Geometry of Multi-Temporal Space-Time: A Unified Model of Subluminal, Luminal, and Superluminal Regimes

¹ University of Engineering & Management Jaipur, India
² Institute of Engineering and Management, Kolkata, India

^* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.

Published online: 29 May 2026

Abstract

Previous proposals on higher temporal dimensions, like String theory, Kaluza-Klein, multi-time cosmologies etc. treat additional axes of time differently (rather specially) from the one that we perceive. They are either tied to perception, entropy, metaphysics, or curved at a quantum level. They are not considered as isotropic or interchangeable. In this proposition, we assume that each temporal axis is physically equivalent i.e. the geometry of time obeys the same tensor symmetries that govern space. We define three kinematic classes of an entity: subluminal, luminal, and super-luminal. We propose a unified framework where the dimensional structure of space-time depends on the three classes. A smooth scalar field ϕ selects the effective dimensional pair (D_s, D_t) that explains the number of spatial and temporal axes. A canonical projection-penalization term enforces the corresponding local signature in the tetrad fields. In the subluminal regime, observers experience the usual (3 + 1) dimensional geometry. In the luminal case, entities travelling at c, they occupy an intrinsic (2 + 2) dimensional regime with two spatial and two temporal coordinates. In this case, a photon is represented by a null world-sheet Σγ(σ, τ) ⊂ ℝ²⁺² satisfying intrinsic null constraints γ_AB∂_αX^A∂_αX^B = 0. We show that the wave-like behaviour of a photon arises from the projection of intrinsic 2D — time oscillations into a 1D — time observational frame. Furthermore, entities in the super-luminal regime are governed by (1 + 3) dimensional system where space-like and time-like dimensions reverse or swap functions. This framework provides a mathematically consistent reformulation of photon kinematics and shows a future path toward a unified geometric theory of subluminal, luminal and superluminal states and higher dimensions of time.