Tensor Geometry and General Relativistic Field Equation Derivation of Taiji Reciprocal Dual Mapping

Abstract

This study establishes the rigorous algebraic structure of the Taiji Reciprocal Dual Mapping based on the fundamental axiom that treats "original" (x_☉) and "dual" (r_⊗) coordinates as pure spacetime position quantities. By deriving a closed dual kernel between the original coordinate x_☉ and the dual coordinate r_⊗, we further compute the infinitesimal differential transformation and substitute it into the flat Friedmann–Lemaître–Robertson–Walker (FLRW) metric. The study successfully constructs a geometric metric tensor g_μν^(Taiji) incorporating the Taiji spatial conformal factors Φ_rr and Φ_Ω. On this basis, we derive the complete Christoffel symbols, Ricci tensor, Ricci scalar, and Einstein tensor (G_μν). Finally, under the assumption of no dark energy (Λ = 0), the covariant conservation law yields an effective Taiji geometric energy density ρ_Taiji ∝ (1+z)^(0.5) (corresponding to a geometric topological index γ = 0.5) and an effective geometric equation of state parameter w_eff = -5/6 ≈ -0.833. This derivation completely endows the accelerated expansion of the universe with a rigorous origin in differential geometry and general relativistic tensor physics.