The Einstein Action for Algebras of Matrix Valued Functions - Toy Models

Details The Einstein Action for Algebras of Matrix Valued Functions - Toy Models The Einstein Action for Algebras of Matrix Valued Functions - Toy Models

1995-10-06

arxiv.org/abs/q-alg/9510007v2

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

9 pages, AMS-LaTeX, serious typesetting problems due to 2.09-2.e incompatibility removed, reference added

Scientific paper

10.1063/1.531662

Two toy models are considered within the framework of noncommutative differential geometry. In the first one, the Einstein action of the Levi-Civita connection is computed for the algebra of matrix valued functions on a torus. It is shown that, assuming some constraints on the metric, this action splits into a classical-like, a quantum-like and a mixed term. In the second model, an analogue of the Palatini method of variation is applied to obtain critical points of the Einstein action functional for $M\sb 4(R)$. It is pointed out that a solution to the Palatini variational problem is not necessarily a Levi-Civita connection. In this model, no additional assumptions regarding metrics are made.

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