Abstract
We study general relativity in the framework of non-commutative differential geometry. As a prerequisite we develop the basic notions of non-commutative Riemannian geometry, including analogues of Riemannian metric, curvature and scalar curvature. This enables us to introduce a generalized Einstein-Hilbert action for non-commutative Riemannian spaces. As an example we study a space-time which is the product of a four dimensional manifold by a two-point space, using the tools of non-commutative Riemannian geometry, and derive its generalized Einstein-Hilbert action. In the simplest situation, where the Riemannian metric is taken to be the same on the two copies of the manifold, one obtains a model of a scalar field coupled to Einstein gravity. This field is geometrically interpreted as describing the distance between the two points in the internal space.
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Communicated by A. Connes
Dedicated to H. Araki
Supported in part by the Swiss National Foundation (SNF)
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Chamseddine, A.H., Felder, G. & Fröhlich, J. Gravity in non-commutative geometry. Commun.Math. Phys. 155, 205–217 (1993). https://doi.org/10.1007/BF02100059
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DOI: https://doi.org/10.1007/BF02100059