Shape Dynamics and GR linked (Gomes Koslowski)

[marcus]

http://arxiv.org/pdf/1101.5974
The Link between General Relativity and Shape Dynamics
Henrique Gomes, Tim Koslowski
14 pages
(Submitted on 31 Jan 2011)
"We show that one can construct two equivalent gauge theories from a linking theory and give a general construction principle for linking theories which we use to construct a linking theory that proves the equivalence of General Relativity and Shape Dynamics, a theory with fixed foliation but spatial conformal invariance. This streamlines the rather complicated construction of this equivalence performed previously. We use this streamlined argument to extend the result to General Relativity with asymptotically flat boundary conditions. The improved understanding of linking theories naturally leads to the Lagrangian formulation of Shape Dynamics, which allows us to partially relate the degrees of freedom"

It's an important paper, with some analogies to AdS/CFT correspondence, particularly (as the authors point out) to the way AdS/CFT was handled by Laurent Freidel in his paper Reconstructing AdS/CFT, http://arxiv.org/abs/0804.0632 . Shape Dynamics and GR are two different theories, with different gauge symmetries, but an equivalence between them, or duality, can be established.

Shape Dynamics may go over to quantum version more easily than GR. Or it may be easier to show it is recovered by QG in the appropriate limit. So it is of particular interest to the Loop community.

The organizers of Loops 2011 have devoted the first parallel session of the conference, on Monday, to four Shape Dynamics talks.

 

[marcus]

Discussion of Loops 2011 here:
https://www.physicsforums.com/showthread.php?t=494720

Conference program with links to abstracts of the talks here:
http://loops11.iem.csic.es/loops11/index.php?option=com_content&view=article&id=75&Itemid=73

First day 23 May Parallel Session A is on Shape Dynamics:
14:45 - 15:05 T. Koslowski Shape Dynamics.
15:05 - 15:25 H. Gomes Causal structure and more in Shape Dynamics.
15:25 - 15:45 J. Barbour Shape Dynamics and Interpretation of the Hamiltonian Constraint.
15:45 - 16:05 S. Gryb Perturbation theory in shape dynamics.
=====abstracts=====
Shape Dynamics.
Tim Koslowski
Central Room. Monday, May, 23rd, 14:45 - 15:05.
Abstract:
Loop Quantum Gravity provides an powerfull framework for the kinematic quantization of General Relativity, however the complicated algebra of the Hamilton constraints poses serious problems for the construction of its dynamics. Shape Dynamics is a gauge theory that is classically equivalent to General Relativity, with a much simpler constraint algebra (using 3-conformal diffeomorphisms). I will review several aspects of Shape Dynamics and hint what one can expect from a Quantization of Shape Dynamics.

Causal structure and more in Shape Dynamics.
Henrique Gomes
Central Room. Monday, May, 23rd, 15:05 - 15:25.
Abstract:
In recent work, a dual theory for ADM gravity was found (http://arxiv.org/abs/1101.5974). This theory is invariant under foliation preserving 3-diffeomorphisms and 3D conformal transformations that preserve the 3-volume (for the spatially compact case), but does not possesses refoliations as generated by gauge transformations. Here we aim to briefly describe how the causal structure arises when one couples a scalar field to the theory, and how that yields in the appropriate limit the same causal structure as GR. Time allowing we will also describe how to couple electromagnetic fields, and talk about the equivalent "Schwarzschild" solution of the theory.

Shape Dynamics and Interpretation of the Hamiltonian Constraint.
Julian Barbour
Central Room. Monday, May, 23rd, 15:25 - 15:45.
Abstract:
In the paper arXiv:0808.1223, Brendan Foster and I showed that a key theorem of Dirac used to argue that observables in quantum gravity must commute with the Hamiltonian constraint becomes invalid precisely in the case of Hamiltonian constraints. To be precise, in a theory with a single Hamiltonian constraint the constraint does not generate a gauge transformation. I will then explain how gravity can be described as a dynamical theory of shapes governed by a single Hamiltonian constraint that generates genuine evolution of the physical state. Shape dynamics leads to a conceptually very clear and simple picture of the classical dynamics of the universe: one shape follows another. This conceptual clarity could be very helpful in the creation of quantum gravity.

Perturbation theory in shape dynamics.
Sean Gryb
Central Room. Monday, May, 23rd, 15:45 - 16:05.
Abstract:
Shape dynamics is a description of the degrees of freedom of the gravitational field in terms of conformally invariant 3-geometry. It is a gauge theory on the ADM phase space and is defined by a totally constrained Hamiltonian consisting of the usual diffeomorphism constraints, the 3-dimensional conformal constraints (which must preserve the 3-volume in the spatially compact case), and one global Hamiltonian constraint. The challenge of shape dynamics is to explicitly construct the global Hamiltonian constraint since it is found by inverting a non-linear partial differential equation. One strategy for solving this equation is to solve it order by order in some perturbative expansion. I will first describe the general framework of shape dynamics then present two different expansions: one in terms of perturbations about a de Sitter background and another in terms of the 3-volume of the universe. Both approaches indicate that the large volume behaviour of gravity on spatial hypersurfaces is conformally invariant and dominated by the cosmological constant. Moreover, these solutions are attractive. This suggests that shape dynamics is a suitable framework for trying to establish a link between conformal field theory in 3 dimensions and gravity in asymptotically de Sitter spacetimes.

Everybody knows who Julian Barbour is. Henrique Gomes came from University of Sao Paolo Brazil to join John Barrett's group at Uni Nottingham around 2007-2008. In 2009 he coauthored important work on spinfoam asymptotics with others of the Nottingham group. He now has a position at Blackett Lab Imperial College London.

Tim Koslowski is at Perimeter, where Gomes was a visitor for part of the time they were collaborating on the Shape Dynamics papers. Before Perimeter he was at Uni Würzburg.

Sean Gryb is at Perimeter and Uni Waterloo. http://arxiv.org/find/gr-qc/1/au:+Gryb_S/0/1/0/all/0/1

 

[marcus]

In post #1 there's a pointer to the January 2011 paper "The Link Between Shape Dynamics and General Relativity"

But actually this October 2010 paper by the same two authors and Sean Gryb explains some things more clearly and at greater length. It is probably the best place to start:

http://arxiv.org/pdf/1010.2481
Einstein gravity as a 3D conformally invariant theory
Henrique Gomes, Sean Gryb, Tim Koslowski
27 pages. Published in Class.Quant.Grav.28:045005,2011
(Submitted on 12 Oct 2010)
"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 Horava-Lifgarbagez 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."

 

[marcus]

These people say that this idea for a theory of geometry which is dual to GR goes back to this September 2010 paper:
http://arxiv.org/abs/1009.3559
Conformal Superspace: the configuration space of general relativity
Julian Barbour, Niall Ó Murchadha
4 pages
(Submitted on 18 Sep 2010)
"It has long been considered that conformal superspace is the natural configuration space for canonical general relativity. However, this was never definitively demonstrated. We have found that the standard conformal method of solving the Einstein constraints has an unexpected extra symmetry. This allows us to complete the project. We show that given a point and a velocity in conformal superspace, the Einstein equations generate a unique curve in conformal superspace."

Superspace is the space of possible 3D geometries. GR is about the evolution of geometry (interacting with matter) and so the theory would determine a CURVE in superspace as geometry of the universe evolves along that curve.

It looks like I'm going to have to print out the Barbour Murchadha. It's remarkably short (4 pages) as some game-changer papers are. Nice thing about physics---that it can happen that way.

So Julian Barbour is speaking on the first day of the Loops 2011 conference in Madrid. Olé!