On the conservation of superenergy and its applications

Abstract

In this work we present a geometric identity involving the Bel-Robinson tensor which is formally similar to the Sparling identity (which involves the Einstein tensor through the Einstein 3-form). In our identity the Bel-Robinson tensor enters through the Bel-Robinson 3-form which, we believe, is introduced in the literature for the first time. The meaning of this identity is that it is possible to formulate a generic conservation law for the quantity represented by the Bel-Robinson tensor (superenergy). We also show how one can use the Bel-Robinson 3-form to estimate the components of the Bel-Robinson tensor which are computed with respect to the causal elements of a frame. This estimate could be useful in a global existence proof of the solutions of a theory of gravitation in dimension four.

Plebański–Demiański solution of general relativity and its expressions quadratic and cubic in curvature: Analogies to electromagnetism

“…In the first line, we used the identity ϑ µ ∧ η ν = δ µ ν η, and in the last line we used the vacuum relation (60). We now show the equivalence of our result to the 3-form introduced by Gómez-Lobo [17]:…”

“…We summarize: The 3-formΣ νρσ is (up to a factor of 2) the energy momentum of Eq. The Bel-Robinson 3-form has been found recently in an interesting paper by Gómez-Lobo [17].…”

Cylindrical gravitational waves: C-energy, super-energy and associated dynamical effects

“…The latter were introduced for vacuum gravitational fields in order to provide a definition of local energy density (super-energy density) and energy flux (super-Poynting vector) in complete formal analogy with electromagnetism. Their properties have been extensively investigated over the years and applications to many different contexts, including gravitational wave spacetimes, have been discussed [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]. A conserved (super-momentum) four vector can be formed also in this case [28,37], carrying the twofold information of the super-energy density and the spatial supermomentum density as measured by a given observer family.…”

A gravitational energy–momentum and the thermodynamic description of gravity

“…BR is completely symmetric, traceless and conserved in vacuum; in analogy with the electromagnetic counterpart, the symmetries of such object allow to define a super-energy density and a super-Poynting vector. The physical interpretation of super-energies is still a matter of debate and research [7][8][9].…”