Invariant definition of rest mass and dynamics of particles in 4D from bulk geodesics in brane-world and non-compact Kaluza-Klein theories

Abstract

In the Randall-Sundrum brane-world scenario and other non-compact Kaluza-Klein theories, the motion of test particles is higher-dimensional in nature. In other words, all test particles travel on five-dimensional geodesics but observers, who are bounded to spacetime, have access only to the 4D part of the trajectory. Conventionally, the dynamics of test particles as observed in 4D is discussed on the basis of the splitting of the geodesic equation in 5D. However, this procedure is {\em not} unique and therefore leads to some problems. The most serious one is the ambiguity in the definition of rest mass in 4D, which is crucial for the discussion of the dynamics. We propose the Hamilton-Jacobi formalism, instead of the geodesic one, to study the dynamics in 4D. On the basis of this formalism we provide an unambiguous expression for the rest mass and its variation along the motion as observed in 4D. It is independent of the coordinates and any parameterization used along the motion. Also, we are able to show a comprehensive picture of the various physical scenarios allowed in 4D, without having to deal with the subtle details of the splitting formalism. Moreover we study the extra non-gravitational forces perceived by an observer in 4D who describes the geodesic motion of a bulk test particle in 5D. Firstly, we show that the so-called fifth force fails to account for the variation of rest mass along the particle's worldline. Secondly, we offer here a new definition that correctly takes into account the change of mass observed in 4D.