Gravity, Non-Commutative Geometry and the Wodzicki Residue

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 $n\ge 4$, $n$ 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.

17p., MZ-TH/93-38

References in corpus (4)

• Gravity in Non-Commutative Geometry
• Grand Unification in Non-Commutative Geometry
• Constraints on the Higss and Top Quark Masses From Effective Potential and Non-Commutative Geometry
• Differential Algebras in Non-Commutative Geometry

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