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Hodge Theory on Compact Two-Dimensional Spacetimes and the Uniqueness of gij with a Specified Rij

Published online by Cambridge University Press:  20 November 2018

Edwin Ihrig
Edwin Ihrig*
Affiliation:
Arizona State University, Tempe, Arizona
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The main question we wish to address in this paper is to what extent does the Ricci curvature of a spacetime determine the metric of that spacetime. Although it is relatively easy to see that the full Riemann curvature uniquely determines the metric for a generic choice of curvature tensors (see [4], [10], [11], [14] and [15], and the references contained therein), very little has been discovered about whether, if ever, Ric (or the stress energy tensor in Einstein's equations for that matter) determines g. Most exact solution techniques for Einstein's equations look only for solutions that have the same symmetries as Ric. It is not true in general that g must inherit the symmetries of Ric. It is not even clear that there is a Ric such that every g with this Ricci tensor is known.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 6 , 01 December 1987 , pp. 1459 - 1474
Copyright
Copyright © Canadian Mathematical Society 1987

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