A relativistic massive particle is described by a timelike worldline with proper-time four-velocity . In signature its normalization is and its momentum is . The relativistic charged-particle action adds electromagnetic coupling, whose variation gives the Lorentz force.
Use metric signature and an arbitrary increasing parameter on a massive particle's timelike curve. Write without yet imposing proper-time normalization. The free relativistic particle action isIts derivative with respect to is . The square root makes the action independent of the chosen increasing parameter. Varying it first preserves the nontrivial velocity dependence that produces the geodesic equation.[ref]
Lecture Notes on General Relativity by Sean M. Carroll. Chapter 3, equations 3.50–3.56, printed pp. 69–70
After variation, choose , so is the four-velocity, and . Substituting into the integrand before an unconstrained variation would instead make it look constant and lose the momentum equation. The choice of proper time must therefore follow the variation, or be handled by an equivalent constrained action.
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A relativistic particle refers to a particle that is moving at speeds close to the speed of light, where the effects of Einstein's theory of relativity become significant. In the realm of classical physics, particles are described by Newtonian mechanics, which assumes that velocities are much less than the speed of light. However, when particles approach relativistic speeds (typically a significant fraction of the speed of light, denoted as \(c\)), their behavior can no longer be accurately described by classical mechanics.