Retarded gravitational field of a Hubble sphere with radially varying density

Birkhoff’s theorem in general relativity states that the metric outside a static,spherically symmetric mass distribution is necessarily the Schwarzschild metric. This work demonstrates that when the cosmological density evolves temporally and spatially due to relativistic contraction along the radial direction, differential propagation delays combined with an overdensity in peripheral regions induce anisotropy in the energy-momentum tensor perturbations. This anisotropy leads to the emergence of a positive, non-zero additional curvature inside the Hubble sphere, manifesting as an effective outward-oriented gravitational field. Numerical simulations reveal accelerations on the order of 10 −10 m/s 2 to 10 −8 m/s 2 in the local universe, sufficient to induce cumulative velocity changes on the order of 10 8 m/s over the Hubble time. This mechanism provides a natural geometric explanation for the acceleration of cosmic expansion without resorting to the dark energy hypothesis. The model’s predictions are in qualitative agreement with observations from the Cosmicflows-4 catalog (Tully et al. 2023), showing marked acceleration up to 300 Mpc followed by gradual weakening. Long-term extrapolation of this model predicts a reversal of the cosmological dynamics, culminating in an ultimate phase of global gravitational collapse.