Einstein's equations from Einstein's inertial motion and Newton's law for relative acceleration

TL;DR

This paper demonstrates that Einstein's gravitational equations for nonrelativistic matter and strong fields are equivalent to Newton's law of relative acceleration, establishing a direct link between general relativity and Newtonian physics through inertial motion and energy-momentum conservation.

Contribution

It shows that Einstein's $R^{ ext{0} ext{0}}$ equation is identical to Newton's law for relative acceleration, derived from inertial motion, Lorentz covariance, and energy-momentum conservation, using local orthonormal bases.

Findings

Einstein's $R^{ ext{0} ext{0}}$ equation matches Newton's relative acceleration law.

Gravitational fields are measured exactly via local orthonormal bases.

The gravitational field equations are linear and identical for inertial observers.

Abstract

We show that Einstein's $R^{\hat{0} \hat{0}}$ equation for nonrelativistic matter and strong gravitational fields is identical with Newton's equation for relative radial acceleration of neighbouring freefalling particles, spherically averaged. These laws are explicitely identical with primary observer's (1) space-time slicing by radial 4-geodesics, (2) radially parallel Local Ortho-Normal Bases, LONBs, (3) Riemann normal 3-coordinates. Hats on indices denote LONBs. General relativity follows from Newton's law of relative acceleration, Einstein's inertial motion, Lorentz covariance, and energy-momentum conservation combined with Bianchi identity. The gravitational field equation of Newton-Gauss and Einstein's $R^{\hat{0} \hat{0}}$ equation are identical and linear in gravitational field for an inertial primary observer.--- Einstein's equivalence between fictitious forces and gravitational forces is formulated as equivalence theorem in the equations of motion. With this, the gravitational field equation of 19th-century Newtonian physics and Einstein's equation for $R^{\hat{0} \hat{0}}$ are identical and bilinear in the gravitational forces for non-inertial primary observers.--- $R^{\hat{0} \hat{0}} = - div \vec{E}_g$ and $2 R^{\hat{i} \hat{0}} = - (curl \vec{B}_g)^{\hat{i}}$ hold exactly for inertial primary observers, if one uses our LONB's. The gravitational $\vec{E}_g, \vec{B}_g $ are measured exactly with quasistatic particles via $(d/dt) p_{\hat{i}}$ and $(d/dt) S_{\hat{i}}$ in correspondence with the electromagnetic $\vec{E}$ and $\vec{B}$. The $(\vec{E}_g, \vec{B}_g)$ are identical with the observer's Ricci connection along his worldline, $(\omega_{\hat{a} \hat{b}})_{\hat{0}}$.