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Dimensionally dependent divergences

Published online by Cambridge University Press:  24 October 2008

Gregory W. Horndeski
Gregory W. Horndeski
Affiliation:
Department of Applied Mathematics, University of Waterloo, Waterloo, Ontario, Canada
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Abstract

It is well known that in an even dimensional Riemannian space there always exists a particular scalar density which is such that its associated Euler–Lagrange tensor vanishes identically. In this note we show that each of these scalar densities can be expressed as the ordinary divergence of a contravariant vector density. Furthermore, there exists an entire class of contravariant vector densities which enjoy this property. These results are discussed with particular reference to the general theory of relativity.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 72 , Issue 1 , July 1972 , pp. 77 - 82
Copyright
Copyright © Cambridge Philosophical Society 1972

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References

REFERENCES

(1)Buchdahl, H. A.On functionally constant invariants of the Riemann tensor. Proc. Cambridge Philos. Soc. 68 (1970), 179–185.CrossRefGoogle Scholar
(2)Lovelock, D.The Einstein tensor and its generalizations. J. Mathematical Phys. 12 (1971), 498–501.CrossRefGoogle Scholar
(3)Lovelock, D.The Euler–Lagrange expression and degenerate Lagrange densities. J. Austral. Math. Soc. (1972). To appear.CrossRefGoogle Scholar
(4)Buchdahl, H. A.On the nonexistence of a class of static Einstein spaces asymptotic at infinity to a space of constant curvature. J. Mathematical Phys. 1 (1960), 537–541.CrossRefGoogle Scholar
(5)Lovelock, D.The Lanczos identity and its generalizations. Atti Accad. Naz. Lincei (VIII) 42 (1967), 187–194.Google Scholar
(6)Lovelock, D.Dimensionally dependent identities. Proc. Cambridge Philos. Soc. 68 (1970), 345–350.CrossRefGoogle Scholar