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. 1. Geometric Foundations & Metric Metric Definition: The framework posits a real Lorentzian metric with signature (+, −, −, −, −): g = c² dt² − c² dτ² − dx² − dy² − dz² (Note: The dτ² term carries the same negative sign as the spatial terms). The τ-Coordinate: Imaginary time τ is treated as a genuine real coordinate—the name follows the historical Wick-rotation convention t → iτ—compactified onto a circle S¹. Intrinsic Evolution: The motion of a particle is not a displacement of a coordinate around the τ-circle, but rather the overall precession of the entire τ-circle (S¹, on which all phases coexist simultaneously) as real time t advances. The quantum phase is associated with a U (1) phase on the fibre τ ∈ S¹, while our observable 4D spacetime is the result of reducing (projecting out) this τ-fibre. Wick Rotation: Geometrically, the Wick rotation is interpreted not as the turning of a single time direction, but as a 90° rotation of the physical section within the (t, τ) plane, mapping from the τ = const (Lorentz) section to the t = const (Euclidean) section. 2. The Null Worldline Axiom The central geometric construction imposes a single null worldline condition on this metric as a postulated dynamical constraint: c² dt² − c² dτ² − |dr|² = 0 This constraint places both massive and massless particles under the same geometric condition, rather than acting as a metric classification. 3. Emergent Relativistic Consequences Taking the null worldline condition as an axiom, several foundational results of special relativity emerge as direct geometric consequences: Time Dilation: Follows naturally from the velocity constraint v² + v_τ² = c². Mass as Invariant Momentum: Taken as a Lorentz-scalar internal momentum, the 𝜏-direction component Pᵀ = mc (contravariant component Pⁿ) is strictly invariant under four-dimensional Lorentz boosts. Energy-Momentum Dispersion: The 5D single null worldline condition yields E = mc² as a special rest case, and rigorously derives the general dispersion relation: E² = (pc) ² + (mc²) ² Sectional Duality: The τ = const section carries the standard Minkowski metric, while the t = const section is negative-definite (Euclidean up to an overall sign). Matter-Antimatter Asymmetry: The distinction between matter and antimatter is proposed to correspond to the orientation of the τ rotation, suggesting a geometric picture of charge conjugation (C) that preserves causal time order without invoking time reversal (T).