Field Equations and Radial Solution in a Noncommutative Spherically Symmetric Geometry

TL;DR

This paper develops a noncommutative gravity theory within torsional spacetime, deriving new spherically symmetric solutions that support noncommutative black hole models and introduce a novel Newtonian force equation.

Contribution

It introduces a noncommutative gravity framework based on teleparallelism, deriving spherically symmetric solutions and a new Newtonian force equation within this context.

Findings

Derived a noncommutative line element for a spherically symmetric mass.

Reaffirmed the coherent state theory for noncommutative Schwarzschild black holes.

First derivation of Newtonian gravity equation in a noncommutative setting.

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.