The Baran Complex Energy Law: A Complex-Plane Decomposition of Relativistic Energy

This work introduces the Baran Complex Energy Law, a decomposition of relativistic energy in the complex plane: E = (γ_gen − 1 + i) mc², i= √-1 The real part carries external (kinetic and gravitational) energy; the imaginary part carries rest energy mc², which is shown to reside naturally on the imaginary axis. The law is derived from the standard relativistic energy-momentum relation, holds for arbitrary gamma factors (Minkowski, Schwarzschild, Schwarzschild+velocity), and recovers classical mechanics in the Newtonian limit: E ≈ ½mv² + GMm/r + i·mc². Two theorems are proven: the total energy relation Re(E) + Im(E) = γ·mc², and the modulus formula. All ideas, mathematical content, and theoretical framework belong to Abdullah Baran(Van-Çatak). Claude (Anthropic) AI assistance was used for LaTeX typesetting, formatting, and mathematical verification.