Article ID Journal Published Year Pages File Type
1898657 Journal of Geometry and Physics 2012 13 Pages PDF
Abstract

Let Tab=Tba=0 be a system of differential equations for the components of a metric tensor on RmRm. Suppose that Tab transforms tensorially under the action of the diffeomorphism group on metrics and that the covariant divergence of Tab vanishes. We then prove that Tab=Eab(L) is the Euler–Lagrange expression of some Lagrangian density LL provided that Tab is of third order. Our result extends the classical works of Cartan, Weyl, Vermeil, Lovelock, and Takens on identifying field equations for the metric tensor with the symmetries and conservation laws of the Einstein equations.

Related Topics
Physical Sciences and Engineering Mathematics Mathematical Physics
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