Report on 2109.11325v1

Abstract

We address a subject that could have been analyzed century ago: how does the universe of general relativity look like when it would have been filled with solid matter?Solids break spontaneously the translations and rotations of space itself.Only rather recently it was realized in various context that the order parameter of the solid has a relation to Einsteins dynamical space time which is similar to the role of a Higgs field in a Yang-Mills gauge theory.Such a "crystal gravity" is therefore like the Higgs phase of gravity.The usual Higgs phases are characterized by a special phenomenology.A case in point is superconductivity exhibiting phenomena like the Type II phase, characterized by the emergence of an Abrikosov lattice of quantized magnetic fluxes absorbing the external magnetic field.What to expect in the gravitational setting?The theory of elasticity is the universal effective field theory associated with the breaking of space translations and rotations having a similar status as the phase action describing a neutral superfluid.A geometrical formulation appeared in its long history, similar in structure to general relativity, which greatly facilitates the marriage of both theories.With as main limitation that we focus entirely on stationary circumstances -the dynamical theory is greatly complicated by the lack of Lorentz invariance -we will present a first exploration of a remarkably rich and often simple physics of "Higgsed gravity".CONTENTS I. Introduction and overview.2 A. An egg of Columbus? 2 B. Why crystal gravity is opportune.3 C.A simple essence: the "wedge fluxoid".4 D. The mathematical machinery: Kleinert's multivalued fields.4 E. The solids on the rigid two dimensional manifolds of the soft matter community.5 F. The overview: the structure of crystal gravity and the main results.5 1.The basics: elasticity and frame fixing (Section II) 6 2. Vortex-boson duality: the analogy (Section III).6 3. Linearized crystal gravity (Section IV).7 4. Translational topology: dislocations andCartan torsion (Section V). 7 5.The topologization of curvature: torque gravitons and the defect density (Section VI). 8 6.Curvature fluxoids and the gravitational obstruction (Section VII).9 7.The Platonic incarnation: the gravitational Abrikosov lattice and the polytopes (Section VIII).9 8.But there is also twist in three dimensions (Section IX). 10 9. The conclusions (Section X). 10 II.Frame fixing: Elasticity as the Higgs field of gravity.10 A. The first law of crystal gravity.10 B. The brutal breaking of Lorentz invariance.13 C. Isotropic elasticity and the shear modulus.III.Intermezzo: Abelian-Higgs duality, the mass of the photon and the fluxoid.A. The superconductor in dual representation.B. The Higgs mechanism.C. Interactions between vortices.D. Fluxoids in the charged superconductor.E. Fluxoids in 3+1 dimensions.IV.Linearized crystal gravity: gravitons and static shear stress.A. Stress-strain duality and gravity.B. Linearized gravity in helical projection.C. The stress action and the spin 2 shear gravitons.D. Coupling static stress with the gravitons: the gravitational Higgs mechanism.E. The mass of the graviton.F. Gravitational hardening of solids.G.The asymmetry of time and the role of the phonons.V. Dislocations, shear stress, torsion and gravitons.A. The dislocation currents sourcing the stress photons.B. Dislocation currents and the space groups.C. Dynamical dislocation currents in a 3+1D space time.D. Gravitons accelerate dislocations.E. Gravitons heat malleable solids.F. Dislocations and the quantization of torsion.

Authors 3

1. Jan Zaanen Aachen Institute-Lorentz for Theoretical Physics

   Leiden University

   Affiliation as printed

   The Institute Lorentz for Theoretical Physics, Leiden University, Leiden, The Netherlands and

2. Floris Balm Aachen Institute-Lorentz for Theoretical Physics

   Leiden University

   Affiliation as printed

   The Institute Lorentz for Theoretical Physics, Leiden University, Leiden, The Netherlands and

3. Aron Beekman

   Keio University

   Affiliation as printed

   Department of Physics and Research and Education Center for Natural Sciences, Keio University, 3-14-1 Hiyoshi, Kohoku-ku, Yokohama 223-8522, Japan