Nonlocal modification of Newtonian gravity

Abstract

The Newtonian regime of a recent nonlocal extension of general relativity (GR) is investigated. Nonlocality is introduced via a scalar "constitutive" kernel in a special case of the translational gauge theory of gravitation, namely, the teleparallel equivalent of GR. In this theory, the nonlocal aspect of gravity simulates dark matter. A nonlocal and nonlinear generalization of Poisson's equation of Newtonian gravitation is presented. The implications of nonlocality for the gravitational physics in the solar system are briefly studied.

Nonlocal gravity: Modified Poisson's equation

“…The physical justification for this supposition is that the implications of the linear form of Eq. (2), for the corresponding linear gravitational potential Φ ℓ , compare favorably with the physics of the flat rotation curves of spiral galaxies [1][2][3][4]. In the linear approximation, the scalar constitutive kernel k is only a function of u and we expect intuitively that even nonlocal gravity would weaken with increasing distance, so that k should go to zero as u := |u| → ∞.…”

“…is modified in the nonlocal theory to read [3,4] ∇ 2 Φ(x) + i ∂k(x, y) ∂x i ∂Φ(y) ∂y i d 3 y = 4πGρ(x) .…”

“…As the nature of dark matter is still unknown, we take the view that what appears as dark matter may very well be a novel aspect of the gravitational interaction. Starting from first principles, a nonlocal generalization of Einstein's theory of gravitation has been developed in recent papers [1][2][3][4], where it is demonstrated that nonlocal gravity can naturally simulate dark matter. However, the main motivation for the development of a nonlocal theory of gravitation came from the principle of equivalence, once the general outline of nonlocal special relativity had become clear [5].…”

“…It can be shown that this theory is amenable to generalization through the introduction of a constitutive kernel. In fact, the simplest case of a scalar constitutive kernel has been employed in [1][2][3][4] to develop a consistent nonlocal generalization of Einstein's theory of gravitation. In this approach to nonlocal gravity, nonlocality survives even in the Newtonian regime and appears to provide a natural explanation for "dark matter" [7][8][9][10][11]; that is, nonlocal gravity simulates dark matter.…”

The Quasi-Maxwellian Equations of General Relativity: Applications to Perturbation Theory

“…We will not enter the details of their works, but we would like to emphasize the question: in a nonlinear regime of gravitational theory, how can we translate the information contained in the nonnull Weyl tensor in terms of the perturbed potential δg ij ? In other words, GR does not contemplate in its boundary conditions any kind of information about intrinsic accelerations of a given configuration on the manifold, which would lead to the development of a nonlocal description of the space-time as suggested by [19,32,88,89]. This type of information shall explicitly be given to the QM formalism, because these equations have partial derivatives of the energy-momentum tensor and contain higher order derivatives of the metric tensor in the dynamics.…”