Transforming the Canonical Lagrangian of a Relativistic Particle for a Smooth Massless Limit: The Einbein Formulation in 1+1 Dimensions

Abstract:This tutorial derives, step by step and without omitting intermediate algebraic manipulations, the reformulation of the canonical square-root action for a relativistic point particle in a generic (1 + 1)-dimensional Lorentzian metric that permits a smooth limit m → 0. The key device is the introduction of an auxiliary world-line scalar density (the einbein). All Euler–Lagrange equations are computed explicitly, the algebraic solution for the einbein is substituted back into the action, and the recovery of the original square-root form is verified by direct calculation. The massless limit is then taken and shown to enforce the null condition while yielding the correct null-geodesic equation. The presentation is self-contained for a reader already familiar with the principles of general relativity, the Lagrangian formalism, and the Euler–Lagrange equations. References to standard textbooks and lecture notes are provided with author names, titles, publication years and, where available, stable URLs.