#### 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

Published by Springer in Communications in Mathematical Physics. Publisher PDF available through link above. Publisher requires a subscription to access article.