Canonical variational completion and 4D GaussBonnet gravity
Abstract
Recently, a proposal to obtain a finite contribution of second derivative order to the gravitational field equations in D =4 dimensions from a renormalized GaussBonnet term in the action has received a wave of attention. It triggered a discussion whether the employed renormalization procedure yields a welldefined theory. One of the main criticisms is based on the fact that the resulting field equations cannot be obtained as the EulerLagrange equations from a diffeomorphism invariant action. In this work, we use techniques from the inverse calculus of variations to point out that the renormalized truncated GaussBonnet equations cannot be obtained from any action at all (either diffeomorphism invariant or not), in any dimension. Then, we employ canonical variational completion, based on the notion of VainbergTonti Lagrangian—which consists in adding a canonically defined correction term to a given system of equations, so as to make them derivable from an action. To apply this technique to the suggested 4D renormalized GaussBonnet equations, we extend the variational completion algorithm to some classes of PDE systems for which the usual integral providing the VainbergTonti Lagrangian diverges. We discover that in D >4 the suggested field equations can be variationally completed, choosing either the metric or its inverse as field variables; both approaches yield consistently the same Lagrangian, whose variation leads to fourthorder field equations. In D =4 , the Lagrangian of the variationally completed theory diverges in both cases.
 Publication:

European Physical Journal Plus
 Pub Date:
 February 2021
 DOI:
 10.1140/epjp/s13360021011530
 arXiv:
 arXiv:2009.05459
 Bibcode:
 2021EPJP..136..180H
 Keywords:

 General Relativity and Quantum Cosmology;
 High Energy Physics  Theory;
 Mathematical Physics;
 58A20;
 53Z05
 EPrint:
 16 pages