ISSN:
1572-9524
Keywords:
general relativity
;
canonical formalism
;
elliptic operators
;
bounded operators
Source:
Springer Online Journal Archives 1860-2000
Topics:
Physics
Notes:
Abstract 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 $$\mathcal{P}$$ of Laplace type. Interestingly, all nonlinearities of the original equations give rise to the potential term in $$\mathcal{P}$$ . 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.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1023/A:1007804205303
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