Symmetric Space Cartan Connections and Gravity in Three and Four Dimensions

Einstein gravity in both 3 and 4 dimensions, as well as some interesting generalizations, can be written as gauge theories in which the connection is a Cartan connection for geometry modeled on a symmetric space. The relevant models in 3 dimensions include Einstein gravity in Chern-Simons form, as well as a new formulation of topologically massive gravity, with arbitrary cosmological constant, as a single constrained Chern-Simons action. In 4 dimensions the main model of interest is MacDowell-Mansouri gravity, generalized to include the Immirzi parameter in a natural way. I formulate these theories in Cartan geometric language, emphasizing also the role played by the symmetric space structure of the model. I also explain how, from the perspective of these Cartan-geometric formulations, both the topological mass in 3d and the Immirzi parameter in 4d are the result of non-simplicity of the Lorentz Lie algebra so(3,1) and its relatives. Finally, I suggest how the language of Cartan geometry provides a guiding principle for elegantly reformulating any 'gauge theory of geometry'.Comment: Article prepared for special journal issue dedicated to Elie Carta

Geometrodynamics and Lorentz symmetry

We study the dynamics of gauge theory and general relativity using fields of local observers, thus maintaining local Lorentz symmetry despite a space/time splitting of fields. We start with Yang--Mills theory, where observer fields are defined as normalized future-timelike vector fields. We then define observers without a fixed geometry, and find these play two related roles in general relativity: splitting fields into spatial and temporal parts, and "breaking" gauge symmetry, effectively reducing the spacetime SO(n,1) connection to an observer-dependent spatial SO(n) connection. In both gauge theory and gravity, the observer field reduces the action to canonical form, without using gauge fixing. In the 4d gravity case, the result is a manifestly Lorentz covariant counterpart of the Ashtekar-Barbero formulation. We also explain how this leads geometrically to a picture of general relativity in terms of "observer space" rather than spacetime---a setting where both spacetime symmetry and the dynamical description are simultaneously available.Comment: 11 pages. Submission for the proceedings of "3Quantum: Algebra, Geometry, Information", Tallinn, July 201

Holographic Special Relativity

We reinterpret special relativity, or more precisely its de Sitter deformation, in terms of 3d conformal geometry, as opposed to (3+1)d spacetime geometry. An inertial observer, usually described by a geodesic in spacetime, becomes instead a choice of ways to reverse the conformal compactification of a Euclidean vector space up to scale. The observer's "current time," usually given by a point along the geodesic, corresponds to the choice of scale in the decompactification. We also show how arbitrary conformal 3-geometries give rise to "observer space geometries," as defined in recent work, from which spacetime can be reconstructed under certain integrability conditions. We conjecture a relationship between this kind of "holographic relativity" and the "shape dynamics" proposal of Barbour and collaborators, in which conformal space takes the place of spacetime in general relativity. We also briefly survey related pictures of observer space, including the AdS analog and a representation related to twistor theory.Comment: 17 pages, 5 illustration

The geometric role of symmetry breaking in gravity

In gravity, breaking symmetry from a group G to a group H plays the role of describing geometry in relation to the geometry the homogeneous space G/H. The deep reason for this is Cartan's "method of equivalence," giving, in particular, an exact correspondence between metrics and Cartan connections. I argue that broken symmetry is thus implicit in any gravity theory, for purely geometric reasons. As an application, I explain how this kind of thinking gives a new approach to Hamiltonian gravity in which an observer field spontaneously breaks Lorentz symmetry and gives a Cartan connection on space.Comment: 4 pages. Contribution written for proceedings of the conference "Loops 11" (Madrid, May 2011

Linking Covariant and Canonical General Relativity via Local Observers

Hamiltonian gravity, relying on arbitrary choices of "space," can obscure spacetime symmetries. We present an alternative, manifestly spacetime covariant formulation that nonetheless distinguishes between "spatial" and "temporal" variables. The key is viewing dynamical fields from the perspective of a field of observers -- a unit timelike vector field that also transforms under local Lorentz transformations. On one hand, all fields are spacetime fields, covariant under spacetime symmeties. On the other, when the observer field is normal to a spatial foliation, the fields automatically fall into Hamiltonian form, recovering the Ashtekar formulation. We argue this provides a bridge between Ashtekar variables and covariant phase space methods. We also outline a framework where the 'space of observers' is fundamental, and spacetime geometry itself may be observer-dependent.Comment: 8 pages; Essay written for the 2012 Gravity Research Foundation Awards for Essays on Gravitatio

Static Spherically Symmetric Kerr-Schild Metrics and Implications for the Classical Double Copy

We discuss the physical interpretation of stress-energy tensors that source static spherically symmetric Kerr-Schild metrics. We find that the sources of such metrics with no curvature singularities or horizons do not simultaneously satisfy the weak and strong energy conditions. Sensible stress-energy tensors usually satisfy both of them. Under most circumstances these sources are not perfect fluids and contain shear stresses. We show that for these systems the classical double copy associates the electric charge density to the Komar energy density. In addition, we demonstrate that the stress-energy tensors are determined by the electric charge density and their conservation equations.Comment: 11 page