The uniqueness of the Einstein field equations in a four-dimensional space
- David Lovelock
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The Euler-Lagrange equations corresponding to a Lagrange density which is a function of g ij and its first two derivatives are investigated. In general these equations will be of fourth order in g ij. Necessary and sufficient conditions for these Euler-Lagrange equations to be of second order are obtained and it is shown that in a four-dimensional space the Einstein field equations (with cosmological term) are the only permissible second order Euler-Lagrange equations. This result is false in a space of higher dimension. Furthermore, the only permissible third order equation in the four-dimensional case is exhibited.
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- Rund, H., Variational problems involving combined tensor fields. Abh. Math. Sem. Univ. Hamburg 29, 243–262 (1966). CrossRef
- Lovelock, D., Degenerate Lagrange Densities for Vector and Tensor Fields. Colloquium on the Calculus of Variations, University of South Africa (1967), 237–269.
- du Plessis, J. C., Invariance Properties of Variational Principles in General Relativity. Ph. D. thesis, University of South Africa (1965).
- Lanczos, C., A remarkable property of the Riemann-Christoffel tensor in four-dimensions. Ann. Math. (2) 39, 842–850 (1938). CrossRef
- Lovelock, D., The Lanczos identity and its generalizations. Atti Accad. Naz. Lincei (VIII) 42, 187–194 (1967).
- Lovelock, D., Divergence-free tensorial concomitants (to appear in Aequationes Mathematicae).
- Thomas, T. Y., Differential Invariants of Generalized Spaces. Cambridge University Press 1934.
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- The uniqueness of the Einstein field equations in a four-dimensional space
Archive for Rational Mechanics and Analysis
Volume 33, Issue 1 , pp 54-70
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- David Lovelock (1)
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- 1. Department of Mathematics, The University, Bristol