Lovelock's theorem

In four dimensional spacetime, any tensor $A^\{\backslash mu\backslash nu\}$ whose components are functions of the metric tensor $g^\{\backslash mu\backslash nu\}$ and its first and second derivatives (but linear in the second derivatives of $g^\{\backslash mu\backslash nu\}$), and also symmetric and divergence-free,

is necessarily of the form

:$A^\{\backslash mu\backslash nu\}=a\; G^\{\backslash mu\backslash nu\}+b\; g^\{\backslash mu\backslash nu\}$

where $a$ and $b$ are constant numbers and $G^\{\backslash mu\backslash nu\}$ is the Einstein tensor.{{cite journal |last=Lovelock |first=David |date= 10 January 1972 |title=The Four-Dimensionality of Space and the Einstein Tensor |journal= Journal of Mathematical Physics |volume=13 |issue=6 |pages=874–876 |doi=10.1063/1.1666069|bibcode=1972JMP....13..874L}}

The only possible second-order Euler–Lagrange expression obtainable in a four-dimensional space from a scalar density of the form $\backslash mathcal\{L\}=\backslash mathcal\{L\}(g\_\{\backslash mu\backslash nu\})$ is

$E^\{\backslash mu\backslash nu\}\; =\; \backslash alpha\; \backslash sqrt\{-g\}\; \backslash left[R^\{\backslash mu\backslash nu\}\; -\; \backslash frac\{1\}\{2\}\; g^\{\backslash mu\backslash nu\}\; R\; \backslash right]\; +\; \backslash lambda\; \backslash sqrt\{-g\}\; g^\{\backslash mu\backslash nu\}$

Lovelock's theorem means that if we want to modify the Einstein field equations, then we have five options.

• Add other fields rather than the metric tensor;
• Use more or fewer than four spacetime dimensions;
• Add more than second order derivatives of the metric;
• Non-locality, e.g. for example the inverse d'Alembertian;
• Emergence – the idea that the field equations don't come from the action.

{{Portal|Physics}}

• Lovelock theory of gravity
• Vermeil's theorem

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Category:General relativity

Category:Theorems in general relativity

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