Embedding Newton–Mach Type Time Evolution into the FRW Metric Phenomenological Implications for the Hubble Tension and Its Limitations

This revised version corrects internal consistency issues, clarifies notation, and improves the presentation of the effective FRW embedding. A central new point of this version is that the Planck-time constraint, together with the working hypothesis alpha = 50/3, yields the precise Newton–Mach value H0^(NM) ≃ 73.4951 km s^-1 Mpc^-1, which is remarkably close to the recent H0DN local-distance-network determination H0 = 73.50 ± 0.81 km s^-1 Mpc^-1. In this sense, the revised analysis strengthens the phenomenological support for the quantization condition alpha = l_pu / l_P ≃ 50/3 within the present Newton–Mach framework. In this work we study an effective cosmological model in which the Friedmann–Robertson–Walker (FRW) metric of general relativity (GR) is kept fixed, while only the time evolution of the scale factor a(t) is prescribed by a Newton–Mach (NM) type ordinary differential equation. Starting from three assumptions—(i) Newtonian mechanics, (ii) a Mach-type relation c^2 = G M(r) / r, and (iii) the local equivalence principle in an FRW background—together with the Planck force F_P = c^4 / G and the dynamics of the physical radius r(t) = a(t) chi_0 of a representative comoving sphere, we derive the NM expansion law dr/dt = c √(1 - 2 ln(r/r0)), or equivalently da/dt = H0 √(1 - 2 ln a). We then reinterpret this time evolution within the FRW metric as an expansion history generated by an effective dark fluid, and uniquely determine the corresponding effective energy density rho_eff(a), pressure p_eff(a), equation-of-state parameter w(a), and deceleration parameter q(a). Our aim is not to propose a new theory of gravity that replaces GR as a whole, but to present, strictly within the FRW+GR framework, a single phenomenological background model of the Universe with NM-type time evolution. Because the resulting expansion history H(a) can modify the inferred low-redshift Hubble parameter relative to standard LCDM, the model may serve as a phenomenological candidate for addressing the H0 tension. At the same time, we do not perform detailed fits to CMB, BAO, or SNe data, nor do we analyse the growth of perturbations, construct an underlying action S[Phi], or develop PPN expansions. These remain limitations of the present work and are left for future detailed studies and observational tests.