SO(16,3) Symmetry: Unification of Gravity and Gauge Forces

This paper proposes and systematizes a four-force unification model based on the non-compact orthogonal Lie group SO(16,3). We demonstrate the mathematical consistency of the group decomposition chain\[SO(16,3) ⊃ SO(6,3) × SO(10) ⊃ SO(3,1) × SO(2,1) × U(1)_Y × SU(5) ⊃ Standard Model,\]and construct a purely geometric MacDowell–Mansouri-type action\[Sfund = 1/2κ^2 ∫ d^4x\,εμνρσ \,ε_{I_1 I_2 I_3 I_4 J_1 ⋯ J₁₅} \,Rμν{}I_1 I_2 \,Eρ{}I_3 \,Eσ{}I_4 \,ΦJ_1 ⋯ Φ^{J₁₅}\], which involves only the SO(16,3) curvature and the frame field, and introduces no fundamental Yang–Mills terms. All low-energy physics—including general relativity, vector/scalar gravity, and the gauge dynamics of the Standard Model—emerge naturally as induced effects following spontaneous symmetry breaking of SO(16,3). In particular, the unified gauge coupling gᵤₙᵢ and the gravitational constant κ are related through a single geometric scale v via\[gᵤₙᵢ = κ/v,\]realizing a genuine dynamical unification. The central innovation of this work lies in interpreting the SO(6,3) subgroup as a generalized theory of gravity encompassing three fundamental gravitational degrees of freedom—tensor gravity (from SO(3,1)), vector gravity (from SO(2,1)), and scalar gravity (from SO(1,1))—and in viewing the SO(10) gauge forces as effective projections of gravitational geometry onto the internal space, thereby establishing a profound unification between gravity and gauge interactions.