Metric-affine cosmological models and the inverse problem of the calculus of variations. Part 1: variational bootstrapping – the method

Abstract

AbstractThe method of variational completion allows one to transform an (in principle, arbitrary) system of partial differential equations – based on an intuitive “educated guess” – into the Euler–Lagrange one attached to a Lagrangian, by adding a canonical correction term. Here, we extend this technique to theories that involve at least two sets of dynamical variables: we show that an educated guess of the field equations with respect to one of these sets of variables only is sufficient to variationally complete these equations and recover a Lagrangian for the full theory, up to boundary terms and terms that do not involve the respective variables. Applying this idea to natural metric-affine theories of gravity, we prove that, starting from an educated guess of the metric equations only, one can find the full metric equations, together with a generally covariant Lagrangian, up to metric-independent terms. The latter terms (which can only involve the distortion of the connection) are then completely classified over 4-dimensional spacetimes, by techniques pertaining to differential invariants.