Archive for Rational Mechanics and Analysis

, Volume 33, Issue 1, pp 54–70 | Cite as

The uniqueness of the Einstein field equations in a four-dimensional space

  • David Lovelock


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.


Einstein Equation Lagrange Density Scalar Density Cosmological Term Einstein Field Equation 
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  1. 1.
    Rund, H., Variational problems involving combined tensor fields. Abh. Math. Sem. Univ. Hamburg 29, 243–262 (1966).zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Lovelock, D., Degenerate Lagrange Densities for Vector and Tensor Fields. Colloquium on the Calculus of Variations, University of South Africa (1967), 237–269.Google Scholar
  3. 3.
    du Plessis, J. C., Invariance Properties of Variational Principles in General Relativity. Ph. D. thesis, University of South Africa (1965).Google Scholar
  4. 4.
    Lanczos, C., A remarkable property of the Riemann-Christoffel tensor in four-dimensions. Ann. Math. (2) 39, 842–850 (1938).CrossRefMathSciNetGoogle Scholar
  5. 5.
    Lovelock, D., The Lanczos identity and its generalizations. Atti Accad. Naz. Lincei (VIII) 42, 187–194 (1967).zbMATHMathSciNetGoogle Scholar
  6. 6.
    Lovelock, D., Divergence-free tensorial concomitants (to appear in Aequationes Mathematicae).Google Scholar
  7. 7.
    Thomas, T. Y., Differential Invariants of Generalized Spaces. Cambridge University Press 1934.Google Scholar

Copyright information

© Springer-Verlag 1969

Authors and Affiliations

  • David Lovelock
    • 1
  1. 1.Department of MathematicsThe UniversityBristol

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