[Paper Review] Dimensional aspects of Lovelock-Lanczos gravity

Aimeric Colléaux|arXiv (Cornell University)|Oct 27, 2020

Cosmology and Gravitation Theories99 references4 citations

TL;DR

This paper presents a systematic framework for regularizing Lovelock-Lanczos gravity in four dimensions, where field equations normally vanish due to dimensional constraints. It introduces covariant, background-independent regularization procedures via metric transformations and dimensional continuation, yielding effective theories with two physical degrees of freedom and non-singular black hole and cosmological solutions.

ABSTRACT

There has recently been an increasing interest in regularizations of Lovelock-Lanczos gravity (LLG) in four dimensions, in which dimensional poles and possibly counter-terms are introduced to compensate the vanishing of the Lovelock field equations in critical and lower dimensions. In this paper, we review and extend some of these results. We first find a class of LLG theories whose perturbative expansion around a given (A)dS vacuum can be regularized up to arbitrary order, the simplest one being close to Lovelock gravities with a unique vacuum. If well-defined, these models might be interpreted as effective field theories of gravitons in four dimensions, or might be combined with other regularization approaches. Among those, we establish the general procedure to obtain $4$D covariant and background independent regularizations from metric transformations. In the conformal (and critical) case, we generalize previous results obtained for the Gauss-Bonnet theory to the full Lovelock series. Similarly, the regularization of Gauss-Bonnet gravity from the breaking of $4$D-covariance down to $3$D is generalized to arbitrary curvature order, seemingly resulting in new Minimally Modified gravities propagating solely the two degrees of freedom of the graviton. Finally, we present general results regarding the minisuperspace regularization of specific sectors of LLG. Non-perturbative (in curvature) regularized theories admitting non-singular black holes as well as non-singular past-dS$_4$ and cyclic closed cosmologies are found. We conclude with the non-uniqueness of these background regularizations by finding inequivalent regularizations of the Bianchi I sector of Lovelock-Lanczos gravity in four dimensions.

Motivation & Objective

• To resolve the vanishing field equations of Lovelock-Lanczos gravity in four dimensions, where higher-order curvature invariants become topological and contribute nothing.
• To develop background-independent, covariant regularization schemes that preserve the second-order nature of the field equations and the two-d.o.f. structure of gravity.
• To construct non-perturbative, regularized models admitting exact non-singular black hole and cyclic cosmological solutions in four dimensions.
• To demonstrate the non-uniqueness of regularization by showing inequivalent regularizations exist even for the Bianchi I sector of the theory.
• To generalize previous results on Gauss-Bonnet regularization to arbitrary curvature order in the full Lovelock series.

Proposed method

• Utilizes dimensional regularization by analytically continuing the spacetime dimension $ d $ to $ d = 4 + \epsilon $, then removing poles via counter-terms to define finite 4D limits.
• Applies metric transformation-based regularizations that preserve diffeomorphism invariance and yield covariant field equations in 4D.
• Employs conformal regularization techniques, particularly in critical dimensions, to regularize the full Lovelock Lagrangian and field equations.
• Introduces $3$-dimensional covariant counter-terms to regularize the Horndeski-Lovelock-Lanczos action, ensuring consistency with lower-dimensional gravity.
• Constructs minisuperspace models for spherically symmetric and Bianchi I spacetimes, using periodic spatial decompositions to define regularized actions.
• Derives explicit expressions for regularized curvature invariants $ Q^{(p)}_{(d-1)} $ in terms of Hubble rates and their time derivatives in $ d=1+3n $, $ d=1+(1+2n) $, and $ d=1+(2+n) $ dimensions.

Experimental results

Research questions

• RQ1 Can Lovelock-Lanczos gravity be consistently regularized in four dimensions to yield non-trivial dynamics despite the vanishing of field equations?
• RQ2 What are the general conditions under which a perturbative expansion of Lovelock-Lanczos gravity around (A)dS vacua can be regularized to arbitrary order?
• RQ3 How can background-independent, covariant regularizations be constructed from metric transformations in four dimensions?
• RQ4 Can the regularization of Gauss-Bonnet gravity via $3$-dimensional covariance be generalized to arbitrary curvature order in the Lovelock series?
• RQ5 Are there non-perturbative, regularized Lovelock models that admit exact non-singular black hole and cosmological solutions in four dimensions?

Key findings

• A class of Lovelock-Lanczos theories exists whose perturbative expansion around (A)dS vacua can be regularized to arbitrary order, suggesting possible effective field theory interpretations in 4D.
• The paper constructs explicit 4D covariant and background-independent regularizations via metric transformations, yielding field equations with two physical degrees of freedom.
• In the conformal and critical case, the regularization procedure generalizes previous Gauss-Bonnet results to the full Lovelock series, producing new minimally modified gravities.
• Non-perturbative regularized models are found that admit exact non-singular black hole solutions and non-singular past-de Sitter and cyclic closed cosmologies.
• The Bianchi I sector of Lovelock-Lanczos gravity admits inequivalent regularizations, demonstrating the non-uniqueness of the regularization procedure even in simple cosmological settings.
• Explicit expressions for regularized curvature invariants $ Q^{(p)}_{(d-1)} $ are derived in $ d=1+3n $, $ d=1+(1+2n) $, and $ d=1+(2+n) $ dimensions, with closed-form formulas for $ p=2 $ and $ p=3 $ in the Kantowski-Sachs sector.