Frequently Asked Questions About Shape Dynamics

Abstract

Barbour’s interpretation of Mach’s principle led him to postulate that gravity should be formulated as a dynamical theory of spatial conformal geometry, or in his terminology, “shapes.” Recently, it was shown that the dynamics of General Relativity can indeed be formulated as the dynamics of shapes. This new Shape Dynamics theory, unlike earlier proposals by Barbour and his collaborators, implements local spatial conformal invariance as a gauge symmetry that replaces refoliation invariance in General Relativity. It is the purpose of this paper to answer frequent questions about (new) Shape Dynamics, such as its relation to Poincaré invariance, General Relativity, Constant Mean (extrinsic) Curvature gauge, earlier Shape Dynamics, and finally the conformal approach to the initial value problem of General Relativity. Some of these relations can be clarified by considering a simple model: free electrodynamics and its dual shift symmetric formulation. This model also serves as an example where symmetry trading is used for usual gauge theories.

Appendices

Appendix A: ADM Formulation

The ADM formulation [8] of GR can be obtained from a Legender transform of the Einstein-Hilbert action on a globally hyperbolic spacetime $\varSigma\times\mathbb{R}$. This Legendre transform can be performed in a generally covariant way (cf e.g. [24]), but we will take a short-cut here starting with the ADM-decomposition of the space-time metric

$$ ds^2=-N^2 dt^2+g_{ab} \bigl(dx^a+\xi^a\,dt\bigr) \bigl(dx^b+ \xi^b\,dt\bigr), $$

(21)

where N denotes the lapse, ξ ^a the shift vector and g _ab the spatial metric. The Legendre transform, after discarding a boundary term, leads to primary constraints that constraint the momenta conjugate to N and ξ ^a to vanish, which are solved by treating N,ξ ^a as Langrange multipliers for the secondary constraints

$$\begin{aligned} S(N) =&\int_\varSigma d^3x \,N\, \biggl( \frac{\pi ^{ab}(g_{ac}g_{bd}-\frac{1}{2}g_{ab}g_{cd})\pi^{cd}}{\sqrt {|g|}}-(R-2\varLambda)\sqrt{|g|} \biggr) \end{aligned}$$

(22)

$$\begin{aligned} H(\xi) =&\int_\varSigma d^3x \,\pi^{ab}( \mathcal{L}_\xi g)_{ab}, \end{aligned}$$

(23)

where π ^ab denote the momentum densities canonically conjugate to g _ab . The total Hamiltonian is a linear combination of the constraints

$$ \mathbb{H}=S(N)+H(\xi). $$

(24)

The constraints H(ξ) generate infinitesimal diffeomorphisms in the direction of ξ ^a. The transformations generated by the scalar constraints S(N) do not have such a simple off-shell interpretation, but on-shell, i.e. on a solutions to Einsteins equations, they generate refoliations. This is the reason why the Poisson algebra of the constraints is called the hyper-surface deformation algebra

$$\begin{aligned} \bigl\{ S(N_1),S(N_2)\bigr\} =&g^{ab}H_b(N_1 \nabla_a N_2-N_2\nabla_a N_1) \end{aligned}$$

(25)

$$\begin{aligned} \bigl\{ S(N),H^a(\xi_a)\bigr\} =&-S(\mathcal{L}_\xi N) \end{aligned}$$

(26)

$$\begin{aligned} \bigl\{ H^a(\xi_a),H^b(\eta_b) \bigr\} =& H^a\bigl([\xi,\eta]_a\bigr). \end{aligned}$$

(27)

This algebra is however not a property of the theory, but a property of the particular choice of constraint functions that we used to describe the theory.

Appendix B: Conformal Spin Decomposition

The technical insight that allowed York to develop his approach to solving the initial value problem of GR was that the spin decomposition of an arbitrary symmetric 2-tensor depends only on the conformal class of the spatial metric, which allowed York to decouple diffeomorphsim constraints (which constrain the longitudinal part of the metric momentum) and scalar constraints (which can be solved by a conformal transformation). To explain this insight in more detail, let us consider a spatial metric g _ab and decompose a symmetric 2-tensor σ ^ab of density weight 0 uniquely into a transverse-traceless part, longitudinal part and trace part $\sigma^{ab}=\sigma ^{ab}_{TT}+\sigma^{ab}_{L}+\sigma^{ab}_{tr}$. The trace part is defined as $\sigma^{ab}_{tr}=\frac{1}{3} \sigma^{cd}g_{cd} g^{ab}$, the longitudinal part is defined as $\sigma ^{ab}_{L}=g^{ca}v^{b}_{;c}+g^{cb}v^{a}_{;c}-\frac{2}{3} g^{ab}v^{c}_{;c}$, where the semicolon denotes the covariant derivative w.r.t. g _ab , and the transverse traceless part satisfies

$$ g_{ab}\sigma_{TT}^{ab}=0\quad\textrm{and}\quad{ \sigma^{ab}_{TT}}_{;b}=0. $$

(28)

The trace σ=g _ab σ ^ab is uniquely determined. It turns out that the vector v ^a is determined up to the addition of a conformal Killing vector of g _ab by the transverse and traceless condition for $\sigma^{ab}_{TT}$, so $\sigma^{ab}_{L}$ and $\sigma ^{ab}_{TT}$ are uniquely determined as well.

The important insight is now that the components of the spin decomposition map into each other under a simultaneous conformal transformation of g _ab and σ ^ab

$$ g_{ab} \to\varOmega^4 g_{ab}\quad\textrm{and}\quad\sigma^{ab}\to \varOmega^{-10}\sigma^{ab}. $$

(29)

In particular, it can be shown that the summands of the spin decomposition transform as

$$ \sigma^{ab}_{tr} \to\varOmega^{-10} \sigma^{ab}_{tr},\quad \sigma ^{ab}_{L} \to\varOmega^{-10}\sigma^{ab}_L \quad\textrm{and} \quad\sigma ^{ab}_{TT} \to\varOmega^{-10} \sigma^{ab}_{TT}. $$

(30)

Appendix C: Initial Value Problem for ADM

The most generic approach to the initial value problem of GR exists for CMC gauge, i.e. when the gauge condition $p=\frac{\pi}{\sqrt {|g|}}=\textit{const}$. is imposed to gauge-fix the scalar constraints of ADM, which is simplest, if we express the ADM constraints in terms of extrinsic curvature $K_{ab}=\frac{1}{\sqrt{|g|}}(g_{ac}g_{bd}-\frac{1}{2} g_{ab}g_{cd})\pi^{ab}$. The gauge condition states that the trace part of g _ab K ^ab is a spatial constant. Inserting a constant trace part into the spin decomposition of K ^ab and using the transverse condition ${K^{ab}_{TT}}_{;b}=0$, we find that the momentum constraint −2π ^ab _;b =0 is a constraint on the longitudinal part $K^{ab}_{L}$, which is required to vanish. This implies that the longitudinal part of π ^ab vanishes, so if the CMC condition and momentum constraints are satisfied, one can write the metric momenta the sum of a transverse traceless part $\pi^{ab}_{TT}$ and a spatially constant trace part

$$ \pi^{ab}\vert _{{\pi^{ab}}_{;b}=0,p=const.}=\frac{1}{3}\, p\, g^{ab} \sqrt{|g|}+\pi^{ab}_{TT}. $$

(31)

The conformal invariance of the spin decomposition now ensures that any conformal transformation of a solution to the momentum constraint is still a solution to the momentum constraint, so one can follow Lichnerowicz and York and consider a conformal transformation of the scalar constraints, which after division by $\varOmega\sqrt{|q|}$ reads

$$ 8 \Delta_g \varOmega= \biggl(\frac{3}{8} \tau^2-2\varLambda \biggr)\varOmega ^5+R\,\varOmega- \frac{\pi^{ab}_{TT}\pi^{TT}_{ab}}{|g|}\varOmega^{-7}, $$

(32)

where $\tau=\frac{2}{3} p$. Solutions to elliptic equations of the form ΔΩ=P(Ω) closely related to the positive roots of P(Ω). In particular, existence of a bounded positive solution Ω(x) follows from the existence of positive constants 0<Ω _i <Ω _s that satisfy P(Ω _i )<0<P(Ω _s ), which bound the solution as Ω _i <Ω(x)<Ω. To investigate the positive roots of the RHS of (32), we multiply it with Ω ⁷, which yields a third oder polynomial in Φ=Ω ⁴:

$$ \biggl(\frac{3}{8} \tau^2-2\varLambda \biggr)\varPhi^3-R\, \varPhi^2-\frac{\pi ^{ab}_{TT}\pi^{TT}_{ab}}{|g|}= \biggl(\frac{3}{8} \tau^2-2 \varLambda \biggr) (\varPhi-\alpha) (\varPhi-\beta) (\varPhi-\gamma). $$

(33)

A detailed inspection of the roots of this polynomial reveal that for $(\frac{3}{8} \tau^{2}-2\varLambda)>0$ one can use theorems for elliptic differential equations that guarantee existence and uniqueness of the solution to (32) except for isolated non-generic cases. Combining the transverse-constant trace condition with the conformal transformation that solves the scalar constraints leads to the York-procedure for constructing generic initial data for General Relativity as the following recipe.

• 1. Choose an arbitrary trial metric $\tilde{g}_{ab}$ and an arbitrary trial $\tilde{K}^{ab}$.

• 2. Derive the transverse-constant trace part of $\tilde{K}^{ab}_{TT}+$. This solves the momentum constraints for π ^ab.

• 3. Solve the scale equation (32) for $\tilde{g}_{ab}$ and $\tilde{\pi}^{ab}_{TT}+\frac{1}{3}\, p\, g^{ab}\sqrt{|g|}$ and hence find the scale factor Ω that solves the scalar- and diffeomorphism- constraints in terms of the rescaled data.

3.1 C.1 Initial Value Problem for SD

The initial value problem for ADM in CMC gauge coincides by construction with the initial value problem of Shape Dynamics in ADM gauge. The initial value problem for SD is however much simpler, because the local Shape Dynamics constraints are linear in the momenta and can be solved by a spin decomposition of the metric momenta. General initial data for Shape Dynamics can thus be obtained as follows:

• 1. Choose an arbitrary spatial metric g _ab and an arbitrary symmetric tensor density π ^ab.

• 2. Project onto the transverse traceless part of π ^ab; this gives initial data $(g_{ab},\pi^{ab}_{TT})$.

Notice that the momentum constraint can be solved before or after solving the conformal constraint, because the spin decomposition of π ^ab is L ²-orthogonal.

Appendix D: ADM Dynamics in CMC Gauge

To the extent of the authors knowledge, the construction of the ADM in CMC dynamical theory presented here is new. For related constructions, see [25] or [18].

To fully gauge fix the CMC constraint, it proves easier to do so from the ground up, by first introducing the following separation of variables:

$$ \bigl(g_{ab},\pi^{ab} \bigr)\rightarrow\biggl(|g|^{-1/3}g_{ab}, |g|,|g|^{1/3} \biggl(\pi ^{ab}-\frac{1}{3}\pi g^{ab}\biggr), \frac{2\pi}{3\sqrt{ |g|}}\biggr)=:\bigl(\rho _{ab},\varphi, \sigma^{ab},\tau\bigr) $$

(34)

where we have denoted the determinant of the metric by |g|, and used φ to denote the physical conformal factor of the metric, as opposed to the auxiliary (Stuckelberg) conformal factor ϕ. However, to simplify matters, we choose a reference metric γ _ij to determine a reference density weight.⁷ Then we define a scalar conformal factor ϕ:=|g|/|γ|, and replace, in (34) φ→ϕ. Nonetheless, this still defines a physical (now scalar) conformal factor, which has the same conformal weight as the densitized version.

The variable τ will give rise to what is commonly known as York time. The inverse transformation from the new variables to the old is given by:

$$ \pi^{ab}=\phi^{-1/3}\biggl( \sigma ^{ab}+\frac {\tau}{3}\rho^{ab}\biggr), \quad \mbox{and}\quad g_{ab}=\phi^{1/3}\rho _{ab} $$

(35)

The non-zero Poisson brackets are given by

$$\begin{aligned} \begin{aligned} \{\rho_{ab}(x),\sigma^{cd}(y)\}&=\biggl(\delta_{(a}^c\delta_{b)}^d-\frac {1}{3}g_{ab}g^{cd}\biggr)\delta(x,y)\\ \{\phi(x),\tau(y)\}&=\delta(x,y)\\ \{\sigma^{ab}(x),\sigma^{cd}(y)\}&=\frac{1}{3}(\sigma^{ab}\rho ^{cd}-\sigma^{cd}\rho^{ab})\delta(x,y) \end{aligned} \end{aligned}$$

(36)

The important point however is that what we will designate as “non-physical” variables τ and ϕ, commute with the physical modes σ ^ab and ρ _ab .

We note that since the Poisson bracket between the σ variables does not vanish, schematically {σ,σ}≠0, and {σ,ρ}≠1, σ and ρ do not form, in the strictest sense of the term, a canonical pair. The Poisson bracket in terms of these variables takes the generalized schematic form:

$$ \{F,G\}= \bigl\{ \xi^\alpha, \xi^\beta\bigr\} (\delta _\alpha F\delta _\beta G- \delta_\beta F \delta_\alpha G ) $$

(37)

where we use the DeWitt generalized summation convention. In our case this becomes

$$\begin{aligned} &\int d^3 x \biggl( \frac{\delta F}{\delta\phi} \frac{\delta G}{\delta\tau} - \frac{\delta G}{\delta \phi} \frac{\delta F}{\delta\tau} + \biggl(\delta_{(a}^c \delta_{b)}^d-\frac{1}{3}g_{ab}g^{cd} \biggr) \biggl( \frac{\delta G}{\delta\sigma^{ab}} \frac{\delta F}{\delta\rho_{cd}} - \frac{\delta F}{\delta\sigma ^{ab}} \frac{\delta G}{\delta\rho_{cd}} \biggr) \\ &\quad{} +\frac{1}{3}\bigl(\sigma^{ij}\rho^{kl}- \sigma^{kl}\rho ^{ij}\bigr) \biggl( \frac{\delta G}{\delta\sigma^{ij}} \frac{\delta F}{\delta\sigma^{kl}} - \frac{\delta F}{\delta\sigma^{ij}} \frac{\delta G}{\delta\sigma ^{kl}} \biggr) \biggr) \end{aligned}$$

(38)

In the reduced phase space, the momentum constraint also decouples from the conformal factor, as we now show, in the smeared form of the diffeomorphism constraint:⁸

$$\begin{aligned} \int d^3 x \bigl( \pi^{ab}\mathcal{L}_\xi g_{ab}\bigr) =& \int d^3 x (\sigma^{ab} \mathcal{L}_\xi\rho_{ab}+\frac{\tau}{3}\bigl(\rho ^{ab}\mathcal{L}_\xi\rho_{ab}+3 \phi^{-1/3}\mathcal{L}_\xi\phi ^{1/3} \bigr) \\ =&\int d^3 x \biggl(\sigma^{ab}\mathcal{L}_\xi \rho_{ab}+\frac{\tau }{3}\mathcal{L}_\xi\ln{\phi} \biggr) =\int d^3 x \bigl(\sigma ^{ab}\mathcal{L}_\xi \rho_{ab} \bigr) \end{aligned}$$

(39)

where in the third equality we used integration by parts and the fact that τ is a spatial constant. In the second equality, since ρ _ab is a tensor density of weight −2/3, for the Lie derivative we have:

$$ \mathcal{L}_\xi\rho _{ab}=\xi^c\rho _{ab|c}-\xi^c_{\phantom{c}|a} \rho_{cb}-\xi^c_{\phantom{c}|b}\rho _{ca}- \frac{2}{3}\xi^c_{\phantom{c}|c}\rho_{ab} $$

(40)

where the solid vertical bar denotes covariant differentiation with respect to ρ ^ab (and thus ξ ^c ρ _ab|c =0). The trace of expression (40) with respect to ρ ^ab vanishes.

Let the ADM action be given by the Legendre transform of the total Hamiltonian

$$ \mathcal{S}=\int dt \int_\varSigma d^3x \bigl(\pi ^{ab}\dot{g}_{ab}-NS- \xi^c H_c\bigr) (x) $$

(41)

Ideally, after reduction we would be able to rewrite (41) as (ignoring the spatial diffeomorphisms for now):

$$ \mathcal{S}=\int dt \int _\varSigma d^3x \bigl(\sigma^{ab}\dot{\rho}_{ab}-\mathcal{H}[\tau, \sigma^{ab},\rho_{ab};x \bigr) $$

(42)

thus being able to identify $\int d^{3} x \mathcal{H}[\tau, \sigma ^{ab},\rho_{ab};x)$ as the true Hamiltonian generating evolution in the reduced phase space. To see that this is possible, we must look only at a rewriting of the symplectic form $\int\pi^{ab}\dot{g}_{ab}$.

Using (38) for calculating $\dot{\rho}_{ab}$ from the Poisson bracket with the Hamiltonian, it is trivial to see that $\dot{\rho}_{ab}$ will contain a traceless projection.⁹ In particular, we get that $\rho^{ab}\dot{\rho}_{ab}=0$. Now, from the inverse transformation (35) we get

$$ \int d^3 x \pi^{ab}\dot{g}_{ab}=\int d^3 x \biggl(\frac {2}{3}\tau \dot{(\ln\phi)}+ \sigma^{ab}\dot{\rho}_{ab}+\frac{\tau}{3}\rho ^{ab}\dot{\rho}_{ab} \biggr)= \int d^3x \bigl(\ln \phi+\sigma^{ab}\dot{\rho}_{ab}\bigr) $$

(43)

where we have used $\rho^{ab}\dot{\rho}_{ab}=0$ and $\dot{\tau}=1$ and integration by parts. But as we have seen, the scalar constraint can be solved in full generality by a unique functional ϕ=F[τ,σ ^ab,ρ _ab ;x).

Thus we can simultaneously do a phase space reduction by defining the variables ϕ:=F[τ,σ ^ab,ρ _ab ;x) and by setting τ to be a spatial constant defining York time, i.e. $\dot{\tau}=1$. Of course, this incorporates the fact that the gauge-fixing τ−t=0 is second class with respect to S(x)=0, and thus we must symplectically reduce to get rid of these constraints. We are then left with a genuine evolution Hamiltonian:

$$ \mathcal{S}=\int dt \int_\varSigma d^3x \bigl(\sigma^{ab}\dot{\rho}_{ab}-\ln{ \phi_o}-\sigma^{ab}\mathcal{L}_\xi \rho_{ab} \bigr) $$

(44)

note that the last term $\sigma^{ab}\mathcal{L}_{\xi}\rho_{ab}$ now only generates diffeomorphisms whose flux is divergenceless (incompressible), since

$$\biggl\{ \int d^3 x \sigma^{cd}\mathcal{L}_\xi \rho_{cd}, \rho_{ab}\biggr\} = \biggl(\delta_{(a}^c \delta_{b)}^d-\frac{1}{3}g_{ab}g^{cd} \biggr)\mathcal{L}_\xi \rho_{cd} $$

One of the drawbacks of using a reference density |γ| so that our variables have a specific form of coordinate-covariance, is that the projected value of any functional F[g,π]→F[ρ,σ] is a priori dependent on the auxiliary (background) metric γ _ij . The only case where this is not so is if the functional F[g,π]=F[ρ,σ] is already conformally invariant.

This holds also if we had chosen not to introduce an auxiliary metric γ _ij , but worked with the original (34) instead. This can be seen as follows: although the replacement (g _ij π ^ij)↦(ρ _ij ,σ ^ij) is local, it is also coordinate-dependent (due to the different density weights) and thus ambiguous. One of the main advantages of Shape Dynamics is that one has diffeomorphism invariance intact for the full variables g _ab ,π ^ab without the need to introduce auxiliary quantities.

Another drawback of using the ADM in CMC formalism is that in order to reconstruct the metric (and all the usual physically meaningful quantities calculated with the full metric), one needs to reinsert the non-local York factor. It is the result of Shape Dynamics that you can have a different theory which also reduces to these variables, but which is expressed (fully locally) in terms of the physically meaningful full 3-metric.

Appendix E: Machian Principles

There are many inequivalent statements of Mach’s principle and for each of these there many inequivalent mathematical implementations; we will therefore provide only aspects of Machian ideas that where relevant for the development of Shape Dynamics. Rather, we start with the prose postulate “There is no absolute space or absolute time. Rather space and time are concepts that are abstracted from the relations of physical objects.” This statement is open to numerous interpretations, so to make it precise we apply it to classical field theory with gravitational and matter degrees of freedom. We then consider the following aspects of relationalism:

• 1. “Equilocality is abstracted from the evolution of physical degrees of freedom.” Using Barbour’s idea of “best matching,” one can readily translate this statement into spatial diffeomorphism invaraince. The canonical formulation of a field theory should thus contain spatial diffeomorphism constraints.

• 2. “Spatial scale is abstracted from local ratios, i.e. ratios with physical rods.” This can be readily translated into the requirement that the theory possess local spatial conformal invariance (in physicist terms: spatial Weyl-invariance), i.e. the canonical formulation possess local spatial conformal constraints.

• 3. “Time is abstracted from the dynamics of local physical degrees of freedom, i.e. the dynamics of physical clocks.” We implement this by requiring that the theory posses local time reparametrization invariance. This means that the canonical formulation of the theory possesses local Hamilton constraints which generate infinitesimal spacetime refoliations.

It is clear that GR implements spatial diffeomorphism- and refoliation-invariance, because the canonical formulation posses spatial diffeomorphism constraints and scalar constraints that generate on-shell refoliations. However, GR does not possess local spatial conformal invariance. Rather, the fact that one can solve the scalar constraints by using spatial conformal transformation, shows that the scalar constraints of GR gauge-fix conformal constraints. In other words: the generators of time reparametrizations in ADM—i.e. the scalar constraints—and conformal constraints form a second class system.

Shape Dynamics on the other hand implements spatial diffeomorphism invariance and local spatial conformal invariance, but it posses a global Hamiltonian and thus fails to implement local time reparametrization invariance. This is of course expected from the equivalence with GR, since otherwise the number of local physical degrees of freedom would not be compatible.

Appendix F: Linking Theory and Symmetry Trading

Physical observables of gauge theories are equivalence classes of gauge invariant phase space functions, where two phase space functions are equivalent if and only if their restrictions to the initial value surface coincide. This dual definition of observables is often necessary for a local description of a field theory, but it is also the reason why gauge symmetries can be traded. The simplest way to see how one gauge symmetry can be traded for another is through a linking theory. An instructive example linking theory is the following: Consider a dynamical system with elementary Poisson brackets $\{ q^{a},p_{b}\}=\delta^{a}_{b}$ for all $a,b \in\mathcal{I}$ and $\{\phi^{\alpha},\pi_{\beta}\}=\delta^{\alpha}_{\beta}$ for all $\alpha,\beta\in \mathcal{A}$ and all other elementary Poisson brackets vanishing. Assume the following first class set of constraints:

$$ \chi_1^\alpha= \phi^\alpha+\phi_o^\alpha(q,p),\quad\quad \chi^2_\alpha =\pi_\alpha-\pi^o_\alpha(q,p), \quad\quad\chi_3^\mu=\chi_3^\mu(q,p), $$

(45)

where $\alpha\in\mathcal{A}$ and μ in an index set $\mathcal{M}$ and where the Hamiltonian is contained in the set $\chi ^{3}_{0}(q,p)=H(p,q)-E$ as an energy conservation constraint. There are two sets of interesting partial gauge-fixings for this system:

$$ \sigma^1_\alpha=\pi_\alpha\quad\textrm{and}\quad \sigma^\alpha_2. $$

(46)

Imposing the gauge fixing conditions $\sigma^{1}_{\alpha}$ leads to the phase space reduction $(\phi^{\alpha},\pi_{\beta})\to(-\phi^{\alpha}_{o}(q,p),0)$ where the Dirac bracket associated with this phase space reduction coincides with the Poisson bracket on the reduced phase space, because. I.e. the elementary Poisson brackets are $\{q^{a},p_{b}\} =\delta^{a}_{b}$. The remaining first class constraints on reduced phase space are

$$ \chi^2_\alpha=-\pi^o_\alpha(q,p),\quad\quad \chi_3^\mu=\chi_3^\mu(q,p). $$

(47)

Imposing on the other hand $\sigma^{\alpha}_{2}$ leads the phase space reduction $(\phi^{\alpha},\pi_{\beta})\to(0,\pi^{o}_{\alpha}(q,p))$. The Dirac bracket again coincides with the Poisson bracket on reduced phase space and the remaining first class constraints are

$$ \chi_1^\alpha=\phi_o^\alpha(q,p),\quad\quad \chi_3^\mu=\chi_3^\mu(q,p). $$

(48)

The set of observables as well as their dynamics coincide for the two reductions, because they are obtained as partial gauge fixings of the same initial system. We call this initial system together with the two partial gauge fixing conditions as linking theory. The linking theory allows us to describe the same dynamical system with two different sets of first class constraints. These two sets of constraints do in general generate two different sets of gauge transformations. This means that linking theories enable us to trade one set of gauge symmetries for another.

Since partial gauge fixing and phase space reduction depend only on the constraint surface and not on the particular set of constraints, one sees that the constraints do not have to take the special form of (45), but any equivalent form of these constraints is admissible for the definition of a linking theory. The only thing that is important for our construction is that the set of canonical pairs (ϕ ^α,π ^α) can be split into two sets of proper gauge fixing conditions ϕ ^α and π _α . This freedom enables us to perform symmetry trading between very general classes of gauge symmetries.

6.1 F.2 Kretschmannization and Best Matching

Very often one is given a particular gauge theory and the goal is to simplify the gauge transformations by trading a complicated set of gauge transformations for a set of gauge transformations that closes on configurations space, i.e. transformations of the form q ^a→Q ^a(q,ϕ), where ϕ ^α denote group parameters. A very useful tool for the construction of a linking theory that proves equivalence between the two systems is given by a canonical implementation of Barbour’s “best matching.” One starts with the original system with first class constraints χ ^μ(q,p)≈0 and extends phase space by the cotangent bundle over the gauge group. To make this extension pure gauge, one introduces additional first class constraints that require the momenta conjugate to the group parameters to vanish π _α ≈0. Then one employs a canonical transform generated by

$$ F=P_aQ^a(q,\phi)+\varPi_\alpha \phi^\alpha, $$

(49)

which generates a canonical transformation that takes q ^a→Q ^a(q,ϕ), $p_{a}\to(Q_{,q}^{-1})^{b}_{a} p_{b}$, ϕ ^α→ϕ ^α and takes the additional constraints to

$$ \pi_\alpha\to\pi_\alpha- Q^a_{,\alpha}\bigl(Q_{,q}^{-1} \bigr)^b_ap_b, $$

(50)

where $(Q_{,q}^{-1})$ denotes the inverse of the matrix $\partial _{q^{b}}Q^{a}(q,\phi)$. We have now Kretschmannized the system, i.e. we have implemented a trivial gauge symmetry in by enlarging the phase space. Canonical best matching is achieved by requiring that the group transformations are pure gauge, i.e. by imposing the conditions

$$ \pi_\alpha\approx0. $$

(51)

Performing a Dirac analysis for the best matching conditions leaves many possibilities; the most interesting in light of the previous subsection is that a subset of the constraints $\chi^{\mu}(Q(q,\phi ),Q_{,q}^{-1}p)$ can be solved for ϕ ^α. This means that this subset of the constraints can be equivalently expressed as $\phi ^{\alpha}-\phi^{\alpha}_{o}(q,p)\approx0$. Moreover, if a group parametrization is chosen s.t. ϕ ^α≡0 denotes the unit element, then $\phi^{\alpha}_{o}(q,p)\approx0$ is equivalent to imposing the original subset of constraints. This means that we have constructed a linking theory through best matching, because imposing ϕ ^α≡0 gauge-fixes the constraints of (50) and the remaining constraints describe the original system. Imposing the best matching condition π _α ≡0 on the other hand turns gauge fixes the constraints $\phi ^{\alpha}_{o}(q,p)$ and allows us to trade them for $Q^{a}_{,\alpha }(Q_{,q}^{-1})^{b}_{a}p_{b}$, which are linear in the momenta and generate the transformations q ^a→Q ^a(q,ϕ) on configuration space. Canonical best matching is thus a way to construct a linking gauge theory, whenever a subset of the constraints $\chi^{\mu}(Q(q,\phi),Q_{,q}^{-1}p)$ can be solved for ϕ ^α.

Appendix G: Poincaré Invariance in the Flat Solution

Here we briefly describe how a solution corresponding to Minkowski spacetime has Poincaré Symmetry. Full conformal symmetry is excised for the standard choice of boundary conditions, but might emerge for a different choice.

In this particular case we will be dealing with the curve of phase space data (g _ab (t),π ^ab(t))=(δ _ab ,0), which thus is already in maximal slicing. One can easily see that the LY equation is simply written as ∂ ² Ω=0, where ∂ ² is the Laplacian for δ _ab . In rectilinear coordinates {x ^a} the set of solutions is given by {1,x ^a}. The lapse fixing equation in the Linking Theory is given by:

$$ e^{-4\phi}\bigl(\nabla^2 N+2g^{ab}\phi _{,a}N_{,b}\bigr) -Ne^{-6\phi} G_{abcd}\pi^{ab}\pi^{cd}=0 $$

(52)

Over our set of data the solutions to this equation can be divided into two sets, one for Ω=1, and one when Ω=x ^a. When Ω=1 the solutions are given by $N_{o}^{(i)}=\{1, x^{a}\}$.

The Hamiltonian for Shape Dynamics is (naively) given by

$$ \mathcal{H}\bigl(N^{(i)}_o, \xi^a, \rho \bigr)=t_{\phi _o}S\bigl(N^{(i)}_o\bigr)+H_a \bigl(\xi^a\bigr)+\pi(\rho) $$

(53)

where only the solutions of the lapse fixing equation are allowed in the smearing. The algebra of constraints emerging from this (ignoring boundary terms) is

$$\begin{aligned} &\bigl[\mathcal{H}\bigl(N_o^{(i)}, \xi ^a, \rho\bigr), \mathcal{H}\bigl(N_o^{(i')}, \xi'^{a'}, \rho'\bigr)\bigr] \\ &\quad= \mathcal{H} \Bigl(\bigl(\xi'^{a'}{N^{(i)}_o}_{,a'}- \xi ^{a}{N^{(i')}_o}_{,a}\bigr), ~ \bigl(g^{cd}\bigl({N^{(i)}_o}_{,c}N_o^{(i')}-{N^{(i')}_o}_{,c}N^{(i)} \bigr)+ \bigl[\mathbf {\xi},\xi'\bigr]^d\bigr), \\ &\quad\quad\quad\ \ \xi'^{a'}\rho_{,a'}-\xi^{a} \rho'_{,a} \Bigr) \end{aligned}$$

(54)

Using the Linking Theory equations of motion, one can check that for Ω=1 the following smearings generate symmetries of the data (i.e $\dot{g}_{ab}=\dot{\pi}^{ab}=0$): time translations and boosts are given respectively by $N_{o}^{(i)}={1, x^{a}}$, translations along the c coordinate $\xi^{a}=\delta^{a}_{(c)}$ and rotations around the d axis ξ ^a=ϵ _ad(c) x ^d. Using the algebra (54) one checks that indeed these reproduce the usual Poincaré algebra. This is clarified in [27].