Vector-Tensor Field Theories and the Einstein-Maxwell Field Equations

The Euler-Lagrange equations corresponding to a Lagrange density which is a function of the metric tensor $g_{ij}$ and its first two derivatives together with the first derivative of a vector field $\Psi _{i}$ are investigated. In general, the Euler-Lagrange equations obtained by variation of $g_{ij...

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Bibliographic Details
Published in:Proceedings of the Royal Society of London. Series A, Mathematical and physical sciences Mathematical and physical sciences, 1974-12, Vol.341 (1626), p.285-297
Main Author: Lovelock, D.
Format: Article
Language:English
Subjects:
Electromagnetic fields
Euler Lagrange equation
Field theory
General relativity
Mathematics
Scalars
Tensors
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Summary:The Euler-Lagrange equations corresponding to a Lagrange density which is a function of the metric tensor $g_{ij}$ and its first two derivatives together with the first derivative of a vector field $\Psi _{i}$ are investigated. In general, the Euler-Lagrange equations obtained by variation of $g_{ij}$ are of fourth order in $g_{ij}$ and third order in $\Psi _{i}$. It is shown that in a four dimensional space the only Euler-Lagrange equations which are of second order in $g_{ij}$ and first order in $\Psi _{i}$ are the Einstein field equations (with an energy-momentum term). Various conditions are obtained under which the Einstein-Maxwell field equations are then an inevitable consequence.
ISSN:1364-5021
0080-4630
1471-2946
2053-9169
DOI:10.1098/rspa.1974.0188