An Energy-Conserving Unstaggered Electromagnetic-Potential Particle-in-Cell Method, Part II: Fully Relativistic Formulation with an Adapted Higuera--Cary Pusher

We develop a fully relativistic extension of the energy-conserving, unstaggered electromagnetic-potential particle-in-cell method introduced in our earlier nonrelativistic work. As in the nonrelativistic generalized-momentum formulation, the fields are advanced by a Crank–Nicolson discretization of the Lorenz-gauge potential equations, the current is deposited from particle orbits, and charge is advanced through the discrete continuity equation. The new ingredient is a generalized-momentum adaptation of the Higuera–Cary relativistic particle pusher. The vector-potential force is decomposed into an orbit-discrete-gradient part, which enforces the same finite-difference chain rule used in our earlier nonrelativistic formulation, and a skew part, which generates a Higuera–Cary magnetic rotation in mechanical momentum. For prescribed fields and frozen orbit data, this generalized-momentum Higuera–Cary map is a unit-Jacobian conjugacy of the mechanical Higuera–Cary map and preserves particle phase volume in $(\text{xx},\text{PP})$. For the fully coupled implicit step, the same orbit data define the current deposit, mesh-to-particle gather, vector-potential chain rule, and relativistic kinetic-energy secant velocity. The resulting particle work cancels the Crank–Nicolson field-energy balance, yielding exact relativistic total-energy conservation up to nonlinear solver tolerance, orbit-quadrature error, and roundoff. A self-adjoint Fourier projection removes modes on even-grid Nyquist planes while preserving the particle–grid work identity. Numerical tests are organized around cold relativistic two-stream and Weibel/filamentation instabilities, with diagnostics for energy drift, Gauss’s law, the Lorenz gauge, particle–grid work, and the orbit chain rule.