[Paper Review] Theoretical and Observational Aspecs in Metric-Affine Gravity: A field theoretic perspective

Adrià Delhom|arXiv (Cornell University)|Jan 24, 2022

Cosmology and Gravitation Theories4 citations

TL;DR

This thesis investigates metric-affine gravity from a field-theoretic perspective, focusing on non-metricity and torsion in higher-curvature gravity models. It proposes a dynamical mechanism for spontaneous Lorentz symmetry breaking via a non-minimal coupling of a vector field (Bumblebee) to the symmetric part of the Ricci tensor, resulting in a non-trivial vacuum expectation value for non-metricity. The key contribution is the first known model where non-metricity acquires a non-zero vacuum expectation value through dynamical symmetry breaking.

ABSTRACT

In this PhD thesis we deal with several theoretical and phenomenological apsects of metric-affine theories of gravity. Concretely, we first give a broad introduction to the necessary tools to understand the framework and elaborate on some subtleties of the minimal coupling prescription between geometry and matter in presence of torsion and nonmetricity. Then we dedicate the central part of the thesis to study the structure of Ricci Based gravity (RBG) theories, which will be of later use to understand generic properties of metric-affine theories. We begin by analysing the structure of the RBG field equations and nontrivial aspects of their solution space. We then analyse the abrosption spectra of some spherically symmetric solutions. Then, we show that, if the projective symmetry in these theories is explicitly broken, then there arise ghost degrees of freedom, and we argue that this will be a generic feature of metric-affine gravity theories. Having done this, we analyse metricafine theories through the EFT lens, showing how the nonmetricity tkes a particular form in generic theories where the symmetrised Ricci tensor appears in the action beyond the Einstein-Hilbert term. This sources effective interactions that we use to place tight constraints to these theories. In the third part of the thesis we present a miscelanea of works which are not so related to the structure of RBG theories. We begin by studying a model for spontaneous breaking of Lorentz symmetry, namely the bumblebee model, in the metric-affine approach. In the following chapter we generalise a conformal invariant definition of proper time given by Perlick to the case with general nonmetricity. Finally, we present arguments that show that the recently proposed D4EGB theory is not well defined in its original form. We finish with a brief outlook.

Motivation & Objective

• To investigate the theoretical structure of metric-affine gravity with non-metricity and torsion from a field-theoretic viewpoint.
• To explore spontaneous Lorentz symmetry breaking via a non-minimal coupling of a vector field (Bumblebee) to the symmetric Ricci tensor.
• To construct a consistent model where non-metricity acquires a non-trivial vacuum expectation value through dynamical symmetry breaking.
• To define a scale-invariant proper time in the presence of generic non-metricity, generalizing Perlick’s Weyl-compatible approach.
• To critically assess the mathematical consistency of four-dimensional Einstein-Gauss-Bonnet gravity (D4EGB), identifying fundamental flaws in its derivation.

Proposed method

• Formalism based on Ricci-Based Gravity (RBG) and projective-invariant metric-affine theories with higher-order curvature invariants.
• Use of non-minimal couplings between matter fields (Bumblebee) and the symmetric part of the Ricci tensor to induce spontaneous Lorentz violation.
• Perturbative analysis of the vacuum structure around a quartic potential for the Bumblebee field, analogous to the Higgs mechanism.
• Generalization of Perlick’s scale-invariant proper time definition to arbitrary non-metricity, ensuring compatibility with the EIH (Ehlers-Pirani-Schild) framework.
• Analysis of the D4EGB model via second-order perturbations around Minkowski spacetime to expose mathematical inconsistencies.
• Regularization of ill-defined equations in D4EGB and proof of non-existence of a diffeomorphism-invariant action yielding the regularized equations.

Experimental results

Research questions

• RQ1 Can a vector field in metric-affine gravity dynamically generate a non-trivial vacuum expectation value for non-metricity?
• RQ2 How does non-minimal coupling between the Bumblebee field and the symmetric Ricci tensor lead to spontaneous Lorentz symmetry breaking?
• RQ3 What is the structure of effective field interactions for scalar and fermionic fields in a background with non-trivial non-metricity?
• RQ4 Can a scale-invariant proper time be consistently defined in generic metric-affine geometries with arbitrary non-metricity?
• RQ5 Is the four-dimensional Einstein-Gauss-Bonnet gravity model mathematically well-defined, particularly in its derivation and solution structure?

Key findings

• The Bumblebee model with non-minimal coupling to the symmetric Ricci tensor yields a stable, non-trivial vacuum with spatial-type Lorentz symmetry breaking.
• The non-trivial vacuum expectation value of the Bumblebee field induces a non-zero vacuum expectation value for the non-metricity tensor, making this the first known dynamical realization of such a configuration.
• Effective interactions for scalar and fermionic fields in the broken phase generate explicit Lorentz-violating terms in their equations of motion.
• A scale-invariant proper time can be consistently defined in generic metric-affine geometries only if the non-metricity satisfies specific compatibility conditions with the conformal structure of light rays and the affine structure of massive particles.
• The D4EGB model is mathematically ill-defined: its equations of motion suffer from a 0/0 indeterminacy in second-order perturbations around Minkowski spacetime.
• No diffeomorphism-invariant action exists that yields the regularized equations of motion of D4EGB, and the claimed spherically symmetric solutions are not valid solutions of either the ill-defined or regularized equations and are geodesically incomplete.