Title
The Classification of Local Generalized Symmetries for the Einstein Equa- tions
Document Type
Article
Journal/Book Title/Conference
Communications in Mathematical Physics
Volume
176
Publication Date
1996
First Page
479
Last Page
539
Abstract
A local generalized symmetry of a system of differential equations is an infinitesimal transformation depending locally upon the fields and their derivatives which carries solutions to solutions. We classify all local generalized symmetries of the vacuum Einstein equations in four spacetime dimensions. To begin, we analyze symmetries that can be built from the metric, curvature, and covariant derivatives of the curvature to any order; these are called natural symmetries and are globally defined on any spacetime manifold. We next classify first-order generalized symmetries, that is, symmetries that depend on the metric and its first derivatives. Finally, using results from the classification of natural symmetries, we reduce the classification of all higher-order generalized symmetries to the first-order case. In each case we find that the local generalized symmetries are infinitesimal generalized diffeomorphisms and constant metric scalings. There are no non-trivial conservation laws associated with these symmetries. A novel feature of our analysis is the use of a fundamental set of spinorial coordinates on the infinite jet space of Ricci-flat metrics, which are derived from Penrose's “exact set of fields” for the vacuum equations.
Recommended Citation
The Classification of Local Generalized Symmetries for the Einstein Equa- tions (with C. Torre), Comm. Math. Physics 176 (1996) 479–539.
Comments
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