A non-trivial connection for the metric-affine Gauss-Bonnet theory in $D=4$

Abstract

We study non-trivial (i.e. non-Levi-Civita) connections in metric-affine Lovelock theories. First we study the projective invariance of general Lovelock actions and show that all connections constructed by acting with a projective transformation of the Levi-Civita connection are allowed solutions, albeit physically equivalent to Levi-Civita. We then show that the (non-integrable) Weyl connection is also a solution for the specific case of the four-dimensional metric-affine Gauss-Bonnet action, for arbritrary vector fields. The existence of this solution is related to a two-vector family of transformations, that leaves the Gauss-Bonnet action invariant when acting on metric-compatible connections. We argue that this solution is physically inequivalent to the Levi-Civita connection, giving thus a counterexample to the statement that the metric and the Palatini formalisms are equivalent for Lovelock gravities. We discuss the mathematical structure of the set of solutions within the space of connections.

arXiv comment13 pages, 1 figure. v2: References added. v3: 14 pages. Minor changes, references added. Version to be published in Phys.Lett.B

References in corpus (27)

• Extended Theories of Gravity
• Dark Energy Dominance and Cosmic Acceleration in First Order Formalism
• Metric-affine f(R) theories of gravity
• Inflation with Non-Minimal Coupling: Metric vs. Palatini Formulations
• Currents and Superpotentials in classical gauge invariant theories I. Local results with applications to Perfect Fluids and General Relativity
• Wormholes and nonsingular space-times in Palatini $f(R)$ gravity
• Lovelock Gravity at the Crossroads of Palatini and Metric Formulations
• Extended Gauss-Bonnet gravities in Weyl geometry
• Nonsingular Black Holes in $f(R)$ Theories
• Spacetimes with vector distortion: Inflation from generalised Weyl geometry
• The Cosmology of Ricci-Tensor-Squared Gravity in the Palatini Variational Approach
• Universality of Einstein Equations for the Ricci Squared Lagrangians
• What is a singular black hole beyond General Relativity?
• On the equivalence of the Einstein-Hilbert and the Einstein-Palatini formulations of general relativity for an arbitrary connection
• Torsion/non-metricity duality in f(R) gravity
• Classical resolution of black hole singularities via wormholes
• Equivalence between Palatini and metric formalisms of f(R)-gravity by divergence free current
• Variational Principles and Cosmological Models in Higher-Order Gravity
• On the (non-)uniqueness of the Levi-Civita solution in the Einstein-Hilbert-Palatini formalism
• Inflationary cosmology with Chaplygin gas in Palatini formalism
• Variational and conformal structure of nonlinear metric-connection gravitational lagrangians
• A Cosmological Solution to Mimetic Dark Matter
• Correspondence between Jordan-Einstein frames and Palatini-metric formalisms
• Consistent Levi Civita truncation uniquely characterizes the Lovelock Lagrangians
• Polynomial $f(R)$ Palatini cosmology -- dynamical system approach
• The cosmological constant filter without big bang singularity
• (Non-)uniqueness of Einstein-Palatini Gravity

Cited by in corpus (6)

• Cosmology in $f(Q)$ geometry
• Big-bounce and future time singularity resolution in Bianchi I: the projective invariant Nieh-Yan case
• On the topological character of metric-affine Lovelock Lagrangians in critical dimensions
• Inflation with the Gauss-Bonnet term in the Palatini formulation
• Jacobson's thermodynamic approach to classical gravity applied to non-Riemannian geometries: remarks on the simplicity of Nature
• Comment on "A comment on metric vs metric-affine gravity"