Nonlocal gravity: Conformally flat spacetimes

TL;DR

This paper explores the structure of nonlocal gravity in conformally flat spacetimes, revealing that only specific symmetric cases satisfy the nonlocal field equations, even in simplified two-dimensional models.

Contribution

It introduces the form of nonlocal gravity field equations in a way similar to general relativity and analyzes their solutions in conformally flat and two-dimensional spacetimes.

Findings

Nonlocal gravity equations are intractable in general conformally flat spacetimes.

Only spacetimes with a Killing or homothetic Killing vector satisfy the nonlocal field equations in 2D.

Smooth 2D spacetimes are conformally flat and satisfy Einstein's equations, but not necessarily nonlocal gravity equations.

Abstract

The field equations of the recent nonlocal generalization of Einstein's theory of gravitation are presented in a form that is reminiscent of general relativity. The implications of the nonlocal field equations are studied in the case of conformally flat spacetimes. Even in this simple case, the field equations are intractable. Therefore, to gain insight into the nature of these equations, we investigate the structure of nonlocal gravity in two-dimensional spacetimes. While any smooth 2D spacetime is conformally flat and satisfies Einstein's field equations, only a subset containing either a Killing vector or a homothetic Killing vector can satisfy the field equations of nonlocal gravity.