Sean Gryb

Einstein gravity as a 3D conformally invariant theory

We give an alternative description of the physical content of general relativity that does not require a Lorentz invariant spacetime. Instead, we find that gravity admits a dual description in terms of a theory where local size is irrelevant. The dual theory is invariant under foliation preserving 3–diffeomorphisms and 3D conformal transformations that preserve the 3–volume (for the spatially compact case). Locally, this symmetry is identical to that of Hoyrava–Lifshitz gravity in the high energy limit but our theory is equivalent to Einstein gravity. Specifically, we find that the solutions of general relativity, in a gauge where the spatial hypersurfaces have constant mean extrinsic curvature, can be mapped to solutions of a particular gauge fixing of the dual theory. Moreover, this duality is not accidental. We provide a general geometric picture for our procedure that allows us to trade foliation invariance for conformal invariance. The dual theory provides a new proposal for the theory space of quantum gravity.

A shape dynamical approach to holographic renormalization

We provide a bottom-up argument to derive some known results from holographic renormalization using the classical bulk–bulk equivalence of General Relativity and Shape Dynamics, a theory with spatial conformal (Weyl) invariance. The purpose of this paper is twofold: (1) to advertise the simple classical mechanism, trading off gauge symmetries, that underlies the bulk–bulk equivalence of General Relativity and Shape Dynamics to readers interested in dualities of the type of AdS/conformal field theory (CFT); and (2) to highlight that this mechanism can be used to explain certain results of holographic renormalization, providing an alternative to the AdS/CFT conjecture for these cases. To make contact with the usual semiclassical AdS/CFT correspondence, we provide, in addition, a heuristic argument that makes it plausible that the classical equivalence between General Relativity and Shape Dynamics turns into a duality between radial evolution in gravity and the renormalization group flow of a CFT. We believe that Shape Dynamics provides a new perspective on gravity by giving conformal structure a primary role within the theory. It is hoped that this work provides the first steps toward understanding what this new perspective may be able to teach us about holographic dualities.

2+1 gravity on the conformal sphere

We show that there are two equivalent first order descriptions of

The Role of Time in Relational Quantum Theories

2+1

A definition of background independence

gravity with a nonzero cosmological constant. One is the well-known spacetime description, and the other is in terms of evolving conformal geometry. The key tool that links these pictures is Cartan geometry, a generalization of Riemannian geometry that allows for geometries locally modeled off arbitrary homogeneous spaces. The two different interpretations suggest two distinct phase space reductions. The spacetime picture leads to the

Symmetry and Evolution in Quantum Gravity

2+1

Right About Time

formulation of general relativity due to Arnowitt, Deser, and Misner, while the conformal picture leads to shape dynamics. Cartan geometry thus provides an alternative to symmetry trading for explaining the equivalence of general relativity and shape dynamics.

The gravity/CFT correspondence

We propose a solution to the problem of time for systems with a single global Hamiltonian constraint. Our solution stems from the observation that, for these theories, conventional gauge theory methods fail to capture the full classical dynamics of the system and must therefore be deemed inappropriate. We propose a new strategy for consistently quantizing systems with a relational notion of time that does capture the full classical dynamics of the system and allows for evolution parametrized by an equitable internal clock. This proposal contains the minimal temporal structure necessary to retain the ordering of events required to describe classical evolution. In the context of shape dynamics (an equivalent formulation of general relativity that is locally scale invariant and free of the local problem of time) our proposal can be shown to constitute a natural methodology for describing dynamical evolution in quantum gravity and to lead to a quantum theory analogous to the Dirac quantization of unimodular gravity.

Shape dynamics and Mach's principles: Gravity from conformal geometrodynamics

We propose a definition for background (in)/dependence in dynamical theories of the evolution of configurations that have a continuous symmetry and test this definition on particle models and on gravity. Our definition draws from Barbours best matching framework developed for the purpose of implementing spatial and temporal relationalism. Among other interesting theories, general relativity can be derived within this framework in novel ways. We study the detailed canonical structure of a wide range of best matching theories and show that their actions must have a local gauge symmetry. When gauge theory is derived in this way, we obtain at the same time a conceptual framework for distinguishing between background-dependent and -independent theories. Gauge invariant observables satisfying Kuchařs criterion are identified and, in simple cases, explicitly computed. We propose a procedure for inserting a global background time into temporally relational theories. Interestingly, using this procedure in general relativity leads to unimodular gravity.

Why gravity codes the renormalization of conformal field theories

We propose an operator constraint equation for the wavefunction of the Universe that admits genuine evolution. While the corresponding classical theory is equivalent to the canonical decomposition of General Relativity, the quantum theory contains an evolution equation distinct from standard Wheeler–DeWitt cosmology. Furthermore, the local symmetry principle—and corresponding observables—of the theory have a direct interpretation in terms of a conventional gauge theory, where the gauge symmetry group is that of spatial conformal diffeomorphisms (that preserve the spatial volume of the Universe). The global evolution is in terms of an arbitrary parameter that serves only as an unobservable label for successive states of the Universe. Our proposal follows unambiguously from a suggestion of York whereby the independently specifiable initial data in the action principle of General Relativity is given by a conformal geometry and the spatial average of the York time on the spacelike hypersurfaces that bound the variation. Remarkably, such a variational principle uniquely selects the form of the constraints of the theory so that we can establish a precise notion of both symmetry and evolution in quantum gravity.