Taking the time-series t as independent variable, the parameter equations Xi(t) of free particle space geodesic can be given. By transforming affine parameter R(t) we achieve homogeneous geodesic differential equations, and derive the first-order differential equations which are satisfied by affine parameter R and the sequence of analytical solutions R marked by rational number Cu. In light of R we define the distance unit of flat four-dimensional coordinate system t,r,θ,φ, and then establish a free particle geodesic affine parameter time-space coordinate system t,ξ,θ,φ. By the study of the diagonalization process of special relativity time-space interval model metric tensor g in t,ξ,θ,φ, we find the spatial and temporal line characteristic quantities t1(t,ξ), τ1(τ,ξ),tt(t,τ,ξ) and ττ1(t,τ,ξ) corresponding to diagonal metric. Derived from these quantities, the dimension of time-space coordinate system is less than 4.