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 = c²dt_C² − dx² − dy² − dz², where t_C = t + iτ; its real part carries 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 from the velocity constraint v² + V_τ² = c². The τ-direction momentum component P^τ = mc is invariant under Lorentz boosts; the five-dimensional null momentum condition yields E = mc² as a special case and the general energy-momentum relation E² = (pc)² + (mc²)². 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 rotation in the complex time plane. The distinction between matter and antimatter is identified with the orientation of τ rotation, offering a geometric basis for charge conjugation that preserves causal time order without invoking time reversal.