Complex Number Energy

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. All ideas, intuitions, and discoveries belong to Abdullah Baran - Van Çatak. Typeset by Claude (Anthropic) acting as amanuensis. In electrical engineering, complex power S = P + iQ separates active power (real, consumed, work-performing) from reactive power (imaginary, stored, non-working). This work extends the same mathematical structure to energy in mechanics and relativistic physics.The central definition is:A = E_active + i·E_reactivewhere E_active is flowing, observable, work-performing energy, and E_reactive is stored, isolated, inaccessible energy. The operator i is interpreted as an isolation operator — it rotates energy by 90 degrees onto the imaginary axis, making it inaccessible, just as 230i volts oscillates electrons without performing net work.Two axioms are established. First, A = E_active + i·E_reactive. Second, E_active + E_reactive = γ_gen·mc², where γ_gen is the Baran generalized Lorentz factor derived from the spacetime metric.Special cases: In relativistic physics, E_active = (γ−1)mc² and E_reactive = mc², giving A = mc²(γ−1+i). Mass is the energy storage — when m = 0 (photon), E_reactive = 0 and A = pc, fully active. In mechanics, A = (1/2)mv² + i·mgh, where kinetic energy flows and potential energy is stored.The differential equation dA/dt = (i−1)·dE_reactive/dt follows directly from Newton's second law and energy conservation — this is not an ansatz. The modulus |A| = η·mc² measures energy balance: minimum at resonance (E_active = E_reactive), maximum when all energy is on one axis.The conjugate A* = E_active − i·E_reactive reverses the imaginary part, with a speculative interpretation as antimatter.To the author's knowledge, extending the active-reactive structure from electrical power to mechanical and relativistic energy in this explicit complex form has not previously appeared in the literature. All ideas belong to Abdullah Baran. Typeset by Claude (Anthropic) acting as amanuensis.