We prove an extension to R^n, endowed with a suitable metric, of the relation between the Einstein-Hilbert action and the Dirac operator which holds on closed spin manifolds. We give first a direct proof of the result on R^4. Then, by means of complex powers and a regularised Wodzicki Residue for a class of operators globally defined on R^n, we show a similar result for n>=4 by using the properties of heat kernels and Dirac operators.
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