Nonlocal Gravity: Fundamental Tetrads and Constitutive Relations

Abstract

Nonlocal gravity (NLG) is a classical nonlocal generalization of Einstein’s theory of gravitation based on a certain analogy with the nonlocal electrodynamics of media. The history dependence enters NLG through a constitutive relation involving a causal kernel that should ultimately be determined via observational data. The purpose of this paper is to reformulate nonlocal gravity such that the nonlocal aspect of the constitutive relation directly connects measurable quantities as in the nonlocal electrodynamics of media. The resulting constitutive relation turns out to coincide with the recent suggestion of Puetzfeld, Obukhov, and Hehl. With the new constitutive relation of NLG, it is possible to show that de Sitter spacetime is not a solution of NLG. Dedicated to Friedrich W. Hehl on his eighty-fifth birthday.

Local Limit of Nonlocal Gravity: Cosmological Perturbations

“…5 https://camb.info/ 2009a, 2009b). For a comprehensive account of NLG, see Mashhoon (2017). The teleparallel equivalent of general relativity (TEGR) is a tetrad theory; that is, the framework of GR is supplemented with a latticework of preferred orthonormal tetrad frame fields that are parallel via the Weitzenböck connection.…”

“…To go forward, we need to specify the exact relationship between N μνρ and C μνρ . By using this approach, a nonlocal relation has thus far led to nonlocal gravity (NLG) theory that is patterned after the nonlocal electrodynamics of media (Hehl & Mashhoon 2009a, 2009bMashhoon 2017). However, the local limit of this nonlocal relation is the main focus of the present investigation.…”

“…) ¢ must be ultimately determined by observation. In this connection, linearized NLG has been treated in detail in Mashhoon (2017) and the Newtonian regime of the theory has been compared with observational data (Rahvar & Mashhoon 2014;Chicone & Mashhoon 2016;Roshan & Mashhoon 2021, 2022a, 2022b. Within the nonlinear regime of NLG, it has been possible to show that de Sitter spacetime is not an exact solution (Mashhoon 2022); indeed, the only known exact solution of NLG is the trivial solution, namely, we find the Minkowski spacetime in the absence of gravitation.…”

“…A detailed explanation for the choice of X μνρ is contained in Mashhoon (2017), which should be consulted for an extensive treatment of the physical basis of NLG. In Equation (30),  C m is the torsion pseudovector defined via the Levi-Civita tensor ò αβγδ by…”

Anisotropic Cosmology in the Local Limit of Nonlocal Gravity

“…The formal similarity between TEGR and electrodynamics can be employed to introduce nonlocality into GR via constitutive kernels [2,3]. In NLG, the gravitational field is local, but the theory involves an average of the field over past events resulting in 16 partial integro-differential field equations [24,25]. No exact nontrivial solution of NLG is known at present [26]; however, the linear regime of the theory has been extensively studied.…”