2000/03/31 by Giampiero Esposito, Cosimo Stornaiolo
Physics and Astronomy · #gr-qc
published as Found.Phys.Lett. 13 (2000) 279-288 · 10 pages, plain Tex. The final version contains new original calculations
arxiv created 2000/05/02 · arxiv updated 2009/11/30
The Arnowitt-Deser-Misner (ADM) evolution equations for the induced metric and the extrinsic-curvature tensor of the spacelike surfaces which foliate the space-time manifold in canonical general relativity are a first-order system of quasi-linear partial differential equations, supplemented by the constraint equations. Such equations are here mapped into another first-order system. In particular, an evolution equation for the trace of the extrinsic-curvature tensor K is obtained whose solution is related to a discrete spectral resolution of a three-dimensional elliptic operator P of Laplace type. Interestingly, all nonlinearities of the original equations give rise to the potential term in P. An example of this construction is given in the case of a closed Friedmann-Lemaitre-Robertson-Walker universe. Eventually, the ADM equations are re-expressed as a coupled first-order system for the induced metric and the trace-free part of K. Such a system is written in a form which clarifies how a set of first-order differential operators and their inverses, jointly with spectral resolutions of operators of Laplace type, contribute to solving, at least in principle, the original ADM system.