Abstract

We briefly review the current status of nonlocal gravity (NLG), which is a classical nonlocalgeneralization of Einstein’s theory of gravitation based on a certain analogy with the nonlocalelectrodynamics of media. Nonlocal gravity thus involves integro-differential field equationsand a causal constitutive kernel that should ultimately be determined from observational data.We consider the stationary gravitational field of an isolated rotating astronomical source in the linearapproximation of nonlocal gravity. In this weak-field and slow-motion approximation of NLG,we describe the gravitomagnetic field associated with the rotating source and compare our resultswith gravitoelectromagnetism (GEM) of the standard general relativity theory. Moreover, we brieflystudy the energy-momentum content of the GEM field in nonlocal gravity.

Highlights

  • The standard formulation of general relativity (GR) involves the extension of classical physics expressed in Minkowski spacetime, with metric dS2 = ημν dX μ dX ν, first to arbitrary curvilinear (“accelerated") coordinates via the locality postulate and to curved spacetime, with metric ds2 = gμν dx μ dx ν, by means of Einstein’s principle of equivalence [1,2,3]

  • We exploit the formal analogy between GR|| and electrodynamics and introduce an average of the gravitational field into the field equations via a causal constitutive kernel [14,15,16]

  • We are interested in the stationary gravitational field of a rotating astronomical body, which is assumed to be confined to a compact region of space

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Summary

Introduction

Implies that each leg of the tetrad field is parallel to itself throughout the manifold, i.e., for each α, Equation (7) is an expression of the parallel transport of the corresponding vector with respect to connection (6) In this theory observers throughout spacetime have access to a global set of parallel vector fields that constitute the components of the tetrad frame field. This circumstance is the essence of teleparallelism; for example, two distant vectors can be considered parallel to each other if they have the same components with respect to the local tetrad frames It follows from Equations (4) and (7) that ∇γ gαβ = 0, so that the Weitzenböck connection is compatible with the metric. The result is the teleparallel equivalent of general relativity, GR|| , to which we turn It follows from Equations (9) and (11) that one can write Einstein’s field equations in terms of the torsion tensor. Lagrangian (26), together with Equation (28), and the attached field momentum (27) were the starting point for a classical nonlocal theory of gravity

Nonlocal Gravity
Nonlocal GEM
Linearized NLG
Kernel of Linearized NLG
GEM in Linearized NLG
Gravitomagnetism in Nonlocal GEM
Nonlocal Contributions to the Metric
Gravitomagnetic Clock Effect in NLG
Gravitational Larmor Theorem in NLG
Gravitational Energy-Momentum Tensor
Discussion
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