Gr-cs clock shift differential

This paper re-examines Einstein’s 1915 field equations without a cosmological constant and interprets cosmological redshift as a clock-rate differential rather than as spatial expansion. A scalar time-flow field v_t(x) (or N(x)) is introduced through g_{00} = -v_t^2(x), defining local proper-time flow d\tau = v_t\,dt. The resulting relation 1+z = v_t^{(\mathrm{obs})}/v_t^{(\mathrm{emit})} reproduces Hubble-like behavior without invoking metric stretching or vacuum energy. This “Einstein + Clock Differential” formulation corresponds to the α → 0 limit of the GR-CS framework, retaining full consistency with General Relativity while eliminating the need for a cosmological constant.