Gravity, non-commutative geometry and the Wodzicki residue |
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Authors: | W Kalau and M Walze |
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Institution: | Johannes Gutenberg Universität, Institut für Physik, 55099, Mainz, Germany |
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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 ≥ 4, n even, the Wodzicki residue Res(D−n+2) is the integral of the second coefficient of the heat kernel expansion of D2. 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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Keywords: | Non-commutative geometry Gravity Wodzicki residue |
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