Invariant Descriptions of Classical Relativistic Particle Motion in 3+1 Dimensions

A Lorentz-invariant formalism for the classical dynamics of a free relativistic particle in 3+1 dimensions is developed within the framework of Lobachevsky hyperbolic geometry using basis gyrovectors. A modified metric representation of the Lorentz-invariant space-time coordinate is introduced, and the general solution for the projective coordinate function is derived via variation of variables, establishing the connection between the projective velocity components and their canonically conjugate variables through the angular integral of motion. The invariant projections of the particle velocity, momentum, energy, and coordinates onto three basis gyrovectors are obtained explicitly as functions of the rapidity. A systematic analysis reveals three distinct kinematic regimes for the transverse momentum: it is absent at low rapidities, real and diverging at an upper boundary in the intermediate regime, and transitions into the imaginary domain in the ultrarelativistic limit, indicating a change of geometry of momentum space. The projective velocity attains a maximum at a finite rapidity value — a physical analogue of the gyrosynchrotron radiation regime. The geometric structure of the particle trajectory in the coordinate plane is analyzed and its conformal mapping onto the unit circle is demonstrated. As a benchmark application, the formalism is applied to a relativistic charged particle in the field of a linearly polarized plane electromagnetic wave, reducing the derivation of transverse momentum to a single projection step versus four sequential steps required by the standard Hamilton–Jacobi approach. Results agree exactly with the known classical solution.