Superpotentials from variational derivatives rather than Lagrangians in relativistic theories of gravity

TL;DR

This paper applies a variational derivative approach to derive superpotentials in Lovelock gravity theories, providing explicit formulas and confirming known results for Einstein and Gauss-Bonnet gravities.

Contribution

It introduces a method to derive superpotentials from variational derivatives in Lovelock gravity, offering explicit expressions and extending previous results.

Findings

Derived general superpotential expressions for Lovelock Lagrangians.

Confirmed the KBL superpotential in Einstein gravity across dimensions.

Provided the superpotential for Gauss-Bonnet and Einstein-Gauss-Bonnet theories.

Abstract

The prescription of Silva to derive superpotential equations from variational derivatives rather than from Lagrangian densities is applied to theories of gravity derived from Lovelock Lagrangians in the Palatini representation. Spacetimes are without torsion and isolated sources of gravity are minimally coupled. On a closed boundary of spacetime, the metric is given and the connection coefficients are those of Christoffel. We derive equations for the superpotentials in these conditions. The equations are easily integrated and we give the general expression for all superpotentials associated with Lovelock Lagrangians. We find, in particular, that in Einstein's theory, in any number of dimensions, the superpotential, valid at spatial and at null infinity, is that of Katz, Bicak and Lynden-Bell, the KBL superpotential. We also give explicitly the superpotential for Gauss-Bonnet theories of gravity. Finally, we find a simple expression for the superpotential of Einstein-Gauss-Bonnet theories with an anti-de Sitter background: it is minus the KBL superpotential, confirming, as it should, the calculation of the total mass-energy of spacetime at spatial infinity by Deser and Tekin.