Lovelock's theorem

Not to be confused with Lovelock theory of gravity.

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

1. Clifton, Timothy; et al. (March 2012). "Modified Gravity and Cosmology". Physics Reports. 513 (1–3): 1–189. arXiv:1106.2476. Bibcode:2012PhR...513....1C. doi:10.1016/j.physrep.2012.01.001. S2CID 119258154.
2. Lovelock, D. (1971). "The Einstein Tensor and Its Generalizations". Journal of Mathematical Physics. 12 (3): 498–501. Bibcode:1971JMP....12..498L. doi:10.1063/1.1665613.
3. Lovelock, David (10 January 1972). "The Four-Dimensionality of Space and the Einstein Tensor". Journal of Mathematical Physics. 13 (6): 874–876. Bibcode:1972JMP....13..874L. doi:10.1063/1.1666069.