Non-relativistic limit of the Einstein equation

TL;DR

This paper derives the non-relativistic limit of Einstein's equations for stationary, axially symmetric space-times, connecting relativistic gravity to Newtonian gravity with additional potentials.

Contribution

It introduces a framework for transitioning from Einstein's equations to Newtonian gravity, including a vector potential differing from classical electrodynamics.

Findings

Reduction of Einstein equations to Newtonian and vector potentials in first-order approximation

Expression of space-time metric in terms of 3-space metric and potentials

Connection between relativistic and Newtonian gravitational fields

Abstract

In particular cases of stationary and stationary axially symmetric space-time passage to non-relativistic limit of Einstein equation is completed. For this end the notions of absolute space and absolute time are introduced due to stationarity of the space-time under consideration. In this construction absolute time is defined as a function $t$ on the space-time such that $\prt_t$ is exactly the Killing vector and the space at different moments is presented by the surfaces $t=\con $. The space-time metric is expressed in terms of metric of the 3-space and two potentials one of which is exactly Newtonian gravitational potential $\Phi$, another is vector potential $\vec A$ which, however, differs from vector potential known in classical electrodynamics. In the first-order approximation on $\Phi/c^2$, $|\vec A|/c$ Einstein equation is reduced to a system for these functions in which left-hand sides contain Laplacian of the Newtonian potential, derivatives of the vector potential and curvature of the space and the right-hand sides do 3-dimensional stress tensor and densities of mass and energy. Subj-class: Classical Physics