Exploring Cartan gravity with dynamical symmetry breaking

Abstract

It has been known for some time that General Relativity can be regarded as a Yang-Mills-type gauge theory in a symmetry broken phase. In this picture the gravity sector is described by an $SO(1,4)$ or $SO(2,3)$ gauge field $A_{b\mu }^a$ and Higgs field $V^a$ which acts to break the symmetry down to that of the Lorentz group $SO(1,3)$. This symmetry breaking mirrors that of electroweak theory. However, a notable difference is that while the Higgs field $\Phi$ of electroweak theory is taken as a genuine dynamical field satisfying a Klein-Gordon equation, the gauge independent norm $V^2\equiv {\eta }_{ab}{V}^a{V}^b$ of the Higgs-type field $V^a$ is typically regarded as non-dynamical. Instead, in many treatments $V^a$ does not appear explicitly in the formalism or is required to satisfy $V^2=const.\ne 0$ by means of a Lagrangian constraint. As an alternative to this we propose a class of polynomial actions that treat both the gauge connection $A_{b\mu }^a$ and Higgs field $V^a$ as genuine dynamical fields with no ad-hoc constraints imposed. The resultant equations of motion consist of a set of first-order partial differential equations. We show that for certain actions these equations may be cast in a second-order form, corresponding to a scalar-tensor model of gravity. One simple choice leads to the extensively studied Peebles-Ratra rolling quintessence model. Another choice yields a scalar-tensor symmetry broken phase of the theory with positive cosmological constant and an effective mass $M$ of the gravitational Higgs field ensuring the constancy of $V^2$ at low energies and agreement with empirical data if $M$ is sufficiently large. More general cases are discussed corresponding to variants of Chern-Simons modified gravity and scalar-Euler form gravity, each of which yield propagating torsion.

2 figures, matches published version

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• $\mathcal{H}$olographic $\mathcal{N}$aturalness and Pre-Geometric Gravity
• Superconducting and Anti-Ferromagnetic Phases of Spacetime
• Emergence of Gravity's Dynamical and Topological Sectors from Pre-geometry
• Comments on the symmetry breaking condition in MacDowell-Mansouri action