What is Energy? A Complex-Number Formulation

Axiom: E = γ_gen·mvc + i·mc², p = mv Definition: γ_gen = 1/√(g₀₀ − v²/c²) where g₀₀ is the time-time component of the spacetime metric and v is the coordinate velocity. The real part carries observable motional energy; the imaginary part carries hidden rest energy. The two are perpendicular in the complex plane: Re(E) ⊥ Im(E). The modulus yields the total energy: |E| = γ_gen·mc². This is verified directly from the gamma definition. The formalism is valid for all spacetimes: Minkowski, Schwarzschild, Reissner-Nordström, Kerr. The gamma definition follows from the Baran proper time convention: c²dT² = −g₀₀c²dt² + dξ², with dξ/dt := v and −dT²/dt² := 1/γ². This gives γ_gen naturally from the metric without additional postulates. The perpendicularity of rest energy and motional energy — analogous to resistance and reactance in electrical impedance Z = R + iX is proven rigorously: Re(γ_gen·mvc · conjugate(i·mc²)) = 0. The complex conjugate E* = γ_gen·mvc − i·mc² reverses the sign of the rest energy on the imaginary axis. One natural interpretation is that this corresponds to antimatter where the rest energy occupies the negative imaginary axis analogous to Dirac's negative-energy solutions. The product E·E* = (γ_gen·mc²)² then represents the annihilation energy. This interpretation is speculative and is noted here as an open question. All ideas, intuitions, and discoveries belong to Abdullah Baran - (Van Çatak). Typeset by Claude (Anthropic) acting as amanuensis. ⭐