Gravity, Non-Commutative Geometry and the Wodzicki Residue

Details Gravity, Non-Commutative Geometry and the Wodzicki Residue Gravity, Non-Commutative Geometry and the Wodzicki Residue

1993-12-21

arxiv.org/abs/gr-qc/9312031v1

J.Geom.Phys. 16 (1995) 327-344

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

17p., MZ-TH/93-38

Scientific paper

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.

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