Variational theory of the Ricci curvature tensor dynamics
Research Center for Theoretical Physics and Astrophysics, Institute of Physics, Silesian University in Opava, Bezručovo nám.13, 74601, Opava, Czech Republic
2 Department of Mathematics and Geosciences, University of Trieste, Via Valerio 12, 34127, Trieste, Italy
Accepted: 16 November 2021
Published online: 25 November 2021
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 rather than the metric tensor . The corresponding Lagrangian function, denoted as , is realized by a polynomial expression of the Ricci 4-scalar and of the quadratic curvature 4-scalar . 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 . 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.
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