Einstein Equations for a Noncommutative Spacetime of Lie-Algebraic Type

TL;DR

This paper derives Einstein equations for a Lie-algebraic noncommutative spacetime, revealing modifications such as a traceless term in the Einstein tensor, which differ from classical formulations.

Contribution

It provides a general formula for the connection, Riemann tensor, and Einstein tensor in a noncommutative Lie-algebraic spacetime framework, highlighting novel features.

Findings

Einstein tensor includes a traceless term not present in classical theory.

Derived Bianchi identities and symmetries for noncommutative spacetime.

Established a consistent framework for noncommutative Einstein equations.

Abstract

A general formula is calculated for the connection of a central metric w.r.t.\ a noncommutative spacetime of Lie-algebraic type. This is done by using the framework of linear connections on central bi-modules. The general formula is further on used to calculate the corresponding Riemann tensor and prove the corresponding Bianchi identities and certain symmetries that are essential to obtain a symmetric and divergenceless Einstein Tensor. In particular, the obtained Einstein Tensor is not equivalent to the sum of the noncommutative Riemann tensor and scalar, as in the commutative case, but in addition a traceless term appears.