Covariant constructive gravity: A step-by-step guide towards alternative theories of gravity

We propose a step-by-step manual for the construction of alternative theories of gravity, perturbatively as well as nonperturbatively. The construction is guided by no more than two fundamental principles that we impose on the gravitational dynamics: invariance under spacetime diffeomorphisms and causal compatibility with given matter dynamics, provided that spacetime is additionally endowed with matter fields. The developed framework then guides the computation of the most general, alternative theory of gravity that is consistent with the two fundamental requirements. Utilizing this framework we recover the cosmological sector of general relativity solely from assuming that spacetime is a spatially homogeneous and isotropic metric manifold. On top of that, we explicitly test the perturbative framework by deriving the most general third-order expansion of a metric theory of gravity that is causally compatible with a Klein-Gordon scalar field. Thereby we recover the perturbative expansion of general relativity. To demonstrate how new physics emerges from our approach, we finally construct the most comprehensive third-order expansion of a theory of gravity that supports general (not necessarily Maxwellian) linear electrodynamics.