Origin of the Minkowski Metric from Complex Spacetime Manifold

We introduce a five-dimensional complex-time manifold in which standard four-dimensional spacetime is extended by a compactified imaginary time coordinate τ. The metric is defined as the holomorphic bilinear form: dZ² = c² dtC² − dx² − dy² − dz² where tC = t + iτ; its real part carries a five-dimensional Lorentzian signature and its imaginary part constitutes a symplectic form encoding quantum phase structure. Within this framework, the worldlines of both massive and massless particles satisfy a unified null geodesic condition with respect to the five-dimensional metric. Several results of special relativity are derived geometrically from this condition: Time Dilation: Follows directly from the velocity constraint v² + V_τ² = c². Invariant Momentum: The τ-direction momentum component P^τ = mc is strictly invariant under Lorentz boosts. Energy-Momentum Relation: The five-dimensional null momentum condition yields E = mc² as a special case, as well as the general relation E² = (pc) ² + (mc²) ². Furthermore, the τ = 0 and t = 0 sections of the metric recover Minkowski and Euclidean signatures respectively, providing a geometric interpretation of the Wick rotation as a literal rotation in the complex time plane. Finally, the distinction between matter and antimatter is identified with the orientation of the τ rotation, offering a geometric basis for charge conjugation (C) that preserves causal time order without invoking time reversal.