THE EINSTEIN ACTION FOR ALGEBRAS OF MATRIX VALUED FUNCTIONS - TOY MODELS

Two toy models are considered within the framework of noncommutative d ifferential geometry. In the first one, the Einstein action of the Lev i-Civita connection is computed for the algebra of matrix valued funct ions on a torus. It is shown that, assuming some constraints on the me tric, this action splits into a classical-like, a quantum-like and a m ixed 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(4)(R). It is pointed out that a solution to the Palat ini variational problem is not necessarily a Levi-Civita connection. I n this model, no additional assumptions regarding metrics are made. (C ) 1996 American Institute of Physics.