A Five-Dimensional Lorentzian Manifold with a Compactified Imaginary-Time Dimension and the Geometry of Special Relativity

We introduce a five-dimensional Lorentzian manifold (Harvey space) in which standard four-dimensional spacetime is extended by a compactified imaginary-time dimension τ. Its metric is the real Lorentzian metric: g = c² dt² − c² dτ² − dx² − dy² − dz² with signature (+, −, −, −, −), where the τ term carries the same negative sign as the spatial terms. Imaginary time τ is treated as a genuine real coordinate—named following the historical Wick-rotation convention t → iτ—compactified onto a circle S¹; a complex structure on the (t, τ) plane renders the Wick rotation a literal rotation within that plane, while the quantum phase is carried by the U (1) holonomy on the fibre τ ∈ S¹. The central geometric construction imposes a single null geodesic condition on this metric: c² dt² − c² dτ² − |dr|² = 0 This places massive and massless particles on the exact same constraint. We emphasize that this is a postulated dynamical constraint rather than a metric classification. Taking this condition as an axiom, several foundational results of special relativity emerge naturally as geometric consequences: Time Dilation: Follows directly from the velocity constraint v² + V_τ² = c². Momentum Invariance: The τ-direction momentum component P_τ = mc remains 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. Wick Rotation: The τ = 0 and t = 0 sections of the metric recover Minkowski and Euclidean signatures respectively, providing a clear geometric interpretation of the Wick rotation as a rotation in the (t, τ) plane. Matter-Antimatter Asymmetry: The distinction between matter and antimatter is identified with the orientation of the τ rotation, offering a geometric picture of charge conjugation that preserves causal time order without invoking time reversal.