The Gravitational Polarization in General Relativity: Solution to Szekeres' Model of Quadrupole Polarization

TL;DR

This paper develops a generalized gravitational potential equation incorporating quadrupole polarization effects, extending Newtonian gravity to a relativistic macroscopic medium with solutions differing inside and outside matter sources.

Contribution

It introduces a biharmonic equation for gravitational potential accounting for quadrupole polarization, providing solutions that extend Newtonian gravity and differ in strong-field regimes.

Findings

The equation reduces to Poisson's equation with renormalized density for weak polarization.

Outside sources, the potential decays exponentially, unlike Newtonian gravity.

Solutions are obtained via Green's function method, highlighting differences from classical gravity.

Abstract

A model for the static weak-field macroscopic medium is analyzed and the equation for the macroscopic gravitational potential is derived. This is a biharmonic equation which is a non-trivial generalization of the Poisson equation of Newtonian gravity. In case of the strong gravitational quadrupole polarization it essentially holds inside a macroscopic matter source. Outside the source the gravitational potential fades away exponentially. The equation is equivalent to a system of the Poisson equation and the nonhomogeneous modified Helmholtz equations. The general solution to this system is obtained by using Green's function method and it does not have a limit to Newtonian gravity. In case of the insignificant gravitational quadrupole polarization the equation for macroscopic gravitational potential becomes the Poisson equation with the matter density renormalized by the factor including the value of the quadrupole gravitational polarization of the source. The general solution to this equation obtained by using Green's function method has a limit to Newtonian gravity.