Lovelock's Theorem

Lovelock's Theorem is a theory of general relativity which 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

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

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 less 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.

References

1 2 3 https://arxiv.org/pdf/1106.2476.pdf
↑ D. Lovelock. The Einstein Tensor and Its Generalizations. Journal of Mathematical Physics, 12:498–501, Mar. 1971.
↑ D.Lovelock.TheFour-DimensionalityofSpaceandtheEinsteinTensor.JournalofMathematical Physics, 13:874–876, June 1972.

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Lovelock's Theorem

Statement

Consequences

See also

References