Variational theory of the Ricci curvature tensor dynamics

Abstract

In this letter a new Lagrangian variational principle is proved to hold for the Einstein field equations, in which the independent variational tensor field is identified with the Ricci curvature tensor R^{mu nu } rather than the metric tensor g_{mu nu }. The corresponding Lagrangian function, denoted as L_{R}, is realized by a polynomial expression of the Ricci 4-scalar Requiv g_{mu nu }R^{mu nu } and of the quadratic curvature 4-scalar rho equiv R^{mu nu }R_{mu nu }. The Lagrangian variational principle applies both to vacuum and non-vacuum cases and for its validity it demands a non-vanishing, and actually also positive, cosmological constant Lambda >0. Then, by implementing the deDonder–Weyl formalism, the physical conditions for the existence of a manifestly-covariant Hamiltonian structure associated with such a Lagrangian formulation are investigated. As a consequence, it is proved that the Ricci tensor can obey a Hamiltonian dynamics which is consistent with the solutions predicted by the Einstein field equations.