 Lovelock's theorem

In four dimensions, a local natural symmetric divergence-free metric tensor involving the metric and its derivatives of at most second order has the Lovelock form $a{G}_{ab}+b{g}_{ab}$ with constant coefficients. Generic nonlinear metric f(R) gravity instead has fourth-order terms and does not meet that derivative-order hypothesis. For affine $f$, the metric field equation is precisely of Lovelock form.

Lovelock's theorem refers to a set of results in the field of geometric analysis and theoretical physics, named after the mathematician David Lovelock. The key results of Lovelock's theorem concern the existence of certain types of gravitational theories in higher-dimensional spacetimes and focus primarily on the properties of tensors and the equations of motion that can be derived from a Lagrangian formulation.