The Time Fibre Bundle and the Geometric Interpretation of Special Relativity

Abstract

We introduce a five-dimensional Lorentzian manifold (Harvey space) in which standard four-dimensional spacetime is extended by a compactified imaginary-time dimension τ. Key Geometric & Physical Framework: Metric Formulation: Its metric is the real Lorentzian metric g = c² dt² − c² dτ² − dx² − dy² − dz² with signature (+, −, −, −, −) (the τ term carries the same negative sign as the three spatial terms). This metric is derived directly by rearranging the special-relativistic proper-time definition rather than introduced as an independent postulate. Interpretation of τ: Imaginary time τ is a genuine real coordinate—named following the historical Wick-rotation convention t → iτ—compactified onto a circle S¹. The intrinsic evolution of a particle is not a displacement of a coordinate around the τ-circle, but the overall precession of the entire τ-circle (S¹, on which all phases coexist simultaneously). Quantum Phase & Spacetime: Quantum phase is associated with a U(1) phase on the fibre τ ∈ S¹, while the four-dimensional spacetime we observe is the result of reducing (projecting out) this τ-fibre. Wick Rotation: Wick rotation is not a turning of a single time direction, but a switch of which direction is taken as vertical—exchanging the τ = const section (Lorentz) for the t = const slice (Euclidean)—whose essence is analytic continuation on the complexified (t, τ) plane. Unified Mechanics & Relativistic Derivations: Within this framework, the worldlines of both massive and massless particles satisfy a single unified null worldline condition with respect to this metric. For massive particles, this null worldline condition is precisely the rearrangement of the four-dimensional proper-time definition under the identification τ = τₚ, extending it universally to all particles (including massless ones). Kinematics & Energy-Momentum: Time dilation follows directly from the velocity constraint v² + Vτ² = c². Taken as a Lorentz-scalar internal momentum, the τ-direction component Pτ = mc is invariant under four-dimensional Lorentz boosts. The five-dimensional null worldline condition yields E = mc² as a special case and the general energy-momentum relation E² = (pc)² + (mc²)². Sections: The τ = const section reduces to the Minkowski metric, while the t = const section is negative-definite (Euclidean up to an overall sign). Matter & Antimatter: The distinction between matter and antimatter is proposed to correspond to the orientation of τ rotation, suggesting a geometric picture of charge conjugation that preserves causal time order without invoking time reversal. Keywords: Five-dimensional Lorentzian manifold, Null worldline, Special relativity, Imaginary time compactification, Wick rotation, Spontaneous symmetry breaking