Lovelock's theorem

General relativity
${\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }={\kappa }T_{\mu \nu }}$
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Lovelock's theorem of general relativity says that from a local gravitational action which contains only second derivatives of the four-dimensional spacetime metric, then the only possible equations of motion are the Einstein field equations.^[1][2][3] The theorem was described by British physicist David Lovelock in 1971.

Statement

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

${\displaystyle A^{\mu \nu }=aG^{\mu \nu }+bg^{\mu \nu }}$

where ${\displaystyle a}$ and ${\displaystyle b}$ are constant numbers and ${\displaystyle G^{\mu \nu }}$ is the Einstein tensor.^[3]

The only possible second-order Euler–Lagrange expression obtainable in a four-dimensional space from a scalar density of the form ${\displaystyle {\mathcal {L}}={\mathcal {L}}(g_{\mu \nu })}$ is^[1] ${\displaystyle E^{\mu \nu }=\alpha {\sqrt {-g}}\left[R^{\mu \nu }-{\frac {1}{2}}g^{\mu \nu }R\right]+\lambda {\sqrt {-g}}g^{\mu \nu }}$

Consequences

Lovelock's theorem means that if we want to modify the Einstein field equations, then we have five options.^[1]

• 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.

See also

• Lovelock theory of gravity
• Vermeil's theorem

References

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