Lovelock's theorem

In four dimensional space, any tensor ${\displaystyle A^{\mu \nu }}$ whose components are function of 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 divergenceless, then field equation in vacuum ${\displaystyle A^{\mu \nu }=0}$, then only possible form of ${\displaystyle A^{\mu \nu }}$ is

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

where ${\displaystyle a}$ and ${\displaystyle b}$ are just simple 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] ${\displaystyle E^{\mu \nu }=\alpha {\sqrt {-g}}\left[R^{\mu \nu }-{\frac {1}{2}}g^{\mu \nu }R\right]+\lambda {\sqrt {-g}}g^{\mu \nu }}$

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.

• Lovelock theory of gravity
• Vermeil's theorem