Complex Numbers Physics - (Abdullah Baran 2026)

Z_emp := F_active + i · F_passive

Force impedance. F_active produces motion (real axis). F_passive is blocked, creates stress (imaginary axis). Both ≥ 0.

F_active — force that accelerates the object. Newton's F = ma uses only this.

F_passive — force that exists but cannot move the object. Stress, pressure, contact load. Newton discards this.

a_emp := Z_emp / m = F_active/m + i · F_passive/m

Impedance acceleration. Real part: Newton's acceleration. Imaginary part: passive acceleration.

|Z_emp| = sqrt(F_active² + F_passive²)

Total mechanical intensity. Active + passive together.

W_emp := W_active + i · W_passive

Impedance energy. Integral of Z_emp over displacement.

W_active = (1/2)mv²

Kinetic energy. Newton sees this.

W_passive = mc²γ

Rest mass energy (and gravitational potential). Newton discards this.

W_emp = pc + i · mc²γ

Full form in flat spacetime (Lorentz). p = mγv, γ = 1/√(1 − v²/c²).

1. Newton writes 0 = 0. This model sees everything.

When you press a wall with 100 N,  Newton says net force = 0, calculation done. But your muscles are working, the wall is under stress, energy is being transferred. The impedance model writes Z = 100i N — the internal stress is quantified. Zero net force does not mean zero physical effect.

2. Rest mass is entirely imaginary.

When you sit still, v = 0, p = 0. The impedance mass is m_emp = i·m. Your entire mass lives in the imaginary layer. Newton writes F = ma = 0 and cannot see you. But you exist. Your mass is real — it is moving through time at speed c. Mass is the temporal motion's correspondence.

E = i·mc², not E = mc².

Einstein gives the magnitude. This framework gives the location. Rest energy is imaginary — blocked, stored, not producing motion. It lives in the imaginary axis. |i·mc²| = mc², so Einstein is correct in magnitude. This framework tells you where it is.

4. The scale measures imaginary mass.

The scale reads the normal force N = mg — a passive, blocked force. In this model, the scale measures m_imaginary. We have never directly measured real mass. We only ever measure the imaginary component. A photon cannot be weighed because its imaginary mass is zero.

5. The speed of light barrier explained by conservation.

The hyperbolic conservation law: W_passive² − W_active² = m²c⁴ = constant. As v approaches c, W_active must go to infinity. But conservation forces W_passive to infinity too. The imaginary layer can never be fully emptied. That is why v = c is unreachable — not because of infinite force, but because of conservation.

6. The stationary object and the photon are mirror images.**

Stationary object: m_emp = i·m — pure imaginary, pure time, no space motion.

Photon: m_emp = E/c² — pure real, pure space, no time experience.

They are complementary extremes of the same complex structure.

7. W_emp is primary; Z_emp is its derivative.

Classical physics puts force first and derives work from it. This model reverses the hierarchy: W_emp is primary, Z_emp = dW_emp/dx is derived from it. Energy comes first. Force is what you get when you differentiate energy with respect to position. This aligns with Lagrangian and Hamiltonian mechanics, where energy is always the fundamental object.

Concept, vision, and physical interpretation: Abdullah Baran

Amanuensis: Claude (Anthropic), ChatGBT