Gravity, Non-Commutative Geometry and the Wodzicki Residue

TL;DR

This paper derives a gravity action within non-commutative geometry using the Wodzicki residue, connecting heat kernel coefficients to Einstein-Hilbert action and extending to non-commutative spaces with a cosmological constant.

Contribution

It establishes a link between the Wodzicki residue and gravity actions, extending classical results to non-commutative geometries with finite-dimensional matrix algebras.

Findings

Wodzicki residue of $D^{-n+2}$ equals the heat kernel coefficient integral.

Reproduction of Einstein-Hilbert action in commutative case.

Derivation of gravity with a cosmological constant in non-commutative setting.

Abstract

We derive an action for gravity in the framework of non-commutative geometry by using the Wodzicki residue. We prove that for a Dirac operator $D$ on an $n$ dimensional compact Riemannian manifold with $n\geq 4$, $n$ even, the Wodzicki residue Res$(D^{-n+2})$ is the integral of the second coefficient of the heat kernel expansion of $D^{2}$. We use this result to derive a gravity action for commutative geometry which is the usual Einstein Hilbert action and we also apply our results to a non-commutative extension which, is given by the tensor product of the algebra of smooth functions on a manifold and a finite dimensional matrix algebra. In this case we obtain gravity with a cosmological constant.