FD-3_Structure of the Spatial Scale k

FD-3 develops the spatial structure generated by the future-direction vector mu, following the time structure established in FD-2. The paper focuses on the spatial component K_ab obtained from the covariant decomposition nabla_b mu_a = - mu_b a_a + K_ab. The symmetric part of K_ab defines the expansion rate theta, which determines the volume change of spatial sections. Using only the minimal geometric assumptions, FD-3 introduces the spatial scale k as a scalar satisfying (1/k)(dk/dtau) = theta,