Einsteins $R^{hat{0} hat{0}}$ equation for non-relativistic sources derived from Einsteins inertial motion and the Newtonian law for relative acceleration

With Einsteins inertial motion (free-falling and non-rotating relative to gyroscopes), geodesics for non-relativistic particles can intersect repeatedly, allowing one to compute the space-time curvature $R^{hat{0} hat{0}}$ exactly. Einsteins $R^{hat{0} hat{0}}$ for strong gravitational fields and for relativistic source-matter is identical with the Newtonian expression for the relative radial acceleration of neighboring free-falling test-particles, spherically averaged.--- Einsteins field equations follow from Newtonian experiments, local Lorentz-covariance, and energy-momentum conservation combined with the Bianchi identity.