic Test Particles: Dynamics in the General Laboratory Frame

TL;DR

This paper investigates how linearized gravity interacts with nonrelativistic test particles within a specially constructed laboratory frame, revealing new insights into tidal effects, Hamiltonian dynamics, and potential quantum interference experiments.

Contribution

It introduces a General Laboratory Frame for analyzing nonrelativistic particles in linearized gravity, clarifies the coupling of gravitational waves to matter, and proposes a gravitational Aharonov-Bohm experiment.

Findings

Hamiltonian reduces to geodesic motion for stationary metrics

Hamiltonian describes geodesic deviation in nonstationary metrics with GWs

Effective fields obey Maxwell-like equations and relate to tidal forces

Abstract

The coupling of gravity to matter is explored in the linearized gravity limit. The usual derivation of gravity-matter couplings within the quantum-field-theoretic framework is reviewed. A number of inconsistencies between this derivation of the couplings, and the known results of tidal effects on test particles according to classical general relativity are pointed out. As a step towards resolving these inconsistencies, a General Laboratory Frame fixed on the worldline of an observer is constructed. In this frame, the dynamics of nonrelativistic test particles in the linearized gravity limit is studied, and their Hamiltonian dynamics is derived. It is shown that for stationary metrics this Hamiltonian reduces to the usual Hamiltonian for nonrelativistic particles undergoing geodesic motion. For nonstationary metrics with long-wavelength gravitational waves (GWs) present, it reduces to the Hamiltonian for a nonrelativistic particle undergoing geodesic \textit{deviation} motion. Arbitrary-wavelength GWs couple to the test particle through a vector-potential-like field $N_a$, the net result of the tidal forces that the GW induces in the system, namely, a local velocity field on the system induced by tidal effects as seen by an observer in the general laboratory frame. Effective electric and magnetic fields, which are related to the electric and magnetic parts of the Weyl tensor, are constructed from $N_a$ that obey equations of the same form as Maxwell's equations . A gedankin gravitational Aharonov-Bohm-type experiment using $N_a$ to measure the interference of quantum test particles is presented.