Field Equations and Radial Solutions in a Noncommutative Spherically Symmetric Geometry

Aref Yazdani

doi:10.1155/2014/349659

Abstract

We study a noncommutative theory of gravity in the framework of torsional spacetime. This theory is based on a Lagrangian obtained by applying the technique of dimensional reduction of noncommutative gauge theory and that the yielded diffeomorphism invariant field theory can be made equivalent to a teleparallel formulation of gravity. Field equations are derived in the framework of teleparallel gravity through Weitzenbock geometry. We solve these field equations by considering a mass that is distributed spherically symmetrically in a stationary static spacetime in order to obtain a noncommutative line element. This new line element interestingly reaffirms the coherent state theory for a noncommutative Schwarzschild black hole. For the first time, we derive the Newtonian gravitational force equation in the commutative relativity framework, and this result could provide the possibility to investigate examples in various topics in quantum and ordinary theories of gravity.

Highlights

Field equations of gravity and radial solutions have been previously derived in noncommutative geometry [1,2,3,4,5]
In order to continue our discussion to achieve to noncommutative field equations, we should show how our model can be coupled with general relativity
There are conceptual differences, in general relativity, curvature is used to geometrize the gravitational interaction, geometry replaces the concept of force, and the trajectories are determined, not by force equations but by geodesics

Summary

Introduction

Field equations of gravity and radial solutions have been previously derived in noncommutative geometry [1,2,3,4,5]. The model leads to the gauge theories of gravitation through an ordinary class of dimensional reductions of noncommutative electrodynamics on flat space, which can be made equivalent to a formulation of teleparallel gravity, macroscopically describing general relativity. This model is developed by the parallel theories of gravitation, giving a clear understanding of Einstein’s principle of absolute parallelism. This connection is known as Wietzenbock geometry on spacetime This model is given appropriately by a noncommutative Lagrangian and introduced by authors in [4, 5].

Equivalence with General Relativity

Noncommutative Field Equations

Noncommutative Line Element

Force Equation in Commutative Limit

Conclusion