Mass of Covariant Systems in External Potentials

Starting from reparameterization invariant action principles, this thesis develops a fully covariant framework for treating the effect of external potentials on particle dynamics using mass-energy equivalence alone. In this framework, all effects of external forces are absorbed into a "proper mass" which corresponds to the total rest frame mass-energy and generates proper time evolution. Furthermore, the notion of inertial rest mass arises naturally as the four-dimensional Legendre transform of the proper mass, which we propose as a novel definition of inertia. Thus, there is a clear distinction between inertia and rest energy in the presence of external potentials. Moreover, it is argued that the very definition of a potential in a relativistic framework must be accompanied by a statement about the rate of change of inertial mass, and therefore that different types of potentials can be characterized by their effect on the inertial mass. As an example, it is shown that if the inertial mass is assumed constant, then any conservative three-force must in the covariant picture be derivable from a four-vector potential satisfying Maxwell's equations. This result is used to distinguish between two models of a relativistic harmonic oscillator found in the literature and clarifies their assumptions. Finally, some ambiguities in the definition of gravitational mass in the relativistic picture are discussed. Under simple assumptions, the gravitational mass is calculated and shown to equal the inertial mass in agreement with the equivalence principle. The framework thus provides a prescription to couple external potentials to gravity and might find uses in table-top experiments which aim to probe gravity on the quantum scale.