The relative energy of homogeneous and isotropic universes from variational priciples

We calculate the relative conserved currents, superpotentials and conserved quantities between two homogeneous and isotropic universes. In particular we prove that their rel- ative \energy" (dened as the conserved quantity associated to cosmic time coordinate translations for a comoving observer) is vanishing and so are the other conserved quanti- ties related to a Lie subalgebra of vector elds isomorphic to the Poincare algebra. These quantities are also conserved in time. We also nd a relative conserved quantity for such a kind of solutions which is conserved in time though non-vanishing. This example pro- vides at least two insights in the theory of conserved quantities in General Relativity. First, the contribution of the cosmological matter uid to the conserved quantities is carefully studied and proved to be vanishing. Second, we explicitly show that our su- perpotential (that happens to coincide with the so-called KBL potential although it is generated dierently) 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 ne in a nite region too and no matching condition on the boundary is required.