The Link between General Relativity and Shape Dynamics

Henrique Gomes; Tim Koslowski

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The Link between General Relativity and Shape Dynamics

Henrique Gomes, Tim Koslowski

General Relativity and Quantum Cosmology · Submitted 31 Jan 2011 · 21 references

Pith's verdict: The linking-theory framework is the real contribution; the asymptotically flat extension leans on an existence argument the authors flag but do not finish.

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arXiv ID1101.5974 (version 3)

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CitationHenrique Gomes, Tim Koslowski. "The Link between General Relativity and Shape Dynamics." arXiv:1101.5974 (2011). https://pith.science/paper/BRZWMST7

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Editor's note

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The thing to know: this is the streamlined version of the Gomes–Gryb–Koslowski equivalence, and the streamlining is genuine. Section 2 turns the Stückelberg/Kretschmannization machinery into two clean propositions — a linking theory with constraints of the form φ−f and π_φ−g, two complementary gauge fixings, and you get two equivalent theories on the reduced phase space. That is a real piece of organization and it is reusable; I expect people coupling matter to Shape Dynamics to cite it. The asymptotically flat extension and the Lagrangian section are the new physics content, and both are worth having in the literature. What holds up: Propositions 1 and 2 are straightforward and correct as stated. The canonical transformation (23) is right, Q = π_φ − 4π is the right charge, and recovering ADM by fixing φ = 0 is immediate. Section 4 is honest, including the negative conclusion that no dynamical relation between lapse and conformal multiplier exists. Where it is soft: §3.3.1. Propositions 3 and 4 need Ker(Δ) on the asymptotically flat slice to be exactly one-dimensional and spanned by N_0. The argument runs Fredholm, then self-adjointness, then 'we know there exists a unique solution to Δρ = 0 given the boundary conditions.' That last step is asserted. The authors themselves flag the non-compact Fredholm subtlety and point to Christodoulou's method of continuity without carrying it out. In fairness, the maximum principle plus the fall-off conditions plausibly does the job, and asymptotically flat Lichnerowicz–York existence theory is in the literature, so this is fillable rather than wrong. But as written the extension rests on it, and the normalization ⟨N_0⟩ = 1 in (34) is a choice, not a derivation. Footnote 11 also waves off the boundary-term regularization. Second, the equivalence is by construction — both theories are gauge slices of the same extended system — so the content is entirely in whether the slice φ = φ_0 is reachable and the reduced constraints close. The authors are upfront that the dictionary is GR in CMC gauge and that Shape Dynamics might admit trajectories with no CMC-foliable GR partner. That is the interesting open question and they do not pretend otherwise. Minor: notation slips (TφH vs TφS in (39)), 'Kretchmannization', 'Brabour'. Copy-editing. Who should read it: anyone in the shape dynamics or canonical gravity corner, and anyone thinking about symmetry trading more generally. Send it out; the analytic gap is the kind of thing a good referee will make them either close or explicitly cite.

Referee report

Significance

If the construction closes, the result is significant: it gives a compact, reusable criterion (Propositions 1 and 2) for when two gauge theories with different symmetries are equivalent, and it extends the GR/Shape Dynamics equivalence from compact Cauchy surfaces to asymptotically flat ones, which is the setting in which most of the classical phenomenology of the correspondence would be tested. The paper is a derivation paper: there is no code, no data, and no numerical claim, so the assessment turns entirely on whether the stated hypotheses are met. The general framework in §2 is elementary but genuinely clarifying, and the explicit 'dictionary' discussion in §3.5 and the conclusion's flowchart (54) are useful contributions independent of the asymptotically flat extension. The Lagrangian analysis of §4 is a modest but honest negative result: it explains why the lapse and the conformal multiplier cannot be related off shell.

Major objections (4)

[§3.3.1, Eqs. (41)–(43)] The function-space argument that carries Propositions 3 and 4 is internally inconsistent as written. The text first states that Δ = ∇² − R 'is an invertible operator' (§3.3, below Eq. 31), then that Δ has a 'unique kernel N_0 ≠ 0' (Eq. 32), and then that 'we know there exists a unique solution to Δρ = 0 given the boundary conditions imposed on the space C_r, which is the domain of ρ. Thus we have a unique solution ρ_0, which therefore must be given by ρ_0 = N_0, as it has the same boundary conditions as the space of smearings.' These three statements cannot all hold: elements of C_r fall off as O(r^{-1}) (Eq. 17), whereas N_0 → c ≠ 0, so N_0 is not in C_r; and if Δ were invertible on C_r the only solution of Δρ = 0 in C_r would be ρ = 0, not N_0. What the Fredholm splitting in Eq. (43) actually requires is the annihilator of Im(δ_C T_φS) inside C^∞(M), spanned by solutions of Δ*N = 0 with N → const, which is a different space from the domain of ρ. Please rewrite Eqs. (41)–(43) with the domain and the annihilator kept explicitly distinct, and state which space N_0 lives in. As written, the step that identifies the one-dimensional complement with N_0 is the step on which the whole asymptotically flat extension rests.
[§3.1, footnote 11; Eqs. (34), (36), (44)] The paper acknowledges that the total Hamiltonian (26) diverges as written and that 'for general purposes' a regularizing boundary term should be added, but omits it on the grounds that it 'does not impinge on either the equations of motion nor on the constraints.' For an asymptotically flat extension this is not a harmless omission: the ADM Hamiltonian requires a boundary term for the variational principle to be well defined and for the energy to be finite, and surface terms are precisely what can spoil the closure of the constraint algebra and change the value of the global charge. The subsequent regularization by volume normalization (Eqs. 34 and 36) and the statement H_gl := T_{φ0}S(N_0) = V T_{φ0}S(x) in Eq. (44) are not enough to settle this, and Eq. (44) in particular needs explanation, since it equates a smeared constraint to V times a scalar density. Please either include the boundary term and show explicitly that the constraint algebra of Eq. (49) closes with it, or state precisely which surface terms are being dropped and why the dropped terms cannot affect the first-class property of ⟨T_{φ0}N_0H⟩.
[§3.1 ('York miracle') and §3.3.1] The existence and uniqueness of φ_0 is the load-bearing hypothesis of the whole construction, and the paper says so: 'The necessary condition that the constraints T_φS can be uniquely solved for φ is a highly nontrivial; we call it privately the York miracle.' The authors further acknowledge in footnote 13 that 'the Fredholm alternative has subtleties when in the non-compact manifold case' and that 'one should instead use Christodoulou's method of continuity,' but that method is not carried out and Christodoulou is not cited. The existence theory for the Lichnerowicz–York equation and the conformal thin sandwich system with asymptotically flat data is a developed subject (Choquet-Bruhat–York; Isenberg; Maxwell's conformal thin sandwich results), and it comes with hypotheses on the trace of the extrinsic curvature and on the sign of the scalar curvature that are not stated here. A concrete test the authors could perform: state the conditions on (g_ab, π^{ab}) under which Δ = ∇² − R has trivial kernel on C_r, and note that the maximum-principle argument requires a sign condition on R that is nowhere imposed. Without such a statement, Proposition 4 is conditional on an unproved analytic hypothesis.
[§3.3.2, Eq. (49)] The constraint algebra of Shape Dynamics is displayed in Eq. (49) but not derived. In particular {⟨T_{φ0}N_0H⟩, π(ρ)} = 0 is the statement that the global Hamiltonian is invariant under the conformal constraint, and it is the property that makes the reduced theory first class; the Dirac-bracket argument of Eq. (46) establishes it only if the corresponding bracket in the linking theory vanishes, and that bracket is not written down. Since the central claim is that the reduced constraints are first class, please show the computation of {⟨T_{φ0}N_0H⟩, π(ρ)} on the surface φ = φ_0, π_φ = 0, or cite the corresponding computation in [1] if it is identical. Eq. (49) also contains a mismatched brace in the third line, '{H^a(ξ^a), π(ρ))}', which should be corrected.

Minor objections (8)

[§3.4] The 'further fixing of the gauge' is described as imposing 'S(x) = 0 in Shape Dynamics and π = 0 in ADM,' but S(x) is not a constraint of Shape Dynamics, whose first-class constraints are {⟨T_{φ0}N_0H⟩, 4π, H_a}; the natural gauge fixing of the conformal constraint is φ_0 = 0. Please state the gauge-fixing conditions in terms of the variables that actually appear in each theory, and explain why φ_0 = 0 and S(x) = 0 are being used interchangeably.
[§4, Eqs. (50) and (53)] The two forms of the action are not obviously equivalent. Eq. (50) contains (ġ_ab − L_ξ g_ab − ρ g_ab), while Eq. (53) contains (ġ_ab − L_ξ g_ab + φ̇ g_ab). Substituting the stated equation of motion φ̇ = −ρ − L_ξφ into Eq. (50) produces an extra term −(L_ξφ) g_ab that is absent from Eq. (53). Please reconcile the two expressions, or state the convention under which the shift term for φ drops out.
[§3.1, Eq. (39)] The derivative in Eq. (39) is written as δT_φH(x)/δφ(y) while the constraint being differentiated is T_φS; the notation T_φS, T_φH, ~T_φS and T_φS(N_0)/V is used inconsistently throughout §3.3. Please fix a single notation and define ~T_φS once, at its first appearance.
[§3.5] The sentence 'the present theory is in no way equivalent to any of previous work' is stronger than the surrounding discussion supports, since the paper itself identifies ADM in CMC gauge as the dictionary between the two theories and the earlier relational work as the source of the conformal superspace picture. A formulation such as 'not equivalent as gauge theories, though sharing the CMC-gauge trajectories' would be more accurate.
[References] Reference [1], on which the compact-Σ equivalence rests, is listed as 'to appear in Class. Quant. Grav. 2011'; please update it to the published version. The appendix only sketches the compact case, so the reader is asked to take the compact-case claims from a reference whose final form is not given.
[§2.3] The claim that 'a necessary and sufficient condition for the existence of a linking theory is that best-matching constraints are first-class amongst themselves ... and second class solely with respect to the constraint that one wants to trade for' is stated without proof. Since this is presented as the criterion that distinguishes the successful cases from the three prototypical failure modes, a short argument or a pointer to where it is shown would help.
[§3.1] The phrase 'we call it privately the York miracle' is out of register for a journal article; 'we refer to this as the York property' or similar would read better. The same paragraph should also note that the procedure is stated to be 'technically inequivalent to York's procedure' without saying in what respect.
[Throughout] Typographical and naming corrections: 'Kretchmannization' and 'Kretschmannization' both appear (e.g. §2.2 and the abstract); 'Brabour' should be 'Barbour' (footnote 2); 'toplogical' in Proposition 3; 'truely' in the appendix; 'conformatlly' in reference [8]; 'In his case' should be 'In this case' (§3.1); 'extraneous fields extraneous' in §3.5.

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The paper

The paper claims

General Relativity and Shape Dynamics are argued to be two gauge fixings of one linking theory, and the argument is extended to asymptotically flat space.

The reading

General Relativity makes the split of spacetime into space and time arbitrary, and that arbitrariness is a gauge symmetry: the equations hold whatever slicing you choose. Shape Dynamics is a candidate reformulation that does the opposite, fixing the slicing once and for all while declaring the spatial metric meaningful only up to local stretching, so its gauge symmetry is local conformal invariance. This paper argues that the two are not competing theories but two partial gauge fixings of a single larger 'linking' theory, built on an extended phase space that adds a conformal field $[the conformal factor]$ and its conjugate momentum to the usual gravitational variables. Fixing the conformal field to zero returns ordinary ADM gravity; fixing its momentum to zero returns Shape Dynamics with a global Hamiltonian and the constraint $4$ times the trace of the momentum vanishing. The authors first give a general recipe for building such linking theories, then carry it through for gravity on asymptotically flat space, extending earlier work that covered only closed compact space, and close with a Lagrangian version of the construction.

What carries the argument

The engine is 'canonical best matching', a phase-space version of Kretschmannization: extend the phase space by an auxiliary field, apply the point transformation generated by $F = int d^3x (g_{ab} e^{4phi} Pi^{ab} + phi Pi_phi)$, and then note that the previously trivial constraint $pi_phi = 0$ becomes the nontrivial charge $Q = pi_phi - 4pi$. The decisive step is solving the transformed scalar constraint $T_phi S$ for $phi$ at the surface $pi_phi = 0$. The functional derivative of $T_phi S$ with respect to $phi$ is the operator $Delta = nabla^2 - R$, and the paper's argument that $Delta$ is invertible under the stated fall-off conditions is what produces a unique $phi_0$ and hence a well-defined reduced theory. A one-dimensional kernel remains, spanned by a lapse $N_0$ normalized by a limiting volume average, and this kernel is exactly what survives as the global Hamiltonian of Shape Dynamics. The paper is explicit that this is not a generic procedure: a linking theory exists only when the best-matching constraints are first class among themselves and second class with respect to precisely the constraint one wants to trade.

Extended reading notes

Core claim

The central claim is that General Relativity and Shape Dynamics are equivalent gauge theories, in the precise sense that each arises from the same linking theory by a partial gauge fixing, so their initial value problems and gauge-fixed trajectories coincide. The linking theory lives on the ADM phase space extended by a conformal pair $(g_{ab}, 4$ times the trace of the momentum$)$ and carries three sets of first-class constraints: the conformally transformed scalar constraint $T_phi S$, the transformed diffeomorphism constraint $T_phi H_a$, and the charge $Q = pi_phi - 4 pi$ that ties the extension to the original variables. The authors show that the gauge fixing $phi = 0$ reduces the theory to ADM gravity, while the gauge fixing $pi_phi = 0$ leaves a purely second-class combination of the scalar constraint, which can be solved for the conformal factor as a functional $phi_0[g, pi]$, and leaves the first-class set consisting of the global Hamiltonian $langle T_{phi_0} N_0 H rangle$, the constraint $4 pi = 0$, and the diffeomorphism constraint. In the authors' words, one trades the freedom to choose the lapse for the freedom to choose the conformal Lagrange multiplier, and General Relativity in constant-mean-curvature gauge becomes the dictionary in which trajectories of the two theories can be matched one to one.

Load-bearing premise

The whole equivalence rests on the operator $Delta = nabla^2 - R$ being invertible on asymptotically flat space with the stated fall-off conditions, so that the transformed scalar constraint can be solved uniquely for the conformal factor; the authors call this the York miracle, prove it through the Fredholm alternative, and themselves note that the Fredholm alternative has subtleties on non-compact manifolds where Christodoulou's method of continuity should be used instead, a step they do not carry out.

Editorial extensions

If this is right

Every General Relativity solution admitting a constant-mean-curvature foliation corresponds to a Shape Dynamics solution, and vice versa, so the two formulations can be used interchangeably on that overlap.
The equivalence extends beyond closed compact Cauchy surfaces to asymptotically flat ones, with the volume-preserving restriction of the compact case replaced by fall-off conditions on the conformal factor and a normalized lapse.
Because equivalence is established through the linking theory rather than through the dictionary, Shape Dynamics may admit trajectories that cannot be translated into constant-mean-curvature-foliable General Relativity, a possibility the authors flag as open.
Shape Dynamics can be treated as a theory in its own right, dropping the requirement that its lapse be positive, since forward evolution is governed instead by the positivity of the multiplier of the global Hamiltonian.
The Lagrangian form of the linking theory shows there is no dynamical relation between the lapse and the conformal multiplier: once $pi_phi = 0$ is imposed, the conformal factor is frozen to $phi_0$ and the two are related only through that fixed functional.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The construction is a template rather than a one-off: any system with a constraint that can be uniquely solved for an auxiliary field after a best-matching transformation should admit a dual description, which suggests the same trading of refoliation for conformal symmetry might be reproducible in matter-coupled or lower-dimensional models.
The authors' remark that Shape Dynamics may contain trajectories with no constant-mean-curvature General Relativity counterpart is the most concrete place to look for a discriminating test, since it predicts a class of Shape Dynamics evolutions that a General Relativity evolution cannot reproduce.
The note that metric Shape Dynamics can be Dirac quantized on the 2+1 dimensional torus while the metric Wheeler-DeWitt equation is not well defined suggests that the practical payoff of the equivalence may lie in quantization rather than in classical solution spaces.

Original abstract

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.

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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.

background contribution method result interpretation main claim

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s1	This streamlines the rather complicated construction of this equivalence performed previously.	contribution 0.46	0.49	0.05	asserted 1.00
s2	We use this streamlined argument to extend the result to General Relativity with asymptotically flat boundary conditions.	contribution 0.58	0.94	0.12	asserted 1.00
s3	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.	contribution 0.55	0.92	0.09	asserted 0.85

What kind of paper the classifier reports this is: theoretical result 0.96 · evidence from proof or derivation 1.00.

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References

Works cited (21)

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

$\delta$-flatness and its natural extension
physics.hist-ph 2026-09 conditional novelty 6.0 of 10

δ-flatness uniquely selects Minkowski spacetime by bounding tidal acceleration, while its minimal generalization δ-MSS selects the maximally symmetric family at the cost of an external constant.
A change of perspective: switching quantum reference frames via a perspective-neutral framework
quant-ph 2018-09 accept novelty 6.0 of 10

Choosing a quantum reference frame is a gauge fixing and switching frames is a symmetry transformation, with Dirac quantization as the frame-neutral bookkeeping and reduced quantization as a single frame's view.
A Shape Dynamics Tutorial
gr-qc 2014-08 accept novelty 2.0 of 10

Shape Dynamics recasts gravity as a gauge theory of spatial conformal geometry, replacing general relativity's relativity of simultaneity with a global time and local shape invariance.

How later papers used it

45 later papers cite it

Evidence: some. 25 of the citing papers share no author with it, and 3 of those carry a confident label from Pith's citation classifier.

Pith read 22 of the 103 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

3 papers by the same authors are listed below: 1 built on it and 2 used its method or tool.

Builds on it 1 of 3 papers with this label · 1 by the same authors

By the same authors

Reviewed and at the same time extended the results on the construction of equivalence of gauge theories obtained in this paper.

Symmetry Doubling: Doubly General Relativity (2012) (same authors) · 0.99 · paraphrased: the licence does not allow quoting

Uses its method or tool 2 of 3 papers with this label · 2 by the same authors

By the same authors

Following this paper, constructed the generating function, found the canonical transformation, and used the transformed variables to construct three sets of constraints.

Coupling Shape Dynamics to Matter Gives Spacetime (2011) (same authors) · 0.87 · paraphrased: the licence does not allow quoting
Used a linking theory from this paper as the most straightforward way of performing this construction.

Time asymmetric extensions of general relativity (2015) (same authors) · 0.98 · paraphrased: the licence does not allow quoting

How the literature uses it

43 arXiv papers cite this in 98 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 63 sentences without a confident label

Builds on it 3 papers

Let us start this paper by reviewing and at the same time extending the results on the construction of equivalence of gauge theories obtained in [ref] .

Symmetry Doubling: Doubly General Relativity · Setup: Equivalence of Gauge Theories · 0.99
It is not difficult to extend previous analyses from shape dynamic [ref] to show the local physical degrees of freedom are unchanged.

Time asymmetric extensions of general relativity · Physical Degrees of Freedom · 0.96
We will adopt the general idea introduced in [ref] to the non–equivariant toy model presented in this chapter.

Shape dynamics and Mach's principles: Gravity from conformal geometrodynamics · Non–equivariance and linking theories · 0.92

Uses its method or tool 3 papers

Following [ref] , we construct the generating function [equation] we find the canonical transformation: [equation] and subsequently use these transformed variables to construct three sets of constraints: the transformed scalar- and diffeomorphism-constraint of GR as well as the transform of $π_ϕ$, [equation]

Coupling Shape Dynamics to Matter Gives Spacetime · Warm-Up: Free Scalar Field · 0.87
The most straighforward way of performing this construction is through the use of a “linking theory" [ref] .

Time asymmetric extensions of general relativity · Physical Degrees of Freedom · 0.98
Because this procedure involved solving $Ω$ for the variable canonically conjugate to $D - D e$, it is easiest in practice to first extend the phase space by introducing a parameter (called $ϕ$ in [ref] ) that parametrizes the flow of the constraint $D - D e =0$.

2+1 gravity on the conformal sphere · The Shape Dynamics Hamiltonian · 0.91

Compares with it 1 paper

The two main differences are the following: First, in the York procedure the constant mean curvature condition is treated as a convenient gauge for decoupling the diffeomorphism from the scalar constraint, but the fact that it defines a first class constraint surface and gauge-fixing properties that enables one to construct a theory[This construction is only possible if one extends phase space first, and then apply phase space reduction [ref] .] with local conformal symmetry does not appear in any way in the York procedure[In fact, the York procedure is only designed as a mathematical device to find valid initial data for GR, in contrast to our approach which deals with the dynamical implementation of conformal symmetry.].

Coupling Shape Dynamics to Matter Gives Spacetime · Introduction · 0.85

Cites it as support 1 paper

Despite this difference in the constraint algebra, it is possible under certain conditions,[The solutions must admit at least one foliation where the spatial Cauchy hypersurfaces have constant mean (extrinsic) curvature – i.e. be `CMC foliable.' One may argue that all physically reasonable solutions will satisfy this condition which, in any case, is not dramatically stronger than the global hyperbolicity assumption that is fundamental to ADM GR.] to prove that the dynamical solutions of SD are gauge equivalent to those of GR [ref] .

The role of time in relational quantum theories · Relational quantization of Shape Dynamics · 0.96

Cites it as background 18 papers

Shape Dynamics (SD) [ref] is a classical theory equivalent to pure gravity in which the spacetime picture that underlies GR is replaced by an intrinsically spatial picture.

Coupling Shape Dynamics to Matter Gives Spacetime · Introduction · 0.92
Shape Dynamics [ref] is a theory of gravity based on spatial conformal symmetry rather than spacetime covariance that turns out to be dynamically equivalent to General Relativity, which means that the physical trajectories of Shape Dynamics and General Relativity coincide.

Observable Equivalence between General Relativity and Shape Dynamics · Introduction · 0.99
Shape Dynamics (SD) [ref] is a classical theory equivalent to pure gravity in which the spacetime picture that underlies GR is replaced by an intrinsically spatial picture.

The Coupling of Shape Dynamics to Matter · Introduction · 0.98

What it cites · 5 sentences without a confident label

Builds on 3 papers

This paper is intended to make the rather involved construction presented in [ref] , which established the equivalence between General Relativity and Shape dynamics in the compact without boundary case, more transparent and at the same time to extend the equivalence of General Relativity and Shape Dynamics to the asymptotically flat case as an example for the general applicability of this procedure.

Einstein gravity as a 3D conformally invariant theory · Conclusion · 1.00
We thus see our work as an extension of references [ref] by giving a clean Hamiltonian description and identifying the underlying reason for the surprising results of the previous papers as the symmetry trading between refoliation- and local spatial conformal symmetry.

The physical gravitational degrees of freedom · Connection with previous work · 0.99
We thus see our work as an extension of references [ref] by giving a clean Hamiltonian description and identifying the underlying reason for the surprising results of the previous papers as the symmetry trading between refoliation- and local spatial conformal symmetry.

Conformal Superspace: the configuration space of general relativity · Connection with previous work · 0.99

Uses the method or tool of 3 papers

In [ref] we took inspiration from [ref] and used the Stückelberg mechanism [ref] and methods from [ref] to construct equivalent gauge theories that have different sets of first class constraints, which turn out to generate different gauge symmetries.

Conformal Superspace: the configuration space of general relativity · Introduction · 0.81
In [ref] we took inspiration from [ref] and used the Stückelberg mechanism [ref] and methods from [ref] to construct equivalent gauge theories that have different sets of first class constraints, which turn out to generate different gauge symmetries.

Einstein gravity as a 3D conformally invariant theory · Introduction · 0.81
In [ref] we took inspiration from [ref] and used the Stückelberg mechanism [ref] and methods from [ref] to construct equivalent gauge theories that have different sets of first class constraints, which turn out to generate different gauge symmetries.

The physical gravitational degrees of freedom · Introduction · 0.81

Compares with 2 papers

This unexpected result relies on the observation that in some cases one is able to find gauges for both systems such that the initial value problems and the equations of motion of both systems coincide, so the two systems do indeed have equivalent trajectories, despite their different gauge symmetries[A similar phenomenon occurs in the AdS/CFT correspondence [ref] , particularly as explored in [ref] .].

Reconstructing AdS/CFT · Introduction · 0.85
This unexpected result relies on the observation that in some cases one is able to find gauges for both systems such that the initial value problems and the equations of motion of both systems coincide, so the two systems do indeed have equivalent trajectories, despite their different gauge symmetries[A similar phenomenon occurs in the AdS/CFT correspondence [ref] , particularly as explored in [ref] .].

The Large N Limit of Superconformal Field Theories and Supergravity · Introduction · 0.85

Discussion

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