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

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