The relative energy of homogeneous and isotropic universes from variational principles

We calculate the relative conserved currents, superpotentials and conserved quantities between two homogeneous and isotropic universes. In particular we prove that their relative "energy" (defined as the conserved quantity associated to cosmic time coordinate translations for a comoving observer) is vanishing and so are the other conserved quantities related to a Lie subalgebra of vector fields isomorphic to the Poincar\'e algebra. These quantities are also conserved in time. We also find a relative conserved quantity for such a kind of solutions which is conserved in time though non-vanishing. This example provides at least two insights in the theory of conserved quantities in General Relativity. First, the contribution of the cosmological matter fluid to the conserved quantities is carefully studied and proved to be vanishing. Second, we explicitly show that our superpotential (that happens to coincide with the so-called KBL potential although it is generated differently) provides strong conservation laws under much weaker hypotheses than the ones usually required. In particular, the symmetry generator is not needed to be Killing (nor Killing of the background, nor asymptotically Killing), the prescription is quasi-local and it works fine in a finite region too and no matching condition on the boundary is required.Comment: Corrected typos and improved forma

Chetaev vs. vakonomic prescriptions in constrained field theories with parametrized variational calculus

Starting from a characterization of admissible Cheataev and vakonomic variations in a field theory with constraints we show how the so called parametrized variational calculus can help to derive the vakonomic and the non-holonomic field equations. We present an example in field theory where the non-holonomic method proved to be unphysical

Further Extended Theories of Gravitation: Part I

We shall here propose a class of relativistic theories of gravitation, based on a foundational paper of Ehlers Pirani and Schild (EPS).All "extended theories of gravitation" (also known as f(R) theories) in Palatini formalism are shown to belong to this class. In a forthcoming paper we shall show that this class of theories contains other more general examples. EPS framework helps in the interpretation and solution of these models that however have exotic behaviours even compared to f(R) theories.Comment: 10 pages. Some refs adde