Geometric Time and Causal Time in Relativistic Lagrangian Mechanics

TL;DR

This paper distinguishes geometric and causal times in relativistic physics, showing that parameterization independence of Lagrangians requires homogeneity in velocity, and introduces ageodesicity to analyze paths like in the twin paradox.

Contribution

It clarifies the roles of geometric and causal times in relativistic Lagrangian mechanics and introduces the concept of ageodesicity for path analysis.

Findings

Homogeneity in velocity ensures Lagrangian independence from causal time.

Parameterization-independent Lagrangians are covariant and homogeneous.

Ageodesicity quantifies deviation from geodesic paths, aiding in twin paradox analysis.

Abstract

In this article, we argue that two distinct types of time should be taken into account in relativistic physics: a geometric time, which emanates from the structure of spacetime and its metrics, and a causal time, indicating the flow from the past to the future. A particularity of causal times is that its values have no intrinsic meaning, as their evolution alone is meaningful. In the context of relativistic Lagrangian mechanics, causal times corresponds to admissible parameterizations of paths, and we show that in order for a langragian to not depend on any particular causal time (as its values have no intrinsic meaning), it has to be homogeneous in its velocity argument. We illustrate this property with the example of a free particle in a potential. Then, using a geometric Lagrangian (i.e. a parameterization independent Lagrangian which is also manifestly covariant), we introduce the notion of ageodesicity of a path which measures to what extent a path is far from being a geodesic, and show how the notion can be used in the twin paradox to differentiate the paths followed by the two twins.