Superpotentials from variational derivatives rather than Lagrangians in relativistic theories of gravity
arXiv:0807.3079 · doi:10.1088/0264-9381/25/17/175024
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.
Scheduled to appear in Class. Quantum Grav. August 2008
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- Covariantized Noether identities and conservation laws for perturbations in metric theories of gravity
- Three types of superpotentials for perturbations in the Einstein-Gauss-Bonnet gravity
- Some globally conserved currents from generalized Killing vectors and scalar test fields
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