[Paper Review] Nonlocal General Relativity

Bahram Mashhoon|arXiv (Cornell University)|Nov 20, 2014

Cosmology and Gravitation Theories17 references3 citations

TL;DR

This paper proposes a nonlocal generalization of Einstein's theory of gravity, where gravitational interactions depend on the past history of spacetime, using a tetrad formalism with both Levi-Civita and Weitzenböck connections. The theory reproduces the phenomenology of dark matter in the Newtonian limit by introducing a nonlocality scale of approximately 1 kpc, effectively replacing the need for dark matter through gravitational memory effects encoded in the torsion tensor.

ABSTRACT

A brief account of the present status of the recent nonlocal generalization of Einstein's theory of gravitation is presented. The main physical assumptions that underlie this theory are described. We clarify the physical meaning and significance of Weitzenböck's torsion, and emphasize its intimate relationship with the gravitational field, characterized by the Riemannian curvature of spacetime. In this theory, nonlocality can simulate dark matter; in fact, in the Newtonian regime, we recover the phenomenological Tohline-Kuhn approach to modified gravity. To account for the observational data regarding dark matter, nonlocality is associated with a characteristic length scale of order 1 kpc. The confrontation of nonlocal gravity with observation is briefly discussed.

Motivation & Objective

• To extend general relativity by incorporating nonlocality based on the hypothesis of locality, accounting for the past history of accelerated observers.
• To reformulate general relativity using a tetrad framework with both Levi-Civita and Weitzenböck connections to introduce nonlocal gravitational effects.
• To demonstrate that nonlocality can reproduce the phenomenology of dark matter in galactic dynamics without invoking unseen matter.
• To establish a physical link between Weitzenböck torsion and the gravitational field, showing its role in encoding nonlocal memory effects.
• To confront the nonlocal gravity model with observational data on galactic rotation curves, showing consistency with the 1 kpc nonlocality scale.

Proposed method

• Adopt a tetrad formalism where spacetime geometry is described by both the Levi-Civita connection (standard GR) and the Weitzenböck connection (nonmetric, torsionful).
• Introduce nonlocal field equations by replacing local partial differential equations with integro-differential equations that depend on the past history of the gravitational field.
• Use the hypothesis of locality as a foundation, then extend it to nonlocality by incorporating the observer's worldline and past acceleration history.
• Model accelerated observers in geodesic coordinates, deriving the orthonormal tetrad frame and associated torsion tensor to characterize inertial and gravitational effects.
• Compute the torsion tensor components from the observer's acceleration and angular velocity, showing that $ C_{0i}{}^0 = -a_i(T) $ and $ C_{ij}{}^0 = -2 ilde{oldsymbol{ ho}} imes \boldsymbol{\omega} $, linking torsion directly to kinematic quantities.
• Derive the effective metric in the linear approximation, showing $ g_{00} = -\mathcal{P}^2 + \mathcal{Q}^2 $, with $ \mathcal{P} = 1 + \mathbf{a} \cdot \mathbf{X} $, to describe nonlocal spacetime structure.

Experimental results

Research questions

• RQ1 Can nonlocality in the gravitational interaction simulate the effects of dark matter in galactic rotation curves?
• RQ2 What is the physical origin of Weitzenböck torsion in a nonlocal generalization of general relativity?
• RQ3 How does the nonlocality scale relate to observed astrophysical data, particularly the 1 kpc scale in galactic dynamics?
• RQ4 Can the hypothesis of locality be consistently extended to nonlocal gravity by incorporating past history of accelerated observers?
• RQ5 How does the nonlocal theory reproduce the Tohline-Kuhn phenomenology of modified gravity in the Newtonian limit?

Key findings

• The nonlocal generalization of general relativity successfully reproduces the phenomenology of dark matter in the Newtonian regime, matching the Tohline-Kuhn approach to modified gravity.
• The theory identifies Weitzenböck torsion as a physical manifestation of gravitational memory, directly linked to the acceleration and rotation of observers.
• In the linear approximation, the torsion tensor components are $ C_{0i}{}^0 = -a_i(T) $ and $ C_{ij}{}^0 = -2\epsilon_{ijk}\omega^k(T) $, showing that torsion encodes inertial and gravitational kinematics.
• The nonlocality scale required to match dark matter observations is approximately 1 kpc, consistent with galactic rotation curve data.
• The effective metric in geodesic coordinates includes nonlocal terms $ \mathcal{P} = 1 + \mathbf{a} \cdot \mathbf{X} $ and $ \mathcal{Q}_i = (\boldsymbol{\omega} \times \mathbf{X})_i $, which introduce nonlocal corrections to spacetime geometry.
• The theory provides a nonlocal alternative to dark matter, where the observed gravitational effects attributed to dark matter arise from the memory of past gravitational interactions.