CERN–TH/97–168 CANONICAL GENERAL RELATIVITY: THE DIFFEOMORPHISM CONSTRAINTS AND SPATIAL FRAME TRANSFORMATIONS

arXiv:gr-qc/9707039v1 16 Jul 1997

M. A. CLAYTON Abstract. The constraint algebra derived in [4] is generated explicitly from a formulation of general relativity in which both the metric and vielbein are independent degrees of freedom. The standard form of the Hamiltonian is composed of the Hamiltonian and momentum constraints, as well as constraints that generate spatial frame transformations; all appearing as primary, first class constraints on phase space. From these results, the conditions that must be satisfied by the generators of any diffeomorphism invariant, non-derivatively coupled, classical field are derived.

Introduction A recent review of actions for general relativity (GR) [18] describes formulations in coordinate and (orthogonal) vielbein frames from variations of Einstein–Hilbert and Einstein–Palatini actions. The notable exception to this list is an action that encompasses both of these cases. Such an action possesses the full general linear group invariance, as well as the usual diffeomorphism invariance, allowing the “limit” to orthogonal vielbein and coordinate frame approaches as different choices of gauge. This structure is represented on the full arbitrariness of the choice of vielbein (16 functions) as well as those of the spacetime metric (10 functions). That one can construct such an action is perhaps no surprise [8], however we will also construct a Hamiltonian for the system in which the spatial GL(3, R) invariance is represented by infinitesimal generators on phase space, and the algebra of these, as well as the generators of diffeomorphisms, are derived (complete with conditions that determine surface contributions that guarantee that the field equations of GR follow properly from the Hamiltonian [20]). As usual in such constructions, the Hamiltonian is far from unique, however we will be focusing on a structure that is; namely the combined algebra of infinitesimal diffeomorphisms and GL(3, R) transformations. In an earlier work [4] we have derived this algebra through a generalization of the geometric argument of Teitelboim [21, 22], which led to a “derivation” of canonical GR [10]. This generalized algebra has appeared in previously in the examination of (orthonormal) tetrad GR [9, 3], and here we extend this type of analysis to the more general case; including an arbitrary tetrad as well as metric degrees of freedom. It is noteworthy that we are not attempting to implement the full set of GL(4, R) generators [3, 7], nor the full set of spacetime diffeomorphisms [12, 13], instead we view the problem as purely geometrodynamic. Data that satisfies the constraints (both spatial metric and vielbein degrees of freedom) is given on the initial value hypersurface; we determine the infinitesimal transformations (and the algebra) of this data that lead to an equivalent physical problem (frame rotations and spatial diffeomorphisms), then determine the conditions on the generator of time evolution that guarantee that the evolution of the system is consistent with spacetime diffeomorphism invariance. Inevitably we end up with a system with a higher degree of redundancy (30 phase space degrees of freedom and 13 primary, first class constraints), and there exists the possibility that new choices of gauge may be useful in numerical relativity. This is not the goal of this work; instead we are interested in exploring the relationship between diffeomorphism invariance, the strong equivalence principle, and the evolution of quantum systems. In particular, whether it is possible to specify some type of quantum evolution that respects diffeomorphism invariance Date: August 25, 2013. PACS: 04.20.Fy, 02.40.-k, 11.10.Ef. 1

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without the need to introduce an (approximate) timelike killing vector as is usually necessary [5, 15]. We would also like to quantify the non-invariance of the conventional curved spacetime field theory, towards an understanding of which results may survive in a more fundamental theory, and which depend strongly on the choice of frame. 1. The General Frame Construction The natural setting for this work is the space of moving (of vielbein) frames above a manifold M, that is, the space of smooth assignments of a frame of reference above each point, equivalent to smooth sections of the general linear frame bundle GLM. Physical fields erected above M (the metric, curvature, scalar fields, etc . . . ) are then associated to this bundle through some representation of GL(n, R), and components may be defined in terms of said frame. Standard presentations of classical general relativity often adopt coordinate (or holonomic) frames from the outset, introducing moving (or vielbein) frames as a ‘generalized’ concept in Riemannian geometry. Even when such frames are given equal footing with holonomic frames, in the variational principle, coordinate components are once again given precedence. In order to show how one may relax this, we proceed to review some results from Riemannian geometry in a general linear frame. 1.1. Moving Frames. A moving frame {eA } above a point in M will be written in terms of a particular coordinate frame as eA := E µ A ∂µ and the coframe that is dual to it as θ A := E A µ dxµ , A implies that the vielbeins satisfy E µ E A = δ µ and where the duality relation θ A [eB ] = δB ν ν A A 1 A µ E B E µ = δB . In this section we consider frames that are chosen to be a smooth section of the general linear frame bundle GLM, however in Section 2 we will consider spatial frames defined by ea := E i a ∂i , where ∂i are the partial derivatives with respect to coordinates on a spatial hypersurface. The volume form defined in a linear frame is defined as [23] Z Z p Z p 0 1 2 3 |g|θ ∧ θ ∧ θ ∧ θ = (1.1) d4 x |g|E, ∗1 = M

M

M

the last form of which will be used here. Using compatibility and vanishing torsion, Gauss’ law takes the form Z Z Z   √ 3 A A 3 d3 x ∂µ E µ A VA d x E∇A [V] = d x E −g∇A [V ] = R ZR ZR (1.2) dSµ E µ A VA = = dS A VA , ∂R

∂R

where the surface measure is defined in terms of the normal vector nA to the boundary ∂R of the region R ⊂ M by SA := ∗1[n]A . We have also made the p definition T := ET for tensors A weighted by the vielbein p density p E := det[E µ ], and T := |g|T for tensors weighted by the spatial metric density |g| := |det[gAB ]|. Under the change of frame determined by M AB ∈ GL(3, R), the frame and coframe transform as θ A → θ B = M BA θ A ,

eA → eB = eA |M −1 |AB ,

(1.3)

A , and similarly for the components of tensors. The where M AC |M −1 |CB = |M −1 |AC M CB = δB A + ω A , and |M −1 |A = δ A − ω A define the generators ∆A of infinitesimal forms M AB = δB B B B B B gl(n, R), acting, for example, on vectors and covectors as B ∆ω˜ [T ]A = −ωA TB , 1

A B ∆ω˜ [T ]A = ωB T ,

(1.4a)

As we are solely interested in retrieving classical general relativity in this work, the possibility that the frame may be degenerate [2] is not considered.

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where ω ˜ represents the matrix ω AB ∈ GL(n, R), implying the action on both types of density √  √ ∆ω˜ γ = −ω AA γ, ∆ω˜ [E] = ω AA E. (1.4b) These generators satisfy the Lie algebra of gl(n, R)

[∆ω˜ 1 , ∆ω˜ 2 ] = −∆[ω1˜,ω2 ] ,

(1.5)

where [ω1 , ω2 ]ab := ω1 ac ω2 cb − ω1 cb ω2 ac . Once one has a definition of parallel transport the covariant derivative operator is defined, which we will assume to act on the components of tensors as, for example C ∇A [V ]B = eA [V B ] + ΓB AC V ,

∇A [V ]B = eA [VB ] − ΓC AB VC ,

(1.6)

we may define what it means for the metric gAB to be compatible with this connection D ∇A [g]BC = eA [gBC ] − gDC ΓD AB − gBD ΓAC = 0.

(1.7)

The non–holonomic coordinate nature of the frames {eA } is reflected by non-vanishing structure constants [16]
[eA , eB ] = CAB C eC , CAB C = E C µ eA [E µ B ] − eB [E µ A ] , (1.8)

and the vanishing of the torsion tensor T (X, Y ) := ∇X [Y ]−∇Y [X]−[X, Y ] (∇X := X A ∇A and [X, Y ] := £X [Y ]) results in the relationship between ΓA [BC] and the structure constants (1.8) A A TBC = ΓA [BC] − CBC = 0.

(1.9)

Throughout we will denote (anti-)symmetrization by [ ] and ( ) respectively, i.e., T[AB] := 1 1 2 (TAB − TBA ) and T(AB) := 2 (TAB + TBA ). This, combined with (1.7), allows an explicit solution of ΓA BC in terms of metric and vielbein components

1 AD eB [gCD ] + eC [gDB ] − eD [gBC ] + 12 gAD gBE CDC E + gCE CDB E + 12 CBC A ; ΓA BC = 2 g (1.10) a combination of the standard coordinate frame and orthogonal frame results. We will also require the Riemann curvature tensor is defined by A E A E A E A RABCD = eC [ΓA DB ] − eD [ΓCB ] + ΓDB ΓCE − ΓCB ΓDE − CCD ΓEB ,

(1.11a)

and the contraction to the Ricci tensor C D C D C D C RAB := eC [ΓC BA ] − eB [ΓCA ] + ΓBA ΓCD − ΓCA ΓBD − CCB ΓDA .

(1.11b)

1.2. The Einstein-Hilbert Action in a General Linear Frame. A direct translation of the action for GR will be written as (we use 16πG = c = 1 throughout) Z p d4x E |g|gAB RAB , Sgr = − (1.12) M

where the Ricci tensor is given by (1.11b). The spacetime is described by a section of Riem M ⊗ GLM, namely, a Riemannian metric and non-degenerate linear frame above M. In a variation of the connection coefficients in RAB (i.e., not considering the variation of the structure constant in the definition (1.11b)) we find that C δΓ RAB = ∇C [δΓ]C BA − ∇B [δΓ]CA ,

(1.13)

and therefore treating ΓA (BC) as independent leads via a Palatini variation to the compatibility conditions (1.7)   δSgr BA (AD B) = 0. (1.14) = −E ∇ [g] − ∇ [g] δ eC eD C δΓC (AB)

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Note that the action is not set up to generate the torsion-free conditions (1.9); instead the variations δΓA [BC] must be determined in terms of the vielbein degrees of freedom through the variation of (1.8)     µ D C δCAB C = ∇A E C µ δE µ B − ∇B E C µ δE µ A + δE µ A E D µ ΓC (1.15) DB − δE B E µ ΓDA . To perform the Einstein–Hilbert variation, we make use of the variation of (1.10) µ D A δΓA BC = δEB E µ ΓDC   + 21 ∇B gAD δgCD + E A µ δE µ C − gAD gCE E E µ δE µ D   + 21 ∇C gAD δgDB − E A µ δE µ B − gAD gBE E E µ δE µ D   − 12 gAD ∇D δgBC − gBE E E µ δE µ C − gCE E E µ δE µ B ,

(1.16)

where we note that the presence of the first term, although implying that δΓA BC is no longer a tensor, is precisely what guarantees that the variation of a covariant derivative will remain covariant. The variation of (1.12) will be performed by treating the densitised components of the inverse metric gAB and the frame degrees of freedom E µ A as independent variables, using     1 RE A µ δE µ A − E∇A ∇B SAB , (1.17a) δR = RAB δgAB + 2RA B E B µ − n−1 where

δSAB := δgAB + gAB δ

p

|g| + 2gAC E B µ δE µ C − 2gAB E C µ δE µ C .

(1.17b)

In computing this we have left the dimensionality n of spacetime arbitrary, and made use of the variational relations:

√

δgAB = −gAC gBD δgCD , √ 1 gAB δgAB , δ −g = n−2

−gδgAB = δgAB −

AB 1 gCD δgCD , n−2 g

δE A µ = −E A ν E B µ δE ν B , δE =

EδE µ A =

A µ 1 n−1 E µ δE A , 1 δE µ A − n−1 E µ A E B ν δE ν B .

(1.18a) (1.18b) (1.18c)

Discarding surface terms we find δSgr = RAB = 0, δgAB δSgr = 2RA B E B µ − δE µ A

(1.19a) A 1 n−1 RE µ

= 0,

(1.19b)

both of which are Einstein’s equations in empty spacetime. (We will consider a minimally– coupled scalar field in Section 3.1 and show that both of (1.19) are equivalent to Einstein’s Equations GAB = RAB − 21 gAB R = 12 TAB .) The equivalence of the variational results from the metric and vielbein degrees of freedom is expected algebraically due to an argument by Floreanini and Percacci [7], and also since we know that Einstein’s equations are covariant under frame transformations. This variational principle is actually a specific case of the more general formalism that includes affine frames, non-metricity and torsion that appears in [8]. 2. Hamiltonian Formalism We turn now to the construction of a Hamiltonian formalism for the system, much of which is a straightforward application of the Bergmann–Dirac procedure for constrained systems described in detail in [11], which we will follow. In order to consider the initial value problem, we must consider the embedding of a family of non-overlapping, spacelike hypersurfaces Σt that foliate M, labelled by some choice of time parameter. The geometry of this scenario has been considered extensively by Kuchaˇr [14]; here we will give results that will be of some use in this work, ignoring global issues (other than the addition of surface contributions to the Hamiltonian) and assuming in all cases that a global section of the frame bundle GLM exists.

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2.1. The Surface-Adapted Basis. The spacetime metric (and inverse, respectively) may be put in surface-normal form g = θ ⊥ ⊗ θ ⊥ − γab θ a ⊗ θ b ,

g−1 = e⊥ ⊗ e⊥ − γ ab ea ⊗ eb ,

(2.1)

where the frame and its dual coframe are given by e⊥ =

1 N ∂t

−

Na N ea ,

ea = E i a ∂i ,

θ a = N a dt + E a i dxi .

θ ⊥ = N dt,

(2.2)

The lapse function N and shift vector N a play the same geometric role as that assigned to them in coordinate frame work [1, 11], namely, the shift vector is the projection of the spacetime vector field that describes the ‘flow of time’ parallel to Σ, and the lapse function the normal projection. (Note that they may either be thought of as a reparameterization of the spacetime metric, or as the E A 0 components of the vielbein in a gauge where E 0 i = 0 in tetrad GR.) The only alteration of this conventional picture is that N a are the components of the shift vector p in the frame {ea } above Σ. It is straightforward to see that the spacetime volume element E |g| √ becomes N E γ where we will write E := det E a i as now representing the determinant of the ~ := N a ea . spatial vielbein. We will also occasionally refer to a spatial vector as, for example N For this surface-adapted frame we find the non-vanishing structure constants from (1.8) [ea , eb ] = Cab c ec ,

[e⊥ , ea ] = C⊥a ⊥ e⊥ + C⊥a b eb ,

(2.3a)

where
Cab c = E c i ea [E i b ] − eb [E i a ] ,

C⊥a ⊥ = ea [ln N ],

(2.3b)

1 ea [N b ] N c + Cac b + E b i ∂t [E i a ]. (2.3c) N N N Taking perpendicular and parallel (to Σ) projections of the compatibility conditions (1.7) results in C⊥a b =

⊥ Γ⊥ ⊥⊥ = Γa⊥ = 0,

Γ⊥ ab := kab ,

Γ⊥ ⊥a := aa ,

Γa⊥⊥ = aa := γ ab ab ,

Γab⊥ = kab := γ ac kbc ,

∇a [γ]bc = 0,

(2.4a) (2.4b)

where we have written ∇a as the covariant derivative operator defined on Σ. The remaining compatibility condition gives the relationship between the extrinsic curvature and the time derivatives of the metric and vielbein degrees of freedom (the Lie derivative £ will be described in more detail in the following section) ∂t [γab ] − 2γ(ac E c i ∂t [E i b) ] − 2N kab − £N~ [γ]ab = 0.

(2.4c)

The projections of (1.9) result in ⊥ ⊥ ⊥ T⊥a = Γ⊥ ⊥a − Γa⊥ − C⊥a = 0 ⊥ ⊥ = Γ⊥ Tab ab − Γba = 0

a T⊥b a Tbc

=

=

Γa⊥b − Γab⊥ − C⊥b a = Γabc − Γacb − Cbc a = 0

0

→ aa = C⊥a ⊥ = ea [ln N ],

(2.5a)

→ k[ab] = 0,

(2.5b)

→

(2.5d)

→

Γa⊥b = kab + C⊥b a , Γa[bc] = Cbc a ,

(2.5c)

which completes the generalization of the results of [11]. The projected components of the Ricci tensor (1.11b) that will appear in the surface reduction of (1.12) are (4)

(2.6a)

(4)

(2.6b)

R⊥⊥ = −de⊥ [k] + ∇a [a]a + aa aa − kab kba ,

Rab = Rab + de⊥ [k]ab − ∇(b [a]a) − aa ab + kkab − 2kac kcb , (4)

where we have written Rab to indicate the projected spacetime tensor, and Rab the intrinsically defined spatial Ricci tensor.

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2.2. Surface-Adapted Derivatives. In order to consider the representation of diffeomorphisms in frames in which the metric is non-dynamical (or prescribed in some non-dynamical manner), we need to consider in some detail the representation of spatial diffeomorphisms. ~ The action of an infinitesimal diffeomorphism of Σ to itself generated by the vector field X, may be written in an arbitrary frame as ab··· ab··· (2.7a) ] := ∇X~ [T ]ab··· £X~ [Tmn··· ˜ [T ]mn··· , mn··· − ∆∇X √ also applicable to densities: £X~ [ γ] = ∇a [X]a . In this form the outcome of the diffeomorphism is represented on the components of tensors as the sum of a covariant derivative ∇X~ := X a ∇a ˜ a := ∇b [X]a ; the frame is unaffected: £ ~ [ea ] = and a spatial frame rotation generated by [∇X] b X 0. This is not necessary though, and while appropriate for considering a fixed frame, is inappropriate for considering diffeomorphisms when constrained to an orthonormal frame since £X~ [γ]ab = ∇(a [X]b) 6= 0. Instead one may define a Lie action that operates on both the frame and tensor components as ′ b £X ˜ [ea ] = −∇a [X] eb , ~ [ea ] = ∆∇X

′ ab··· ab··· £X ~ [Tmn··· ] = ∇X [T ]mn··· ,

(2.7b)

so that when the action on a tensor (not just the components) is considered, one finds the same result as (2.7a). This representation is more appropriate for considering orthonormal frames since the action of £′ on the components of the metric £′~ [γ]ab = ∇X [γ]ab vanishes due X to compatibility. Similarly, the definition of the derivative off of Σ that is surface-covariant is defined to act on the components of tensors as [11] ab··· ab··· d⊥ [Tmn··· ] := e⊥ [Tmn··· ] + ∆C˜⊥ [T ]ab··· mn··· ,

(2.8)

where C˜⊥ is the matrix with components C⊥a b defined in (2.3). This operator defines the derivative normal to Σ that ‘follows’ the vielbein (since d⊥ [E i a ] = 0 by definition), and describes the evolution of all quantities in terms of the original frame. (Note that the definition of C⊥a b involves time derivatives of the frame.) As with the case of the Lie derivative, we may also consider the opposite case, namely where the normal derivative is defined so that the metric does not ‘evolve’ and the frame does. Let us introduce an operator that considers the evolution of the vielbein more generally d′⊥ [E i a ] := e⊥ [E i a ] + ∆D˜ ⊥ [E i a ] = e⊥ [E i a ] − D⊥a b E i b ,

(2.9)

ab··· e ⊗ e · · · θ m ⊗ θ n · · · ] = ˜ ⊥ is undefined as yet. We want to require that d′ [Tmn··· where D a b ⊥ ab··· m n d⊥ [Tmn··· ea ⊗ eb · · · θ ⊗ θ · · · ] (so that the operators acting on the tensors are identical), so we find that the action of d′⊥ on the components of a tensor must be given by ab··· ab··· ab··· ], ] = e⊥ [Tmn··· ] + ∆C˜ ′ [Tmn··· d′⊥ [Tmn··· ⊥

a

′ where C⊥b := C⊥b a − D⊥b a + E a i e⊥ [E i b ]. (2.10)

If we now require that d′⊥ [γ]ab = 0 (so that the normal derivative operator does not affect the components of the metric) then we may choose 1 1 1 ac γ ∂t [γbc ] + eb [Na ] − N c E a i eb [E i c ], (2.11) D⊥b a = − 2N N N ′ a = − 1 γ ac e [γ ]. (This is far from unique since any choice of D and therefore C⊥b ⊥ bc ⊥[ab] will 2 ′ result in d⊥ [γ]ab = 0, however we have chosen what we consider to be a ‘simplest choice’ which corresponds to the operator considered in Section 2.3 as well as [4].) In terms of these operators, the compatibility condition (2.4c) takes on either of the two forms d⊥ [γ]ab = 2kab ,

or

E a i d′⊥ [E i b ] = −kab .

(2.12)

The total time derivative operators dt = N de⊥ + £N~ and d′t = N d′e⊥ + £′~ , correspond to N the definition of the total derivative of the tensor, and are not equivalent to the time derivative of the components. For example: dt [X a ea ] = d′t [X a ea ]∂t [X a ea ] 6= ∂t [X a ]ea .

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The same ‘game’ that has been applied to the Lie derivative and the normal derivative ab··· ] = X a e [T ab··· ], and could also be played with the covariant derivative, defining ∇X~ [Tmn··· a mn··· ∇X~ [ea ] = X b Γcba ec . Indeed, this is how parallel transport in a principle bundle is transferred to associated bundles [16]. This we do not pursue since the resulting action is not covariant with respect to general linear frame transformations when acting on the frame or tensorial components alone. 2.3. The Hamiltonian. From the results of the previous two sections, the GR action (1.12) is reduced to Z Z Sgr = dt d3 x N(γ ab Rab − R⊥⊥ ) Z ZΣ (2.13)
= dt d3 x N de⊥ [k] + γ ab de⊥ [k]ab − 2(∇a [a]a + aa aa ) + R + k2 − kab kba . Σ

Using compatibility, we find that ∇a [a]a + aa aa = (γ ab ∇a ∇b [N ])/N is a surface term in the action which will be dropped, and furthermore
N de⊥ [k] + γ ab de⊥ [k]ab = 2∂t [k] − (kab + γab k)∂t [γγ ab ] − 2kab E b i ∂t [E i a ] (2.14) − 2Na ∇a [k] − 2E∇a [kab N b ] + 2Na ∇b [k]ba . R Dropping the total time derivative and writing Sgr = dt Lgr , the Lagrangian is given by Z 
d3 x − kab + γab k ∂t [γγ ab ] − 2kab E b i ∂t [E i a ] Lgr = Σ (2.15)

 2 a b a b + N R + k − k b k a + 2N ∇b [k] a − ∇a [k] .

In this form the Lagrangian is most easily treated in Palatini form, that is, by considering the extrinsic curvature kab as a tensor of Lagrange multipliers that enforce (2.4c). The configuration of the system at any instant in time is described by a section of Riem Σ ⊗ GLΣ, however due to the form of (2.15) it is convenient to choose instead the densitised canonical coordinates γ ab and E i a , and so we are considering sections of Riem Σ ⊗ GLΣ. Determining the conjugate momenta via δLgr = −(kab + γab k), π ab : = (2.16a) ∂t [γγ ab ] δLgr = −2kab E b i , (2.16b) pa i : = ∂t [E i a ] the canonical coordinate–momentum pairs are  I  (Q , PI ) := (γγ ab , π ab ), (E i a , pa i ) , (2.17) corresponding to phase space T ∗ Riem Σ ⊗ GLΣ). We use the standard form of the Poisson bracket  Z X δF δG δG δF 3 d x {F, G} := − , (2.18) δQI (x) δPI (x) δPI (x) δQI (x) Σ I

where F and G are functions of the phase space variables. (The results herein could be globalized along the lines of [6] in which case we would introduce the related weak symplectic form on phase space, however a local treatment is sufficient for this work.) In addition to the phase space coordinates we have the Lagrange multipliers {N, N a , N ab , kab , λ ab } where the R P Hamiltonian determined from Hgr = Σ dx I Q˙ I PI − Lgr is written as
¯)pa i Eb i Hgr = λ ba kab + a ¯(π ab − 14 δba π) + 21 (1 − a
(2.19) + N ba 2γγ ac π bc − π δba − pa i E i b + pδba  
+ N −R + kab kba − k2 + 2Na ∇b −kba + δab k .

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In this form, λ ab and N ab enforce (2.16a) and (2.16b), and we have indicated the arbitrariness in assigning kab by a parameter a ¯. The tensor of constraints enforced by by N ab J ab := 2γγ ac π bc − π δba − pa i E i b + pδba , are uniquely specified by the requirement that tions on phase space, and it is straightforward  ab c d γ , J d [ω c ] = ∆ω˜ [γγ ]ab ,  i E a , J bc [ω cb ] = ∆ω˜ [E i a ],

(2.20)

they generate infinitesimal gl(3, R) frame rotato show that  π ab , J cd [ω dc ] = ∆ω˜ [π]ab , (2.21a)  a (2.21b) p i , J bc [ω cb ] = ∆ω˜ [pa i ],

where the infinitesimal GL(3, R) frame rotations ∆ω˜ are defined in (1.4). These generators (the phase space representation of ∆ab of [4]) satisfy the gl(3, R) Lie algebra  a c d  a J b , J d [ω c ] = ∆ω˜ J b , (2.22)

which differs from (1.5) by a sign due to the fact that ∆ operates from the left while J operates from the right. Here and throughout we make use of the notation, for example Z b a d3 x ω abJ ba . (2.23) J a [ω b ] := Σ

It is important to note that J ab is not a symmetric tensor of constraints, even though from the results (2.16) one expects it to be (actually it is weakly vanishing). In fact it is precisely these antisymmetric components that generate SO(3) rotations in a local orthonormal frame. While J ab ≈ 0 imposes the consistency of (2.16), the constraint imposed by λab explicitly relates the extrinsic curvature to the conjugate momenta through ¯)γac pc i Eb i , kab ≈ −¯ a(πab + 41 γab π) − 12 (1 − a

(2.24)

and in the form given, the variation of (2.19) with respect to the extrinsic curvature would then determine λab in some complicated manner. So, although treating the extrinsic curvature as a tensor of Lagrange multipliers was convenient in order to pass to the Hamiltonian, it is inconvenient to carry around the Lagrange multipliers λab since the constraints that they impose merely generate the ambiguity in determining kab from the conjugate momenta. This may be circumvented in a simple manner. When passing to the Hamiltonian Hgr , we may replace each occurrence of the extrinsic curvature in the Hamiltonian with some combination of the canonical momenta consistent with (2.24) so that the only place kab will occur is in the constraint enforced by λab . The variation of kab will then enforce λab ≈ 0, which will play no further role and may therefore be dropped consistently. The Hamiltonian determined in this way is identical to that which would occur if one had replaced the extrinsic curvature by (2.4c) in the Lagrangian (2.15). In this way we see that the GR Hamiltonian can always be written as Z
H + N aH a + N abJ ba + Egr , (2.25a) d3 x NH Hgr = Σ

where H := −R + H,

H a := E∇b [H]ba ,

(2.25b)

and H and H ab correspond to some combination of the canonical variables that are equivalent to kab kba − k2 and −2kab + 2δba k respectively using (2.24). In order for these field equations to properly follow from the Hamiltonian, it was necessary to add the surface contributions Egr to the Hamiltonian (2.25a), which are required to satisfy [20] δEgr + SR + Sk = 0

(2.26a)

DIFFEOMORPHISMS AND FRAME TRANSFORMATIONS

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in order for the field equations to be properly recovered from the Hamiltonian, where Z
(2.26b) dS a N ∇b [δSab ] − ∇b [N ]δSba SR := ∂Σ

comes from the variations of the surface Ricci curvature term, and Z Z a bc 1 dS a N H δγbc + dS a N b δHab Sk : = − 2 ∂Σ ∂Σ Z  (2.26c)  dS a N a Hbc E c i δE i c − N b Hab E c i δE i c + N b Hcb E a i δE i c + Nb Hac E b i δE i c , + ∂Σ

comes from variations of the momentum constraint. This result covers the choices made in the following section upon replacement of H ab with the appropriate form corresponding to (2.29) or (2.31). 2.4. The Constraint Algebras. If we consider the following parameterisation of H ab appearing in the momentum constraints (2.25b) (we are considering the arbitrariness in choosing kab only, and not considering arbitrary mixing of the constraints):

(2.27) Hab = a 2γγ ac π bc − δbaπ + (1 − a) pa i E i b − δba p + bγbcJ [ac] , we are lead to the following action on the canonical coordinates  i  ab i i E a , H c [f c ] = (1 − a)∆∇f γ , H c [f c ] = a£f~[γγ ]ab , ˜ [E a ], ˜ [E a ] + b∆∇f []

(2.28)

˜ is the matrix ∇b [f ]a , and ∇f ˜ where ∇f [ ] corresponds to the antisymmetric components ac γ ∇[b [f ]c] . We easily retrieve the action of £f~ (2.7a) when a = 1, b = 0, and that of £~′ (2.7b) f when a = b = 0. We will therefore consider the operator that represents the action of £ on phase space H a := E∇b [2γγ bc πac − δab π ],

(2.29)

which corresponds to the choice H ab = 2γ ac πbc − δba π and acts on phase space as   i  ab π ab , H c [f c ] = E£f~[π]ab , E a , H c [f c ] = 0, (2.30a) γ , H c [f c ] = £f~[γγ ]ab ,  a
p i , H c [f c ] = −E b i f aH b − 12 δba f cH c + E b i £f~[2γγ ac πbc − 21 δbaπ ] ≈ E b i £f~[pa j E j b ]. (2.30b)

The operator that represents the action of £′ is

J ]ba , H ′a := E∇b [pb i E i a − δba p] = H a − E∇b [J

which corresponds to choosing H ab = pa i E i b − δba p and acts as  ab ′ c  i i γ , H c [f ] = 0, E a , H ′c [f c ] = ∆∇f ˜ [E a ],
 π ab , H ′c [f c ] = E Vab [f ] − γab V cc [f ] ≈ £f~′ [π]ab ,  a a c ′ a b a 1 a p i , H ′c [f c ] = ∆∇f ˜ [p i ] + E i U b [f ] − 2 E i U c [f ] ≈ £f~[p i ],

(2.31)

(2.32a) (2.32b) (2.32c)

where

Vab [f ] : = 12 ∇c [fa Hcb + fc Hab − fa Hbc ],       Uab [f ] : = ∇b [f ]c Hac + γbd ∇c f c H(ad) − γbd ∇c f a H(cd) + ∇c fb H[ca] .

(2.32d) (2.32e)

Note that since the conjugate momenta encode information about the extrinsic curvature, it is not surprising that the action of these generators is somewhat non-trivial [4]. It is the strong action on the canonical coordinates that is of the most relevance for non-derivatively coupled theories. We will also consider two choices of the Hamiltonian constraint, however the general parameterisation of the ambiguity is far too complicated to be particularly useful. Instead we will note that in order to match the action of the surface normal operators d⊥ and d′⊥ on the

DIFFEOMORPHISMS AND FRAME TRANSFORMATIONS

10

canonical coordinates strongly, we must consider the following two choices. The generator that represents the action of d⊥ on phase space
(2.33) H := −R + E π ab π ba − 38 π 2 ,

acts on phase space as (note that for a scalar we still have ∇a ∇b φ = ∇b ∇a φ and we define ∇2 := γ ab ∇a ∇b )  ab
γ , H [N ] = 2N π ab − 83 γ ab π , (2.34a)  i E a , H [N ] = 0, (2.34b) 
2 π ab , H [N ] = −E ∇a ∇b [N ] + γab ∇ [N ] + N Rab

− 2N π ac π cb − 83 ππab + N γab π cd π dc − 83 ππ , (2.34c)
 a b ac b ac 1 p i , H [N ] = −2E iγ ∇c ∇b [N ] + N E iγ 2Rcb − 2 γ cb R
(2.34d) + 21 E a i N π cd π dc − 38 π 2 . It is relatively straightforward to check from the unprimed results that  kab , H [N ] ≈ −N Rab + 2N kac kcb − N kkab + ∇a ∇b [N ],

(2.35)

(and therefore reproduces the field equations (2.6b)) using either of (2.34c) or (2.34d). We also find (using (2.34a)) that {γab , H [N ]} = 2N kab , reproducing the dynamical compatibility condition (2.4c). Defining pE ab := γ(ac pc i E i b) , we also consider the generator corresponding to the action of ′ d⊥
H ′ : = −R + 14 E pE ab pE ba − p2 (2.36)
= H − 12 pE abJ ab − 14 J (ab)J (ab) − 21 J aaJ bb , which acts as

 ab ′ γ , H [N ] = 0,  i i i E a , H ′ [N ] = − 21 N ∆pE ˜ [E a ] ≈ N ∆k ˜ [E a ],
 π ab , H ′ [N ] = −E ∇a ∇b [N ] + γab ∇2 [N ] + N Rab
+ 41 N γ ab pE cd pE dc − p2 ,  a
p i , H ′ [N ] = −2E b iγ ac ∇c ∇b [N ] + N E b iγ ac 2Rcb − 12 γ cb R

− 21 N pE ab pb i − ppa i + 18 NE a i pE bc pE cb − p2 .

(2.37a) (2.37b) (2.37c) (2.37d)

Using the above variations (or inferring the result from the tensorial character of each) the Poisson brackets of the gl(3, R) generators with these four operators may be determined   H , J ab [ω ba ] = H ′ , J ab [ω ba ] = 0, (2.38a)   H ]a , H ′ ]a . (2.38b) H ′a , J bc [ω cb ] = ∆ω˜ [H H a , J bc [ω cb ] = ∆ω˜ [H

The algebra of the unprimed constraints may also be determined Z 
H [f ], H [g] = d3 x γ ab f ∇a [g] − g∇a [f ] H b , ZΣ  H ], d3 x f £~g [H H [f ], H c [gc ] = Σ Z  H ]a , d3 x f a £~g [H H a [f a ], H a [ga ] = Σ

(2.39a) (2.39b) (2.39c)

DIFFEOMORPHISMS AND FRAME TRANSFORMATIONS

11

which, in the limit of a spatial coordinate frame (E i a = δai ), agrees with the Dirac algebra [21, 10, 22, 6]. The algebra for the primed system Z 
  ′ J ][ab] , d3 x f ∇a[g] − g∇a [f ] γ abH ′b + E∇b [J (2.40a) H [f ], H ′ [g] = Σ Z      ′ ′ c H ′ ]a + ∇c f pE ab g(aJ [b)c] , H ′ ] + 12 f ga ∆pE d3 x f £~g [H (2.40b) H [f ], H c [g ] = ˜ [H Σ Z  ′ a H a [f ], H ′a [ga ] = − d3 x f a gb Rc dabJ dc , (2.40c) Σ

when specialised to an orthonormal spatial frame (γab = δab ) is equivalent to that in [3, 9]. (If one instead had used the non-symmetric pE ab = pa i E i b in (2.36) then to (2.40b) given one would need to add 41 (gc ∇c [f ] − f ∇c [g]c )J[ab]J [ab] and remove the symmetrizers the relation (2.36).) These strong results agree with [4]2 up to the overall sign which is due to the fact that we are considering a right (Poisson) action in this work, whereas in [4] we derived the left action [10]. Either of the algebras (2.39) or (2.40) is a local representation of the Lie algebra LDiff 0 M (the connected component since the exponential map is not onto) of the spacetime diffeomorphism group Diff M. R H+ Considering the Hamiltonian constructed from the unprimed constraints: Hgr = Σ d3 x NH
N aH a + N abJ ba , we find that the weak evolution equations ∂t [γγ ab ] = {γγ ab , Hgr } ≈∆N˜ [γγ ]ab + £N~ [γγ ]ab − N ∆k˜ [γγ ]ab , i

i

i

∂t [E a ] = {E a , Hgr } ≈∆N˜ [E a ],


π ab , Hgr } ≈∆N˜ [π]ab + E£N~ [π]ab − E ∇a ∇b [N ] + γab ∇2 [N ] ∂t [π ab ] = {π + N Rab − N E(2kac kcb + kkab ) + N Eγab kcd kdc ,

∂t [pa i ] = {pa i , Hgr } ≈∆N˜ [pa i ] − 2E b i £N~ [kab ] − 2E b iγ ac ∇b ∇c [N ] + 2N E b iγ ac Rcb .

(2.41a)

(2.41b) (2.41c) (2.41d)

Using compatibility, (2.41a) may be written as ∆ω˜ [γγ ]ab ,

where ω ab = N ab − ∇b [N ]a − N kab ,

(2.42)

which is consistent with the easily verified result that the weak evolution equations for the R 3
′ ′ ′ = ′a ′ ′a ′b primed system with Hamiltonian Hgr Σ d x N H + N H a + N bJ a is equivalent to inserting N = N ′,

a

Na = N′ ,

a

N ab = N ′ b + ∇b [N ]a + N kab ,

(2.43)

in (2.41). In either of these formulations, the system is parameterised by the 30 fields (2.17), J ab with either H and H a , or H ′ and H ′a ), and the 13 with 13 primary, first–class constraints (J a associated Lagrange multipliers (N , N and N ab ), and we find two configuration space degrees of freedom per spacetime point as expected [11]. Choosing N ab ≈ 0 results in a system in which the spatial vielbeins do not evolve, and solving the GL(3, R) constraints by choosing pa i ≈ E b i (2γγ ac πbc − 12 δbaπ ) we can consider the evolution of {γγ ab , π ab } sector exclusively. Instead choosing N ′ ab = 0, we find a system in which γ ab does not evolve and one may play the opposite game, solving the GL(3, R) constraints by π ab ≈ 21 γac (pc i E i b + pδbc ) and considering the evolution of {pa i , E i a }. Note that from (2.42) one need only require that
(2.44) N (ab) = γ c(a ∇c [N ]b) − N π ab − 41 γ ab π = γ c(a ∇c [N ]b) + N kab , in order to find a system in which the metric degrees of freedom do not evolve, allowing other gauge choices in this case. 2

Due to an unfortunate typesetting error, the right hand sides of equations (28b-e) of [4] should read [∆ax , ∆y ], [δnx , ∆y ], [δnx , ∆ay ], and [δax , ∆y ] respectively. Also the rather obvious typo in the last line of (10), which should read ∇b [k]ac − gad ∇d [k]bc .

DIFFEOMORPHISMS AND FRAME TRANSFORMATIONS

12

One may treat any choice of initial data in either of these two ways. By transforming the initial data by ∆ω˜ defined so that ∆ω˜ [E i a ] = δai (which is uniquely determined from the inverse as [˜ ω ]ai = E a i ), we transform the initial data to a physically equivalent set for which the frame is holonomic. If instead we choose ∆ω˜ [γ]ab = δab and therefore diagonalize the spatial metric, ω ˜ is defined only up to arbitrary spatial rotations which are generated by J [ab] . 3. Application to Matter Fields For matter fields that are non-derivatively coupled to GR, one adds the matter Lagrangian Lm to that of GR, and upon passing to the Hamiltonian one finds that the total Hamiltonian may be written in the standard form (2.25a) where the generators are the sum of GR and m a gr a ma matter contributions: H = H gr + H m , H a = H gr a + H a and J b = J b + J b ; the GR generators are either the primed or unprimed constraints of Section 2.4. Requiring that the matter sector be non-derivatively coupled excludes one very important case, namely that of a fermion [17]. The conditions on such a theory are not so straightforward to derive since the presence of the extrinsic curvature in the matter action leads to a mixing of the gravitational and matter phase space; indeed the Palatini variation does not result in metric compatibility. Although the resulting generators must still satisfy either of the algebras (2.39) or (2.40), it is a nontrivial matter to derive conditions on the matter sector alone as we shall
do in Section 3.2. The phase space of the combined system is of the form T ∗ Riem Σ ⊗ GLΣ ⊗ T ∗ Q (we have written Q as the configuration space of the matter degrees of freedom), and the Poisson brackets are split into the form {F, G} = {F, G}gr + {F, G}m where {F, G}gr is determined from (2.18) and {F, G}m is the contribution from the matter degrees of freedom. It is J ab and H a that generate frame transformations and spatial diffeomorphisms in this extended phase space, and H which completes the generators of LDiff 0 M. Thus although we cannot assume that the GR generators act in any particular manner on functionals of T ∗ Q, the total generators do. Using this information we can expand the algebra (of either the primed or unprimed form), replacing the Poisson brackets with respect to the GR generators with the known results or functional derivatives, and thereby end up with conditions on the matter generators written solely in terms of Poisson brackets in T ∗ Q. The resulting conditions will have to hold even if one is considering evolution of the matter fields on a fixed GR background, since only in that way will the resulting evolution be consistent with that derived from a diffeomorphically equivalent or frame transformed background geometry. These conditions we derive in Section 3.2, however first we consider the specific case of a classical scalar field. 3.1. Classical Scalar Field. As a straightforward example of the coupling of matter, we consider the Lagrangian density of a classical scalar field Lφ = 12 EgAB ∇A [φ]∇B [φ] − EV [φ] (3.1)
2 1 E ∂t [φ] − N a ea [φ] − Nγγ ab ∇a [φ]∇b [φ] − NV [φ]. = 2N 2 R In computing the variations of the action Sφ = M d4 x Lφ , we find the field equations for φ   δSφ = EgAB ∇A ∇B [φ] − Eδψ [V ], δφ

(3.2)

and
δSφ = 21 E∇A [φ]∇B [φ] − 12 EgAB V [φ] = 12 E TAB − 21 gAB T , AB δg
δSφ √ B AC ∇C [φ] − 13 E A µ 21 gBC ∇B [φ]∇C [φ] + γV [φ] µ = E µ ∇B [φ]g δE A = TA B E B µ − 13 TE A µ ,

(3.3a)

(3.3b)

DIFFEOMORPHISMS AND FRAME TRANSFORMATIONS

where the stress–energy tensor is defined by √ δSφ 1 TAB = 2 AB = −g∇A [φ]∇B [φ] − gAB gCD ∇C [φ]∇D [φ] + gAB V [φ]. δg 2

13

(3.3c)

It is straightforward to show that these variations are consistent with the Gravitational field equations (1.19), and lead to Einstein’s equations. The momentum conjugate to φ is given by
E ∂t [φ] − N a ea [φ] , (3.4) Pφ := N for which it is straightforward to show that the Hamiltonian of the combined system is the φ sum of the GR and scalar field Hamiltonians, H = H gr + H φ and H a = H gr a + H a where gr

H φ = 21 E(Pφ )2 + 21 Nγγ ab ∇a [φ]∇b [φ] + NV [φ],

H gr a

H φa = Pφ ∇a [φ],

(3.5)

and H and are either of the primed or unprimed constraints of GR determined in Section 2.4. No alteration of J ab is necessary since we have parameterised T ∗ Q by fields that do not transform. The Poisson brackets are the sum over gravitational and scalar field coordinates (with obvious notation) {F, G} = {F, G}gr + {F, G}φ . There is also an additional contribution to (2.26a) of the form Z
S a N a Pφ + Nγγ ab ∇b [φ] δφ. Sφ =

(3.6)

(3.7)

∂Σ

It is then straightforward to derive Hamilton’s equations

{φ, H} = {φ, H φ }φ = N Pφ + N a ∇a [φ],   {Pφ , H} = {Pφ , H φ }φ = E∇a Nγγ ab ∇b [φ] − Nδφ [V ] + E∇a [N a Pφ ].

(3.8a) (3.8b)

One may show that H and H a satisfy either of the algebras (2.39) or (2.40) (depending on the choice of GR constraints), however the algebra of the scalar field constraints alone Z
 φ φ d3 x f ∇a [g] − g∇a [f ] γ abH φb , H [f ], H [g] = (3.9a) Σ Z   
  φ d3 x f Pφ ∇a [ga Pφ ] − ga ∇a [φ] ∇b fγγ bc ∇c [φ] − f EδV [φ] , (3.9b) H [f ], H φa [ga ] = ZΣ  φ a φ a H φ ]a , d3 x f a £~g [H H a [f ], H a [g ] = (3.9c) Σ

(since the theory is not derivative–coupled, the Poisson brackets in (3.9) may be replaced by {·, ·}φ ) and although the first and last of (3.9) are familiar, the form of the mixed commutator (3.9b) is obscure. The reason that it is not of a simple form and may not be given in terms of the generators, is due to the cross terms that come from the commutators of the GR constraints with those of the matter sector [10]. We will now make this more explicit. 3.2. Generalization. Assume a generic, non-derivative coupled theory, and therefore that the matter constraints do not depend on the extrinsic curvature (they may depend on both the spatial metric and vielbein in what follows). We  know that the generators are of the form m where χm = χm [Q, P ; γ ab , E i ] ∈ H , H , J a , and that the algebra has the χI = χgr + χ a a I I I b I χ . Expanding the generators in the algebra we find form χI , χJ = κK K IJ  gr gr  m m   gr m  m gr
gr m χI , χJ = χI , χJ gr + χI , χJ m + χI , χJ + χI , χJ = κK IJ χK + χK , (3.10a)

and cancelling the known GR result we therefore have that  m m  gr m  m gr m χI , χJ m = κK . IJ χK − χI , χJ − χI , χJ

(3.10b)

DIFFEOMORPHISMS AND FRAME TRANSFORMATIONS

14

The final two terms may be written in terms the known action of the GR generators on the gravitational phase space and functional derivatives as  m gr  • χI , χJ = π ab , χgr J

δ  m   a gr δ  m χ + p , χ • χI , i I J δγγ ab E ia

(3.11)

where the notation (for example) 

π ab , χgr J



δ   • ab F := δγγ

Z

Σ

 d3 x π ab (x), χgr J

δ δγγ ab (x)

  F ,

(3.12)

has been adopted. The action of χgr is known from (2.21a), (2.30), (2.32), (2.34) or (2.37), and once replaced with the appropriate result we have reduced the algebra to a conditions that must be satisfied by the action of the matter generators on T ∗ Q. There are additional conditions that may be derived from the form of these results; since χm does not depend on the extrinsic curvature, we are able to identify contributions that must separately vanish. To begin, we note that the matter contribution to the gl(3, R) generators will not depend on the gravitational degrees of freedom (see the discussion in [4]), and we find the conditions  a  ma mc d J b , J d [ω c ] m = ∆ω˜ J m b ,  m δ h m i δ h m i a ab γ H [f ] = 0, H H [f ], J m b [ω ba ] m + ∆ω˜ [E i a ] • [f ] + ∆ [γ ] • ω ˜ δγγ ab δE i a  m a δ h m ai δ h m ai b ab γ H H a [f ], J m c [ω cb ] m + ∆ω˜ [E i a ] • [f ] + ∆ [γ ] • H [f ] ω ˜ a δγγ ab a δE i a Z H m ]a , d3 x f a ∆ω˜ [H =

(3.13a) (3.13b)

(3.13c)

Σ

from the brackets of the generators of gl(3, R) with themselves, the momentum constraint and Hamiltonian constraint respectively. If one is working in an unprimed GR system, from the diffeomorphism algebra one derives  m H [f ], H m [g] m =  m a H [f ], H m a [g ] m =  m a b H a [f ], H m b [g ] m =

Z Z Z


d3 x f ∇a[g] − g∇a [f ] γ abH m b,

(3.14a) (3.14b)

Σ

H m ] − E£~g [γγ ]ab • d3 x Ef £~g [H

Σ

H m ]a d3 x f a £~g [H

Σ

+ E£f~[γγ ]ab •

δ h m i H [f ] , δγγ ab

δ h m ai δ h m ai ab γ [g ] − E£ [γ ] • H H [f ] , ~ g δγγ ab a δγγ ab a

(3.14c)

with the extra conditions
δ h m i δ h m i ab 3 ab H H [f ] , [g] = g π − γ π • 8 δγγ ab δγγ ab i
δ h a [g ] = 0. f π ab − 38 γ ab π • ab H m a δγγ


f π ab − 38 γ ab π •

(3.15a) (3.15b)

These last two are satisfied if we are working with a system in which spatial derivatives of the metric do not enter into the Hamiltonian generator, and the metric does not appear explicitly in the momentum generator; this is the usual case.

DIFFEOMORPHISMS AND FRAME TRANSFORMATIONS

In the primed case one finds Z
 m m d3 x f ∇a[g] − g∇a [f ] γ abH m H [f ], H [g] m = b, ZΣ  m δ h m i 3 a i m H d x f [g ] = ] • H [f ] , H [f ], H m £ [H ] − E∆ [E ˜ ~ g a a ∇g m E ia Σ Z  m a d b H a [f ], H m d3 x f a gb Rc dabJ m c [g ] = − b m

15

(3.16a) (3.16b)

Σ

δ h m ai δ h m ai i ] • H H a [f ] , [g ] − E∆ [E ˜ a a ∇g Eia Eia

(3.16c)

δ h m i 1 δ h m i i [g] − g∆ [E ] • H H [f ] ˜ a pE 2 Eia Eia Z
J m ][ab] , d3 x E f ∇a [g] − g∇a [f ] ∇b [J = Σ h  i Z   (a m[b)c]  δ i m m a 3 a 1 1 H f f . ] • H ∆ [E g ∆ [H d x f pE g J [g ] = ] + ∇ ˜ ˜ a c a ab a pE pE 2 2 Eia Σ

(3.17a)

i + E∆∇f ˜ [E a ] •

with extra conditions i 1 ˜ [E a ] • 2 f ∆pE

(3.17b)

It is straightforward to see that all of the conditions (refeq:J conditions–3.17) are satisfied by the generators (3.9) for the scalar field. These general conditions on non-derivatively coupled classical field theories (3.13–3.17) guarantee that the system is diffeomorphism invariant (at least for the connected component of the diffeomorphism group). This does not imply that there is no preferred frame or pregeometry, merely that the whole system is written in a diffeomorphism–invariant manner. In order to remove these possibilities one would have to require that the generators for the scalar field (for example) satisfy the above conditions without having to require the existence of additional fields (a timelike vector field for instance), nor should it be necessary to require a symmetry of the background geometry (a timelike killing vector). Furthermore, the form of the conditions does not depend on how we have parameterized the gravitational phase space. Conclusions We have shown that by treating the components of the metric and the frame on an equal footing we not only have a consistent variational principle, but the generalized setting encompasses both the coordinate and orthonormal frame approaches via a choice of gauge. From the Einstein–Hilbert actionR for general relativity we have
derived a Hamiltonian in the (extended) H + NaH a + N abJ ba , where H and H a are the generators (of standard form Hgr = Σ dx NH the connected component) of spacetime diffeomorphisms, and J ab are the generators of gl(3, R) which represent infinitesimal changes of frame on the spatial hypersurface Σ. This is an explicit construction of the algebra derived more geometrically in [4], from which we have derived the general conditions on the generators arising from any classical, non-derivatively coupled matter action that guarantee diffeomorphism invariance. Using this knowledge we plan to extend this analysis to the quantum regime, specifically considering the possibility of diffeomorphism invariant semi-classical quantum–geometric evolution [19]. However we are also looking to quantify the degree which a reduced or semi-classical quantisation respects both diffeomorphism invariance and the strong equivalence principle. Acknowledgements The author acknowledges support from the Natural Sciences and Engineering Research Council of Canada in the form of a postdoctoral fellowship.

DIFFEOMORPHISMS AND FRAME TRANSFORMATIONS

16

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