We propose a step-by-step manual for the construction of alternative theories of gravity, perturbatively as well as in the exact setting. The construction is guided by no more than two fundamental principles that we impose on the gravitational dynamics: their invariance under spacetime diffeomorphisms and the compatibility of their causal structure with given matter dynamics, provided that spacetime is additionally endowed with a matter field. 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 framework in the perturbative setting, 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. Moreover, we construct the most general third-order perturbative theory of gravity that is capable of supporting a general (not necessarily Maxwellian) linear theory of electrodynamics.
We present a method of constructing perturbative equations of motion for the geometric background of any given tensorial field theory. Requiring invariance of the gravitational dynamics under spacetime diffeomorphisms leads to a PDE system for the gravitational Lagrangian that can be solved by means of a power series ansatz. Furthermore, in each order we pose conditions on the causality of the gravitational equations, that ensure coevolution of the matter fields and the gravitational background is possible, i.e. gravitational equations and matter equations share the same initial data hypersurfaces.
Several physically inspired problems have been proven undecidable; examples are the spectral gap problem and the membership problem for quantum correlations. Most of these results rely on reductions from a handful of undecidable problems, such as the halting problem, the tiling problem, the Post correspondence problem or the matrix mortality problem. All these problems have a common property: they have an NP-hard bounded version. This work establishes a relation between undecidable unbounded problems and their bounded NP-hard versions. Specifically, we show that NP-hardness of a bounded version follows easily from the reduction of the unbounded problems. This leads to new and simpler proofs of the NP-hardness of bounded version of the Post correspondence problem, the matrix mortality problem, the positivity of matrix product operators, the reachability problem, the tiling problem, and the ground state energy problem. This work sheds light on the intractability of problems in theoretical physics and on the computational consequences of bounding a parameter.
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