Archive for Rational Mechanics and Analysis

, Volume 33, Issue 1, pp 54-70

First online:

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

  • David LovelockAffiliated withDepartment of Mathematics, The University

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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.