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

TL;DR

This paper derives Einstein's $R^{ ext{0} ext{0}}$ equation for non-relativistic sources directly from Newtonian physics and inertial motion principles, showing its exactness and connection to Newtonian relative acceleration.

Contribution

It provides a derivation of Einstein's $R^{ ext{0} ext{0}}$ equation from Newtonian and inertial principles, emphasizing its validity for strong fields and relativistic matter.

Findings

$R^{ ext{0} ext{0}}$ matches Newtonian relative acceleration for non-relativistic sources

Geodesics can intersect repeatedly in non-relativistic regimes

Einstein's field equations can be derived from Newtonian experiments and conservation laws

Abstract

With Einstein's 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. Einstein's $R^{\hat{0} \hat{0}}$ 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.--- Einstein's field equations follow from Newtonian experiments, local Lorentz-covariance, and energy-momentum conservation combined with the Bianchi identity.