Unconstrained Lagrangian Variational Principles for the Einstein Field Equations

This paper deals with the problem of establishing a systematic theoretical formulation of variational principles for the continuum gravitational field dynamics of classical General Relativity (GR). In this reference, the existence of multiple Lagrangian functions underlying the Einstein field equations (EFE) but having different physical connotations is pointed out. Given validity of the Principle of Manifest Covariance (PMC), a set of corresponding variational principles can be constructed. These are classified in two categories, respectively, referred to as constrained and unconstrained Lagrangian principles. They differ for the normalization properties required to be satisfied by the variational fields with respect to the analogous conditions holding for the extremal fields. However, it is proved that only the unconstrained framework correctly reproduces EFE as extremal equations. Remarkably, the synchronous variational principle recently discovered belongs to this category. Instead, the constrained class can reproduce the Hilbert–Einstein formulation, although its validity demands unavoidably violation of PMC. In view of the mathematical structure of GR based on tensor representation and its conceptual meaning, it is therefore concluded that the unconstrained variational setting should be regarded as the natural and more fundamental framework for the establishment of the variational theory of EFE and the consequent formulation of consistent Hamiltonian and quantum gravity theories.