ADM formulation and Hamiltonian analysis of Coincident General Relativity
Fabio D'Ambrosio, Mudit Garg, Lavinia Heisenberg, Stefan Zentarra
aa r X i v : . [ g r- q c ] J u l ADM formulation and Hamiltonian analysis of Coincident General Relativity
Fabio D’Ambrosio, ∗ Mudit Garg, † Lavinia Heisenberg, ‡ and Stefan Zentarra § Institute for Theoretical Physics, ETH Zurich, Wolfgang-Pauli-Strasse 27, 8093, Zurich, Switzerland (Dated: July 8, 2020)We consider a simpler geometrical formulation of General Relativity based on non-metricity,known as Coincident General Relativity. We study the ADM formulation of the theory and performa detailed Hamiltonian analysis. We explicitly show the propagation of two physical degrees offreedom, as it should, even though the role of boundary terms and gauge conditions is significantlyaltered. This might represent an alternative promising new route for numerical relativity and canon-ical quantum gravity. We also give an outlook on the number of propagating degrees of freedom innon-linear extension of non-metricity scalar.
I. INTRODUCTION
General Relativity (GR) is the most successful classi-cal theory of gravitation that we have at hand. Basedon the equivalence principle and Special Relativity, Ein-stein postulated that gravitation, caused by any formof energy, manifests itself as the curvature of spacetime.Given a distribution of matter, the Riemannian geometryof spacetime is determined by the Einstein field equa-tions. This theory has withstood intense observationaland experimental scrutiny and its most spectacular pre-dictions, such as the existence of black holes and gravi-tational waves, have been confirmed, opening up a newera in GR and relativistic astrophysics. Within its inter-pretation as the geometrical property of spacetime, thequestion arises whether equivalent geometrical formula-tions of GR exist. In fact, the much richer structure of theaffine sector allows two additional and fully equivalentformulations of GR based on flat geometries, attributinggravitation to either torsion [1] or non-metricity [2]. Thisgeometrical trinity of gravity [3] (see also [4]) offers newperspectives on the computation of gravitational energy-momentum, the entropy of black holes, and on canonicalquantization [5, 6].One of the most important fundamental theoretical de-velopments in GR was its Hamiltonian formulation. Instandard mechanics, the Hamiltonian of a system is sim-ply the Legendre transform of the system’s Lagrangianfunction, H := P i p i ˙ q i − L . This transformation elimi-nates velocities and renders the Hamiltonian a functionof position variables q i and conjugate momenta p i . Allequations of motion that follow from a variational prin-ciple can be formulated as equivalent Hamiltonian equa-tions, which are really just dynamical equations for q i and p i . Similar considerations are true for field theo-ries and GR can indeed be cast into a Hamiltonian form.However, the transition from a Lagrangian to a Hamilto-nian formulation is now more subtle due to the presence ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] of constraints and Dirac’s procedure [7] for generalizedHamiltonian systems has to be employed. Even then,special care is needed due to the theory’s general covari-ance.Significant progress in understanding GR’s Hamilto-nian formulation has been made by the pioneering workof Arnowitt, Deser and Misner (ADM) [8] who intro-duced a new set of variables, which nowadays form thebedrock of numerical relativity. In this formalism, space-time is foliated into a family of spacelike hypersurfacesΣ t and the metric g µν can be conveniently decomposedinto a lapse function N , a shift vector field N i , and athree-dimensional spatial metric γ ij . While lapse andshift enter into the Hamiltonian formulation as Lagrangemultipliers, the spatial metric and its conjugate momen-tum govern the dynamics of GR. These variables allow tocast Einstein’s field equations in the form of Hamilton’sequations. Apart from numerical relativity, the ADMformalism is at the heart of studies concerning canonicalquantum gravity and the causal structure of GR.The Dirac procedure of constrained systems relies onthe degeneracy det H µν = 0 of the Hessian matrix H µν = δ S δ ˙ O µ δ ˙ O ν , where O represents the configuration fields ofthe theory under consideration. The degeneracy of theHessian matrix in constrained systems translates into thefact that some of the degrees of freedom do not propa-gate. In the language of the Hamiltonian formulation, theHessian prevents the invertibility of configuration spaceinto phase space and its degeneracy restricts the physicalphase space to a constrained surface. In order to studyhow many constraints there are, we need to study the fullconstraints algebra and the nature of the constraints. So-called first-class constraints point toward the existence ofgauge symmetries and remove twice the number of de-grees of freedom. The ADM decomposition of standardGR reveals that lapse and shift generate such first-classconstraints, thereby removing eight degrees of freedom inconfiguration space and hence giving rise to two propa-gating degrees of freedom.The study of black holes, gravitational waves, neutronstars, and other similarly strong gravitational phenomenarely strongly on numerical relativity, which heavily usesthe ADM 3 + 1 decomposition. In this way, Einstein’sfield equations are treated as a constrained initial valueproblem, where the initial gravitational fields on somehypersurface are numerically evolved to neighboring hy-persurfaces. Stability and convergence of the numericalsolutions are indispensable for such a numerical analysisand therefore much effort has been given to coordinates,gauge conditions and reformulations of Einstein’s equa-tions in terms of adequate variables. In this Letter weperform a detailed ADM decomposition and Hamiltoniananalysis of a representation of General Relativity basedon non-metricity, known as Coincident General Relativ-ity (CGR) [2]. Since the role of boundary terms andgauge conditions in CGR are altered, this representationcould offer an alternative new route for numerical rela-tivity and canonical quantum gravity. In addition, wegive an outlook on the number of propagating degrees offreedom in f ( Q ). II. COINCIDENT GENERAL RELATIVITY
In this work we consider an exceptional class of sym-metric teleparallel theories of gravity which are consis-tent with a vanishing affine connection. In this geometri-cal formulation of GR, gravity is deprived of any inertialcharacter and purged from the boundary terms. The es-sential starting point is the condition of teleparallelism.Vanishing curvature, R αβµν = 2 ∂ [ µ Γ αν ] β + 2Γ α [ µ | λ | Γ λν ] β ! = 0 , (1)forces the affine connection to have the formΓ αµβ = (Λ − ) αρ ∂ µ Λ ρβ , (2)where Λ αβ ∈ GL (4 , R ). The second requirement of thesymmetric teleparallelism is the vanishing of the torsiontensor, T αµβ = 2Γ α [ µβ ] = 2(Λ − ) αρ ∂ [ µ Λ ρβ ] ! = 0 . (3)This further restricts the form of the connection to be apure diffeomorphism,Γ αµβ = ∂x α ∂ξ λ ∂ µ ∂ β ξ λ and Λ αβ = ∂ β ξ α , (4)with arbitrary ξ α . As it becomes clear from the above ex-pression, the gauge choice ξ α = x α , known as coincidentgauge [2], trivializes the connection Γ αµβ = 0. The fun-damental geometric object of CGR is the non-metricitytensor Q αµν := ∇ α g µν = ∂ α g µν − − ) λρ ∂ α Λ ρ ( µ g ν ) λ . (5)At quadratic order, there are five independent contrac-tions of the non-metricity tensor, Q := c Q αµν Q αµν + c Q αµν Q µνα + c Q µ Q µ + c ¯ Q µ ¯ Q µ + c Q µ ¯ Q µ , (6) where the two independent traces are denoted by Q µ := Q µαα and ¯ Q µ := Q ααµ . In terms of the non-metricityscalar Q , the action of symmetric teleparallel gravity canbe written compactly as S [ g, Γ] = Z d x √− g Q . (7)Only for the parameter choices c = − , c = 12 , c = 14 , c = 0 , c = − , (8)is symmetric teleparallelism equivalent to GR. Onecan establish a duality relation between the standardEinstein-Hilbert Lagrangian and the quadratic non-metricity Lagrangian, R = Q − D α ( Q α − ¯ Q α ) , (9)where Q := Q | c i → GR , the curly Ricci scalar R is asso-ciated to the Levi-Civita connection of g and D α is themetric-compatible covariant derivative. For later conve-nience, let us rewrite the CGR action as S [ g, Γ] = Z d x √− g Q = 14 Z d x √− g ( − g αρ g βµ g σν + 2 g αν g βµ g σρ + g αρ g βσ g µν − g αβ g µν g σρ ) Q αβσ Q ρµν . (10)The theory described by (10) has been shown to be com-pletely equivalent to GR [2] and hence represents a new,alternative way of studying the Hamiltonian formulationof GR, with potentially important implications for bothnumerical relativity and canonical quantum gravity. III. ADM DECOMPOSITION
In this section we perform the 3 + 1 decompositionof the CGR action (10) in terms of ADM variables. Wework exclusively in the coincident gauge, Γ αµβ = 0. Con-sequently, the non-metricity tensor simplifies to Q αµν = ∂ α g µν and the action (10) becomes purely a functionalof the metric. Furthermore, one can easily verify thatthe coincident gauge and the ADM decomposition aremutually compatible. We can therefore follow the stan-dard ADM formalism and foliate spacetime by spacelikehypersurfaces Σ t of constant time t . Then, we choosethe ADM decomposition in which the four dimensionalmetric g µν splits in the usual way, g µν = (cid:18) − N + N i N i N i N j γ ij (cid:19) , (11)where N denotes the lapse function, N i the shift vec-tor field, and γ ij the three dimensional spatial metricinduced on Σ t . While the latter describes the intrin-sic geometry of each hypersurface, lapse and shift de-termine how these hypersurfaces are connected to eachother. More precisely, the shift vector field measures howmuch a given trajectory at constant spatial coordinatesis non-orthogonal to the hypersurface and the lapse de-notes the proper time per unit coordinate time measuredby an observer moving orthogonal to the slices. Sincethe spatial metric is symmetric, it contains six indepen-dent components, the shift vector has three componentsand the lapse function only one, giving a total of ten in-dependent components for the metric g µν , as expected.Imposing g µα g αν = δ µν , the components of the inversemetric are easily determined to be g = − N , g i = N i N and g ij = γ ij − N i N j N . The determinant of the four di-mensional metric simply decomposes into √− g = N √ γ ,with γ denoting the three dimensional metric determi-nant.We further introduce a vector n µ which is orthog-onal to the leaves of the foliation and which satisfiesthe normalization condition n µ n µ = −
1. Its compo-nents are then found to be n µ = (cid:0) /N, − N i /N (cid:1) whilethe components of its associated one-form are given by n µ = ( − N,~ γ ij lives on the spatial hypersur-faces we have γ αβ n β = 0, fulfilling the relation γ µν = g µν + n µ n ν . (12)Using the normal vector, we can introduce the extrinsiccurvature of the hypersurface, K ij := 12 N ( D i N j + D j N i − ˙ γ ij ) , (13)where D i is the metric compatible covariant derivativeof γ ij . The extrinsic curvature provides information onhow the hypersurface is curved with respect to the man-ifold in which it is embedded. With these geometricalquantities, we are well equipped to perform the 3 + 1 de-composition of the CGR action and express it in termsof ADM variables, S [ g ] = Z d x √ γN (cid:8) (3) Q − γ in γ mj n α Q αnm n ν Q jiν + 14 ˜ γ ijmn [ n α Q αjm n ρ Q ρin + 2 n β Q iβm n µ Q jµn − Q inm n µ n ν Q jµν + 2 n α Q αnm n ν Q jiν ]+ 2 γ ij γ mn Q nij n α n β Q αβm + 2 γ ij n α n β Q αβi n µ n ν Q jµν − γ ij n σ Q ijσ n ρ n µ n ν Q ρµν (cid:9) , (14)where ˜ γ ijmn = γ ij γ mn − γ in γ mj and we denoted the threedimensional non-metricity scalar by (3) Q := 14 ( − γ il γ jm γ kn + 2 γ in γ jm γ kl + γ il γ jk γ mn − γ ij γ mn γ kl ) Q ijk Q lmn . (15)As apparent from the above expression of the action (14),it is convenient to calculate the projections of the non-metricity tensor along the normal vector. This will allowus to straightforwardly express the action in terms of lapse, shift, the extrinsic curvature, and the intrinsic non-metricity scalar of the hypersurfaces. These projectionsare explicitly given by n α Q αjk = γ ik ∂ j N i + γ ij ∂ k N i N − K jk ,n α Q iαk = 1 N γ jk ∂ i N j ,n α n β Q αβk = γ jk N ( ˙ N j − N i ∂ i N j ) ,n α n β Q iαβ = − ∂ i NN ,n α n β n σ Q αβσ = 2 N ( N i ∂ i N − ˙ N ) . (16)Using (16), the CGR action (14) simplifies to S [ g ] = Z d x √ γ n N ( (3) Q + K ij K ij − K ) + K∂ i N i + ˙ N (cid:18) ∂ i N i N (cid:19) + (cid:20) N ( N γ ij Q kij − ∂ k N ) (cid:21) ˙ N k + γ ij γ kl ( Q jkl − Q kjl ) ∂ i N − ∂ i N i N j ∂ j NN + N i ∂ j N ∂ i N j N − ∂ i N j N (2 ∂ j N i + N i γ mn Q jmn ) o . (17)At this point it is worth mentioning that integrations byparts with subsequent omissions of boundary terms areallowed, without running the risk of altering the sym-plectic structure of the underlying theory [9]. We makeuse of this fact to rewrite the first term in the third lineof (17). A simple integration by parts yields γ ij γ kl ( Q jkl − Q kjl ) ∂ i N ≡ ( (3) Q i − (3) ˜ Q i ) D i N = D i [ N ( (3) Q i − (3) ˜ Q i )] − N D i ( (3) Q i − (3) ˜ Q i ) , (18)where in the first line we introduced (3) Q k := γ ij Q kij and (3) ˜ Q k := γ ij Q ikj . We shall subsequently drop the totalderivative term on the second line of (18). Similarly, thelast line of (17) can be expressed compactly as √ γ (cid:18) N i ∂ j N ∂ i N j N − ∂ i N j N (cid:0) ∂ j N i + N i γ mn Q jmn (cid:1)(cid:19) = − ∂ i N j ∂ j (cid:18) N i √ γN (cid:19) . (19)Next, we pay special attention to the second line of (17),where time derivatives acting on lapse and shift appear.The term multiplying the time derivate of the shift vectorcan be brought into the form ∂ k ( √ γ/N ). Analogously,the terms proportional to K , ˙ N and ∂ j N can be broughtinto the suggestive form √ γ K − N j ∂ j NN + ˙ NN ! = − ∂ µ ( √ γn µ ) . (20)We use all these prearrangements to express our CGRaction (17) as S [ g ] = Z d x (cid:26) √ γN [ (3) Q − D i ( (3) Q i − (3) ˜ Q i ) + K ij K ij − K ] − ∂ i N i ∂ µ ( √ γn µ ) + ∂ µ N i ∂ i ( √ γn µ ) (cid:27) . (21)In the above expression, the last two terms in the secondline are in fact identical, which immediately becomes ap-parent after performing two partial integrations to swapthe derivatives ∂ µ and ∂ i , and dropping the resultingboundary terms. Thus, they cancel each other and theaction can be further simplified to S [ g ] = Z d x √ γN [ (3) Q − D i ( (3) Q i − (3) ˜ Q i ) + K ij K ij − K ] . (22)It is worth to emphasize that lapse and shift appear nownon-dynamically in this final action, i.e. there are noterms proportional to ˙ N and ˙ N i , respectively. The onlydynamical variables are γ ij , with a priori six degrees offreedom. However, not all of these modes propagate asour Hamiltonian analysis will show in the next sectionfollowing the standard Dirac procedure. IV. HAMILTONIAN ANALYSIS
In this section we perform the Hamiltonian analysisof the action (22), following [10]. Let us reiterate thatwe can always perform partial integrations and removeboundary terms from an action without alteration to thesymplectic structure of the theory it describes [9]. Thismeans that the actions (22) and (17) describe the sametheory. However, notice that (17), while completely ex-pressed in terms of ADM variables, contains terms pro-portional to ˙ N and ˙ N i and would therefore lead to rathercomplicated primary constraints which contain partialderivatives of field variables. This represents a strongdeparture from the standard assumptions at the base ofDirac’s procedure and complicates the Hamiltonian anal-ysis significantly.However, in the action (22) lapse and shift enter non-dynamically and these problems are therefore avoided.In fact, the momentum densities conjugate to lapse andshift are simply given by ˜ π N := δ S δ ˙ N = 0 and ˜ π i := δ S δ ˙ N i =0, respectively, and ˜ π µ := (˜ π N , ˜ π i ) constitute the fourprimary constraints which define the primary constraintsurface Γ P . We use Ashtekar’s tilde notation to indicate tensor densities ofweight one, with the exception of Lagrangian and Hamiltoniandensities for which we use L and H , respectively. The only non-vanishing conjugate momentum densitiesare those associated to the spatial metric,˜ π ij := δ S δ ˙ γ ij = √ γ (cid:0) Kγ ij − K ij (cid:1) . (23)They do not represent constraints since K ij explicitlycontains the velocities ˙ γ ij . The non-vanishing Poissonbrackets (PBs) between the phase space variables are { N ( x ) , ˜ π N ( y ) } = δ (3) ( ~x − ~y ) , { N i ( x ) , ˜ π j ( y ) } = δ ij δ (3) ( ~x − ~y ) , { γ ij ( x ) , ˜ π mn ( y ) } = δ m ( i δ nj ) δ (3) ( ~x − ~y ) . (24)This immediately implies that PBs between the primaryconstraints are strongly zero: { ˜ π µ , ˜ π ν } = 0. The primaryHamiltonian density of the system is given by H := λ ˜ π N + λ i ˜ π i + ˜ π ij ˙ γ ij − L , (25)where λ and λ i are arbitrary Lagrange multipliers. Using˙ γ ij = D i N j + D j N i − N K ij from the definition of theextrinsic curvature (13) together with K ij = 1 √ γ (cid:18) ˜ π ij − (cid:0) ˜ π kk (cid:1) γ ij (cid:19) , (26)which can easily be inferred from (23) by taking its trace,the Hamiltonian density can be brought into the form H = − √ γ ( N (cid:2) (3) Q − D i ( (3) Q i − (3) ˜ Q i ) (cid:3) − Nγ (cid:18) ˜ π ij ˜ π ij −
12 (˜ π ii ) (cid:19) + 2 N i D j ˜ π ji √ γ ! ) + λ ˜ π N + λ i ˜ π i . (27)Next we introduce the Hamiltonian H := R Σ t H ( x, t ) d x ,which generates evolution in the phase space. In orderfor this evolution to be consistent with the constraints,the PBs between ˜ π N , ˜ π i , and H need to vanish on Γ P .This leads straightforwardly to the secondary constraints˜ C := −√ γ " (3) Q − D i ( (3) Q i − (3) ˜ Q i ) − γ (cid:18) ˜ π ij ˜ π ij −
12 (˜ π ii ) (cid:19) ˜ C i := − D j ˜ π ji . (28) The equal time PB between two functions F ( x ) and G ( x ) of thephase space variables { N, N i , γ ij , ˜ π N , ˜ π i , ˜ π ij } is defined as { F ( x ) , G ( y ) } := Z Σ t d z X k (cid:18) δF ( x ) δ Φ k ( z ) δG ( y ) δ ˜Π k ( z ) − δG ( y ) δ Φ k ( z ) δF ( x ) δ ˜Π k ( z ) (cid:19) , with the placeholders Φ k = { N, N i , γ ij } and ˜Π k = { ˜ π N , ˜ π i , ˜ π ij } . Notice that these constraints do not depend on lapse andshift and the Hamiltonian can therefore be rewritten inthe suggestive form H = Z Σ t d x (cid:16) λ ˜ π N + λ i ˜ π i + N ˜ C + N i ˜ C i (cid:17) , (29)where lapse and shift are recognized to act as Lagrangemultipliers. The simultaneous vanishing of the primaryand secondary constraints defines the secondary con-straint surface Γ S ⊆ Γ P . Demanding that this surfaceis preserved at all times leads us to check for tertiaryconstraints. That is, we require the PBs { ˜ C ( x ) , H } = Z Σ t d y ( N ( y ) { ˜ C ( x ) , ˜ C ( y ) } + N i ( y ) { ˜ C ( x ) , ˜ C i ( y ) } ) , { ˜ C i ( x ) , H } = Z Σ t d y ( N ( y ) { ˜ C i ( x ) , ˜ C ( y ) } + N j ( y ) { ˜ C i ( x ) , ˜ C j ( y ) } ) (30)to vanish on Γ S . Notice that the PBs { ˜ π µ , ˜ C ν } do not ap-pear in the above expression because they all vanish dueto the absence of lapse and shift in the secondary con-straints. Hence, our task is to compute the PBs { ˜ C µ , ˜ C ν } in order to check whether there are tertiary constraints.To that end, and in order to avoid delta distributions inour computations, we introduce the smeared scalar andvector constraints C S ( N ) := Z Σ t N ˜ C d xC V ( ~N ) := Z Σ t N i ˜ C i d x, (31)where ~N := N i ∂ i . First of all, the PBs of the vectorconstraint with the dynamical phase space variables yield { γ ij ( x ) , C V ( ~N ) } = Z Σ t d x δ m ( i δ nj ) δC V δ ˜ π mn ( x )= L ~N γ ij ( x ) = 2 D ( i N j ) { ˜ π ij ( x ) , C V ( ~N ) } = − Z Σ t d x δ ( im δ j ) n δC V δγ mn ( x )= − π k ( i D k N j ) + D k (˜ π ij N k )= L ~N ˜ π ij ( x ) , (32)where L ~N denotes the Lie derivative with respect to thevector field ~N . Note that the PBs between the vectorconstraint C V ( ~N ) and all other phase space variables Constraints are only imposed after PBs have been computed. Note that the smearing is valid for arbitrary test functions, notjust for lapse and shift. vanish on the constraint surface Γ S . Therefore, the vec-tor constraint generates the spatial diffeomorphisms forany arbitrary function F of γ ij and ˜ π ij , { F ( γ ij , ˜ π ij ) , C V ( ~N ) } = L ~N F ( γ ij , ˜ π ij ) , (33)and treat all other phase space variables as constants.Next, we pay special attention to the PBs between thesmeared scalar and vector constraint, making use of (33): { C S ( N ) , C V ( ~N ) } = Z Σ t d x N ( x ) { ˜ C ( x ) , C V ( ~N ) } = − Z Σ t d x ˜ C ( x ) L ~N N ( x )= − C S ( L ~N N ) , (34) { C V ( ~N ) , C V ( ~N ) } = Z Σ t d x N i ( x ) { ˜ C i ( x ) , C V ( ~N ) } = − Z Σ t d x ˜ C i ( x ) L ~N N i ( x )= − C V ( L ~N ~N ) = − C V ([ ~N , ~N ])= C V ([ ~N , ~N ]) , (35)where we have used integration by parts and the Liebracket [ ~N , ~N ] defined by[ ~N , ~N ] = ( N j ∂ j N i − N j ∂ j N i ) ∂ i . (36)Since C S and C V both vanish on Γ S independently oftheir argument, we find that the above PBs are preservedby the evolution generated by the Hamiltonian H . Whatremains to be checked is the PB between two smearedscalar constraints. To that end, it is convenient to split C S ( N ) into algebraic and non-algebraic parts with re-spect to γ ij and ˜ π ij : C S ( N ) = − Z Σ t d x N √ γ " (3) Q − D i ( (3) Q i − (3) ˜ Q i ) | {z } non-algebraic − γ (cid:18) ˜ π ij ˜ π ij −
12 (˜ π ii ) (cid:19)| {z } algebraic =: C S alg ( N ) + C S non-alg ( N ) . (37)One immediate observation is that the algebraicpart gives { C S alg ( N ) , C S alg ( N ) } = 0 as well as { C S non-alg ( N ) , C S non-alg ( N ) } = 0. The non-trivial con-tribution to the PB between the scalar constraints comesfrom { C S alg ( N ) , C non-alg S ( N ) } + { C S non-alg ( N ) , C alg S ( N ) } = − Z Σ t d z ˜ π ij [ N D i D j N − N D i D j N ]= Z Σ t d z [ N D i N − N D i N ] ˜ C i = C V (cid:0)(cid:0) N ∂ i N − N ∂ i N (cid:1) ∂ i (cid:1) , (38)where we have used integration by parts (omittingboundary terms) and in the second line we recognized thesmeared vector constraint. Summarizing, we concludethat the PBs between the smeared secondary constraintsare given by { C S ( N ) , C V ( ~N ) } = − C S ( L ~N N ) (39) { C V ( ~N ) , C V ( ~N ) } = C V ([ ~N , ~N ]) { C S ( N ) , C S ( N ) } = C V (cid:0)(cid:0) N ∂ i N − N ∂ i N (cid:1) ∂ i (cid:1) , which all vanish on the constraint surface Γ S . Hence,there are no tertiary constraints to take into consider-ation. Moreover, we have shown that { ˜ π µ , ˜ π ν } = 0, { ˜ π µ , ˜ C ν } = 0, and { ˜ C µ , ˜ C ν } ≈ , which implies thatwe have in total four first-class primary constraints andfour first-class secondary constraints. This removes a to-tal of 2 × × Q which defines the CGR action satisfies the duality re-lation (9) and is therefore equivalent to the Einstein-Hilbert action up to a boundary term. Moreover, onecan prove a further duality relation between the intrinsicnon-metricity scalar and the intrinsic curvature, (3) Q − D i ( (3) Q i − (3) ˜ Q i ) = (3) R , (40)by virtue of which one recognizes (22) as the ADMaction of GR. Hence, CGR possesses the same sym-plectic structure and propagates the same number ofdegrees of freedom as GR, but the role of boundaryterms and gauge conditions are altered. The physicalHamiltonian of CGR therefore differs from the one ofGR and CGR may therefore offer an alternative newroute for numerical relativity and canonical quantumgravity [5]. V. OUTLOOK ON f ( Q ) So far we focussed our attention on CGR described bythe Lagrangian L = √− g Q . However, non-linear exten-sions of the form L = √− g f ( Q ) with f ′ = 0 have beenstudied in the literature and interesting consequences forcosmology have been described [11, 12]. It is thereforenatural to wonder, how many degrees of freedom f ( Q ) Two phase space functions are said to be weakly equal, F ( q, p ) ≈ G ( q, p ), if and only if F ( q, p ) = G ( q, p ) up to an arbitrary linearcombination of constraint. theory propagates and how this number depends on theform of f .At the beginning of the previous section, we empha-sized the importance of having removed terms propor-tional to ˙ N and ˙ N i from the action in order to pro-ceed with the Hamiltonian analysis. Also in f ( Q ) lapseand shift are non-dynamical. We can venture an ed-ucated guess on the propagating number of degrees offreedom based on insights gained in [11] from studyingcosmological perturbations around a FLRW backgroundin conjunction with the fact that lapse and shift are non-dynamical variables. For later reference, we recall thatthe metric perturbations assume the form δg = − a φ (41) δg i = δg i = a ( ∂ i B + B i ) δg ij = 2 a (cid:20) − ψδ ij + (cid:18) ∂ i ∂ j − δ ij ∂ k ∂ k (cid:19) E + ∂ ( i E j ) + h ij (cid:21) and using this ansatz it was shown in [11] that f ( Q ) prop-agates at least two additional degrees of freedom. Sincelapse and shift are non-dynamical, they lead to four pri-mary constraints in the Hamiltonian theory. Hence, f ( Q )could a priori propagate four, five, or six degrees of free-dom. However, the perturbation analysis has uncoveredthat only ψ , E , E i , and h ij can propagate. Hence, inorder to obtain five propagating degrees of freedom, ei-ther φ or B need to become dynamical. But this is incontradiction with the fact that lapse and shift are non-dynamical, thus leaving us with the possibility of havingfour or six degrees of freedom.Attaining six degrees of freedom would only be possibleif the vector perturbation E i , subjected to the constraint ∂ i E i = 0, would propagate its two degrees of freedom.Evidence from cosmological perturbation theory suggeststhat either E i do not propagate at all or there is a strongcoupling problem for the cosmological background, dueto which kinetic term of E i vanishes [11].We can consider this counting problem also from a dif-ferent vantage point and reach the same conclusion: lapseand shift give rise to four primary constraints. These con-straints can either be first- or second-class. Excluding thetrivial case of GR where all constraints are first-class, andalso excluding for the time being the case where all con-straints are second-class, leaves us with two possibilities.Either the constraint pertaining to the non-dynamicallapse is first-class, while the constraint pertaining to theshift vector is second-class, or vice versa. In the first caseone would obtain 10 − × − × − × − × f ( Q ) is currentlyunderway in order to further investigate the above argu-ments. VI. CONCLUSION
In this Letter we performed the ADM decompositionand the Hamiltonian analysis of Coincident General Rel-ativity (CGR), thereby providing an independent proofthat CGR propagates two physical degrees of freedom.We started with the quadratic non-metricity action in thecoincident gauge with vanishing connection and foliatedthe spacetime into spacelike hypersurfaces. In this 3 + 1decomposition, the metric can be represented in terms ofthe ADM variables – lapse, shift, and the spatial metric– since the coincident gauge and the ADM decomposi-tion are mutually compatible. After making extensiveuse of projections between the normal vector and thenon-metricity tensor, and applying integrations by parts,we accomplished to express the CGR action in terms oflapse, shift, extrinsic curvature, and the non-metricityscalar of the spacelike hypersurfaces. The resulting ac-tion does not carry any dynamics for the lapse and shiftfields and it can be shown to be equivalent to the ADMaction of GR. To complete our analysis and explicitlyshow that CGR propagates two degrees of freedom, wereviewed the Hamiltonian analysis of GR in ADM vari-ables. We showed that lapse and shift generate first-class constraints which eliminate eight degrees of freedom inconfiguration space.This is no surprise since the non-metricity scalar Q satisfies the duality relation (9) with the Ricci scalar.Hence, the CGR action and the Einstein-Hilbert actiononly differ by a boundary term which has no influence onthe symplectic structure of the theory. However, the roleof boundary terms and gauge conditions are altered inCGR and the physical Hamiltonian of CGR differs fromthe one of GR. This difference may offer an alternativenew route for numerical relativity and canonical quantumgravity. We leave the exploration of these consequencesto future work.In the final section of this Letter we also ventured aneducated guess about the number of physical degrees offreedom of f ( Q ) theory and found strong indications forsix propagating modes. ACKNOWLEDGEMENTS
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