The uniqueness of the Einstein field equations in a four-dimensional space
Abstract
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.
- Publication:
-
Archive for Rational Mechanics and Analysis
- Pub Date:
- January 1969
- DOI:
- 10.1007/BF00248156
- Bibcode:
- 1969ArRMA..33...54L