Eur. Phys. J. Plus (2021) 136: 180
https://doi.org/10.1140/epjp/s13360-021-01153-0

Regular Article

Canonical variational completion and 4D Gauss–Bonnet gravity

¹ Laboratory of Theoretical Physics, Institute of Physics, University of Tartu, W. Ostwaldi 1, 50411, Tartu, Estonia
² Faculty of Mathematics and Computer Science, Transilvania University, Iuliu Maniu Str. 50, 500091, Brasov, Romania

^c nico.voicu@unitbv.ro

Received: 10 December 2020
Accepted: 23 January 2021
Published online: 5 February 2021

Abstract

Recently, a proposal to obtain a finite contribution of second derivative order to the gravitational field equations in dimensions from a renormalized Gauss–Bonnet term in the action has received a wave of attention. It triggered a discussion whether the employed renormalization procedure yields a well-defined theory. One of the main criticisms is based on the fact that the resulting field equations cannot be obtained as the Euler–Lagrange 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 Gauss–Bonnet 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 Vainberg–Tonti 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 Gauss–Bonnet equations, we extend the variational completion algorithm to some classes of PDE systems for which the usual integral providing the Vainberg–Tonti Lagrangian diverges. We discover that in 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 fourth-order field equations. In , the Lagrangian of the variationally completed theory diverges in both cases.