Shape Dynamics
Shape Dynamics is a new theory of gravity that, for most situations, is completely indistinguishable from General Relativity, but which is unlike General Relativity in that it is based on a completely different symmetry principle and notion of time. In Shape Dynamics, there is no notion of relative simultaneity: simultaneity is absolute. Instead, it is local spatial scale that is relative. Despite these distinctly different features, it can be proved that, for a specific choice of local scale, one can almost always reproduce a spacetime that solves the Einstein equations. Some known exceptions to this include spacetimes with singularities, in which case the analogous solutions in Shape Dynamics are singularity free. Although it is not known how general this result is, it is hoped that these differences, along with the presence of a new symmetry principle, may help pave a new route towards the quantum theory. For a more detailed description of Shape Dynamics and suggestions for further reading, please see below. This page is intended for researchers or beginning graduate students looking to find an introduction to the basics of Shape Dynamics. For a more elementary introduction to the ideas behind Shape Dynamics, please explore my Outreach page, where I have indexed several popular science expositions of ideas directly or indirectly related to Shape Dynamics.
Brief History
The idea for Shape Dynamics originated from Julian Barbour, who believes that angles between spacial configurations represent the fundamental ontology of Nature. These angles are meant to represent the scaleinvariant information about the “shape” degrees of freedom of a system (see here to get Barbour's view on Shape Dynamics). Since Barbour's original proposal, the field has grown rapidly into a new approach to gravitational physics and, in particular, a new starting point for Quantum Gravity.
Early versions of Shape Dynamics (see, for instance, this early paper) involved fixing a preferred notion of simultaneity in General Relativity such that the freely specifiable initial data of the system are locally scaleinvariant. The authors argued that, since the theory only feeds on scaleinvariant information, the whole framework should be considered scaleinvariant. Later, it was shown — in a paper with Gomes, Koslowski, and myself — that the preferred notion of simultaneity used in the earlier versions of Shape Dynamics could be used to construct a gauge theory, defined on the same phase space as General Relativity, but with a different symmetry, namely: local scaleinvariance. The cost of having this extra symmetry is the failure of the formalism to reproduce a spacetime with all the symmetries of General Relativity. Thus, what we have is a kind of symmetry trading procedure that allows us to trade a subset of the full spacetime symmetries with local scaleinvariance. For more details on this paper, expand this link.
The realization that General Relativity can be reinterpreted as a scaleinvariant gauge theory has sparked a great deal of recent activity, including: understanding the implications to the Problem of Time in Quantum Gravity, eliminating certain singularities in GR, identifying gaugeinvariant observables in cosmology, providing a new perspective on AdS/CFT, and exploring a new theory space for Quantum Gravity. See below for a brief discussion of some of these exciting possibilities.
Motivation
Shape Dynamics is an attempt to describe gravity using a framework that is invariant under the symmetry
which is often called conformal invariance by mathematicians or local Weyl invariance by physicists. In the above, g_{ab} is the spatial metric and φ(x) is a conformal factor that can vary arbitrarily from point to point in space. This symmetry represents arbitrary xdependent rescalings of infinitesimal arclengths as measured by the spatial metric, g_{ab}. One can motivate this symmetry through different means. Some of these are listed below.
 Measurements of length are ultimately local comparisons: one is always effectively doing a local comparison between some reference rod and the length that one is trying to measure. Thus, the intrinsic length of any object would appear to have no empirical meaning. However, the length of an infinitesimal segment of an object is determined by the spatial metric. It must then follow that part of the spatial metric itself, specifically its conformal factor, should also have no empirical meaning. This reasoning leads us to the conclusion that the local scale factor of the metric should be treated as a pure gauge quantity in a theory that faithfully represents reality. What Shape Dynamics shows is that, up to a potential global scale, this requirement is compatible with General Relativity. That scale appears to emerge in the framework is seen to be a choice one makes in the formalism to describe physics locally using spacetime.
 One the most common ways for quantum field theories to be deemed meaningful is through the presence of a UV fixed point in the RG flow of the theory. Conventional wisdom from condensed matter systems suggests that such fixed points exhibit local Weyl invariance. One can understand this in simple terms by noting that fixed points are characterized by dimensionless couplings with vanishing βfunctions. This is another way of saying that the couplings of the theory do not vary with scale; and is, thus, a form of scale invariance. One might then expect that any UV completion of GR should exhibit Weyl invariance in this way. Given that perturbative searches for a standard UV fixed point in GR have come up negative, it may be worth investigating a slightly different framework that implements only spatial Weyl invariance instead, while still reproducing GR at low energy.
 More pragmatically, given that no attempt to find a completely adequate theory of quantum gravity has been successful, it is perhaps useful to investigate different formulations of the classical theory (in this case, one with a different symmetry principle) which could lead to insights into the quantization. The alternative symmetry principle offered by Shape Dynamics provides a new theory space for quantum gravity and differs from standard GR on a global level. This could change the whole structure of the quantum theory.
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The Basics of Shape Dynamics
Shape Dynamics is a Hamiltonian theory of Weylinvariant geometry, and is most naturally described in terms of phase space. This means that it is most closely connected with the 3+1 formulation of General Relativity developed by ADM (see a nice detailed review of their work here). This means that Shape Dynamics can only correspond to spacetimes that are globally hyperbolic. Furthermore, it is most natural for conceptual purposes (though not strictly necessary for the formalism) to fix the spatial topology to be closed. In principle, we would like to implement the phase space analogue of the symmetry (1) as a canonical transformation so that the momentum, π^{ab}, conjugate to g_{ab} transforms as
As it turns out, however, GR can crucially only be obtained if the conformal factor φ is restricted such that the volume of the original metric is equal to the volume of the transformed metric. Such transformations we will call Volume Preserving Conformal Transformations (or VPCTs).
It is now possible to describe the structure of the theory by first describing its symmetries and then by specifying how the evolution is computed on phase space. We first demand that spatial lengths be invariant under how one chooses to lay down physical rods in order to build physical reference frames representing some choice of coordinates. This implies that we want invariance under local spatial diffeomorphisms. Infinitesimally, this symmetry is generated by the ADM diffeomorphism constraint, H_{a}(x) = 0, which is imported directly into Shape Dynamics. As discussed above, we also want gauge invariance under VPCTs. This is obtained by imposing the additional constraint D(x) = 0, whose definition one can find (for instance) in the original paper. The Diff constraint, H_{a} = 0, and the VPCT constraint, D = 0, form a simple first class algebra. Together, they constitute all the local gaugegenerating constraints of the theory. This means that the local physical degrees of freedom are simply the degrees of freedom associated with the conformal geometry. However, because of the global volume preserving restriction on φ(x), there are still two global degrees of freedom that are, as yet, unconstrained. These are the total volume, V, of the spatial geometry and its canonical conjugate: the York time, τ (up to a numerical factor).
To have the same degrees of freedom as GR, we must impose an additional first class constraint to kill the global variables (V,τ). This can be done using a unique global constraint whose integral curves can be used to reconstruct a solution to the Einstein equations. There is a simple way to visualise how this Shape Dynamics Hamiltonian, H_{SD}, can be obtained. First, imagine the constraint surface formed by D = 0 as a constraint surface on the ADM phase space. Now imagine the constraint surface formed by H’ = H_{ADM}  H_{CMC}, where, H_{ADM} is the ADM Hamiltonian constraint and, for the moment, H_{CMC} is just some smearing of it. In 1972, York and O'Murchadha (see here) showed that there is a unique smearing such that D = 0 intersects the surface H’ = 0 exactly once. In other words, it is a good gauge fixing (or choice of foliation) for GR. This choice of foliation is called Constant Mean (extrinsic) Curvature (CMC) gauge (because D = 0 represents surfaces of constant mean extrinsic curvature). This leads to the picture drawn below.
The last step is to motivate the definition of H_{SD} by making use of the above diagram. The easiest way to define a VPCT invariant Hamiltonian that is equivalent to GR in CMC gauge is simply to gauge invariantly lift the flow of H_{CMC} everywhere onto the D = 0 surface. To do this, we have to find the particular conformal factor φ_{0} that will take you from some arbitrary point on the D = 0 surface to the intersection with H’ = 0. This can be achieved by solving an elliptic partial differential equation called the Lichnerowicz—York equation (see here for a modern perspective on this equation).
A word of caution is in order when trying to take too literally the above diagram. Drawing infinite dimensional surfaces on a two dimensional sheet necessarily requires certain simplifications. In more physically relevant situations, such as when adding matter fields or when considering asymptotically flat spacetimes, the split of H_{ADM} into H’ and H_{CMC} is not always unique. This can lead to ambiguities in the definition of H_{SD} that are important to understand. Furthermore, the existence of H_{SD} is not obvious. To study these issues, it is convenient to introduce the Linking Theory formalism, where the ADM phase space is extended to include the conformal factor, φ, as a gauge field with conjugate momentum, π_{φ}. Using this framework, it is possible to show that the ADM theory and SD arise as different gauge fixings of this Linking Theory, and are therefore equivalent. Subsequent gauge fixings can be performed in each theory in order to obtain evolution in CMC gauge. This gives a dictionary for comparing both theories. The general picture is illustrated in the diagram below. For more details about the construction of the Linking Theory, see the paper by Gomes and Koslowski.
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Advanced Topics
The advent of the new version of Shape Dynamics has sparked some recent activity in several research directions aimed at understanding the classical and quantum theory in more detail. Here is a short — and certainly nonexhaustive list — outlining these different approaches in no particular order. I have added extra detail to projects that I have participated in.

Conformal Cartan Variables:

Black Holes:

The Problem of Time:

AdS/CFT:

Quantization:
The biggest obstruction towards quantization in Shape Dynamics is finding a way to understand the quantum action of the Shape Dynamics Hamiltonian, which is constructed by solving a nonlinear elliptic differential equation. The inherent difficulty comes from two sources: i) the polynomial nature of the Lichnerowicz—York equation and ii) the inversion of the spatial Laplancian. The first difficulty originates from the varying conformal weights of the different terms of the Hamiltonian constraint. The second difficulty originates from the inhomogeneous transformation properties of the Ricci scalar under Weyl transformations. Thus, all problems result from the simple fact that the natural densities and curvature invariants of Riemannian geometry have unpredictable and seemingly unnatural transformation properties under Weyl transformations. What I believe is needed is a better set of mathematical tools for identifying natural Weylinvariant structures. Trying to do Shape Dynamics without such tools is like trying to do General Relativity without differential geometry. In principle, it would be possible to express everything in GR in terms of simple vector calculus, but then it would be extremely difficult to identify the natural structures, such as the Einstein tensor.
Fortunately, Élie Cartan developed a set of mathematical tools which might have exactly the required properties. He developed a generalization of Riemannian geometry, which is now named after him, where one generalizes the notion of tangent space to include any homogeneous space with the same dimension as the geometry you want to describe. Because a homogeneous space can always be constructed by forming the quotient of two group manifolds, the fibres of the tangent bundle have the structure of a G/Hspace, where G is some large Lie group and H is some subgroup which stabilizes G. A connection, A, on this fibre bundle then splits into two pieces: one which corresponds to the broken part of the large symmetry group G, and another which corresponds to the unbroken symmetry H. The former is identified with the frame field, e, of the geometry, while the latter is identified with the generalized connection, ω,
A simple example to help understand the general structure is Riemannian geometry itself. In this case, the large group, G,is the Galilean group (or the Poincaré group in the case of a pseudoRiemannian manifold) and the stabilizing group, H, is the rotation group (or the Lorentz group for the pseudoRiemannian case). The components of the translational subgroup of A, which is broken by the quotienting, corresponds to the usual frame field, e. The curvature of this part of A is then the usual torsion. The components of the unbroken rotational subgroup correspond to the usual SO(D) connection (where D is the dimension of space), ω. Its curvature is the usual Riemannian curvature tensor.
An even simpler (but new) example is the Cartan geometry formed by quotienting SO(3) by SO(2). The result of the quotient is the homogeneous 2sphere. One can picture the role of the different components of A by imagining using a 2sphere as the tangent space of some arbitrary 2dimensional manifold, as shown below. There are 2 distinct ways that this 2sphere can be manipulated: i) it can be rotated about the point of contact and ii) it can be rolled (without slipping) in the two independent directions of the manifold. THe first manipulation is a choice of SO(2) rotation that stabilizer the full SO(3) isometries of the 2sphere. The second class of manipulations corresponds to broken part of the SO(3) symmetries and represents translations on the 2sphere.
Describe conformal Cartan geometry.
Describe how it might apply to Shape Dynamics.
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Suggested Reading
(Give general links for reading more about this stuff...)