Archive for Rational Mechanics and Analysis

, Volume 33, Issue 1, pp 54–70

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

  • David Lovelock

DOI: 10.1007/BF00248156

Cite this article as:
Lovelock, D. Arch. Rational Mech. Anal. (1969) 33: 54. doi:10.1007/BF00248156


The Euler-Lagrange equations corresponding to a Lagrange density which is a function of gij and its first two derivatives are investigated. In general these equations will be of fourth order in gij. 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.

Copyright information

© Springer-Verlag 1969

Authors and Affiliations

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