Nonlocal Gravity: Modified Poisson's Equation

TL;DR

This paper investigates a nonlocal modification of Poisson's equation in gravity, exploring its implications for dark matter simulation, solving the inverse problem, and analyzing specific examples and solutions.

Contribution

It introduces a method to determine the nonlocal kernel from observational data and provides explicit examples and solutions in the linear regime.

Findings

Nonlocal gravity can mimic dark matter effects.

Conditions for the existence of the nonlocal kernel are discussed.

Explicit kernels are numerically evaluated for simple cases.

Abstract

The recent nonlocal generalization of Einstein's theory of gravitation reduces in the Newtonian regime to a nonlocal and nonlinear modification of Poisson's equation of Newtonian gravity. The nonlocally modified Poisson equation implies that nonlocality can simulate dark matter. Observational data regarding dark matter provide limited information about the functional form of the reciprocal kernel, from which the original nonlocal kernel of the theory must be determined. We study this inverse problem of nonlocal gravity in the linear domain, where the applicability of the Fourier transform method is critically examined and the conditions for the existence of the nonlocal kernel are discussed. This approach is illustrated via simple explicit examples for which the kernels are numerically evaluated. We then turn to a general discussion of the modified Poisson equation and present a formal solution of this equation via a successive approximation scheme. The treatment is specialized to the gravitational potential of a point mass, where in the linear regime we recover the Tohline-Kuhn approach to modified gravity.