Hamiltonian Formulation Of General Relativity

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Massachusetts Institute of TechnologyDepartment of PhysicsPhysics 8.962Spring 2002Hamiltonian Formulation of GeneralRelativityc 2005 Edmund Bertschinger. All rights reserved.Revision date June 5, 20051IntroductionThe usual approach to treating general relativity as a field theory is based on the Lagrangian formulation. For some purposes (e.g. numerical relativity and canonical quantization), a Hamiltonian formulation is preferred. The Hamiltonian formulation of a fieldtheory, like the Hamiltonian formulation of particle mechanics, requires choosing a preferred time variable. For a single particle, proper time may be used, and the Hamiltonianformulation remains manifestly covariant. For a continuous medium, the Hamiltonianformulation requires that a time variable be defined everywhere, not just along the pathof one particle. Thus, the Hamiltonian formulation of general relativity requires a separation of time and space coordinates, known as a 3 1 decomposition. Although the formof the equations is no longer manifestly covariant, they are valid for any choice of timecoordinate, and for any coordinate system the results are equivalent to those obtainedfrom the Lagrangian approach.It is convenient to decompose the metric as follows:g00 α2 γ ij βi βj , g0i βi , gij γij ,(1)where γ ij is the inverse of γij , i.e. γ ik γjk δ i j . This 3 1 decomposition of the metricreplaces the 10 independent metric components by the lapse function α(x), the shiftvector βi (x), and the symmetric spatial metric γij (x). The inverse spacetime metriccomponents are1βiβ iβ jg 00 2 , g 0i 2 , g ij γ ij 2 ,(2)ααα1

where β i γ ij βj . From now on, except as noted otherwise, all Latin (spatial) indicesare raised and lowered using the spatial metric. The determinant of the four-metric isg α2 γ where γ is the determinant of γij .The 3 1 decomposition separates the treatment of time and space coordinates. Inplace of four-dimensional gradients, we use time derivatives and three-dimensional gradients. In these notes, the symbol i denotes the three-dimensional covariantderivative with respect to the metric γij . We will not use the four-dimensionalcovariant derivative. Thus,1ii j Ai j Ai γjkAk , γjk γ il ( j γkl k γjl l γjk ) .(3)2In these notes we choose units so that 16πG 1. We assume a coordinate basis throughout.These notes first consider a general metric and then specialize to a perturbed RobertsonWalker spacetime.2Curvature and Gravitational ActionsIn the 3 1 approach, spacetime is described by a set of three-dimensional hypersurfacesof constant time t x0 propagating forward in time. These hypersurfaces have intrinsiccurvature given by the three-dimensional Riemann tensor,(3) iRlkmiiinin k γlm m γkl γknγlm γmnγkl.(4)Contractions define the three-dimensional Ricci tensor (3)Rij (3)Rkikj and Ricci scalar,(3)R γ ij (3)Rij . In addition to the intrinsic curvature, the hypersurface of constanttime has an extrinsic curvature Kij arising from its embedding in four-dimensionalspacetime:1Kij ( i βj j βi t γij ) .(5)2αThe full spacetime curvature is related to the intrinsic and extrinsic curvature of theconstant-time hypersurfaces by the Gauss-Codazzi equations1 ( k Kjl l Kjk )α(6) (3)Ri jkl (4)R0jkl β i K i k Kjl K i l Kjk .(7)(4) 0R jkland(4) iRjklMTW and other sources give these relations assuming an orthonormal basis, for whichβ i 0 and α 1. Equations (6) and (7) are exact for any coordinate basis. The othercomponents of the four-dimensional Riemann tensor follow from 11 1(4) 0 j (β k Kik ) Kjk i β k(8)R i0j t Kij Ki k Kjk i j α ααα2

and(4) kR i0j (4) kR ilj β l (4) 0 l (4) 0 k R ilj β R i0j β α k Kij i K kj .(9)These equations may be combined to give:(4)Rijkl (3)Rijkl Kik Kjl Kil Kjk ,(4)R0jkl (4)Rijkl β i α( k Kjl l Kjk ) (4)Rkl0j ,(4)R0i0j α t Kij α2 Ki k Kjk α i j α αβ k k Kij α i (β k Kjk ) α j (β k Kik ) (4)Rkilj β k β l ,(10a)(10b)(10c)and(4) ijRkl(4) 0jR(4) 0iRkl0j (3) ijRkl K i kK j l K i lK j k 4 [ij]β [k K l] ,α(11a)1 ( k K j l l K j k ) ,(11b)α1111 t K i j K i k K kj i j α j (β k K i k ) ( k β i )K kj . (11c)ααααFrom these one obtains the four-dimensional Einstein tensor components H ij2(3)γKK K R,,H ij2α2 γαHi β i H , Hi 2 γ j (K ij Kγ ij ) , 22α γβ iβ j H11 2 t ( γ P ij ) (3)Rij (3)Rγ ij2α γ α γ2 11 ( i j γ ij 2 )α k β i P jk β j P ik β k P ijα α 11 2ijkijkl 2P k P P P Pkl P P γ ij ,22G00 (12a)G0i(12b)Gijwhere K γij K ij andP ij Kγ ij K ij , P γij P ij .(12c)(13)(Components of the four-dimensional Riemann and Einstein tensors are raised and lowered using the four-dimensional metric; components of all other quantities, includingKij and the three-dimensional Riemann tensor, are raised and lowered using γij .) Thefour-dimensional Ricci scalar obeys (4) g R α γ Kij K ij K 2 (3)R 2 t ( γ K) 2 i γ(Kβ i i α) . (14)3

Equation (14) provides an expression for the Einstein-Hilbert Lagrangian in the 3 1decomposition. This expression includes two derivative terms that make no contributionto the equations of motion. We may therefore define a new action involving the intrinsicand extrinsic curvatures of the hypersurfaces of constant coordinate time. The result isthe ADM action [1]:Z SADM [α, βi , γij ] d4 x LADM (α, βi , γij ) , LADM α γ Kij K ij K 2 (3)R . (15)The intrinsic curvature term may be integrated by parts to giveZZ ij k 3 3(3)illγd xRα γd x (γ γjk γ jk γjk) i α αγ ij γlik γkj γijk γklplus a surface term3H(16)ikα(γ jk γjk γ ij γjk)dSi , where dSi is the covariant surface element.ADM FormulationIn the Lagrangian approach, the classical equations of motion follow from extremizing thetotal action with respect to the metric fields α(x), βi (x), γij (x) and any matter fields.The matter action SM also depends on the metric fields. The functional derivative isdefined by the integrand of a variation, neglecting any boundary terms arising from totalderivatives, e.g. ZδS4δS[γij ] lim S[γij (x) δγij (x)] S[γij (x)] d xδγij (x) ,(17)δγij 0δγijwhere there variation is carried to first order in δγij . The four-dimensional stress-energytensor is given by2 δSMT µν .(18) g δgµνUsing equation (1), this givesδSM α2 γ T 00 ,δαδSM α γ γ ij T 0j ,δβiδSM1 α γ (T ij β i β j T 00 ) .δγij2(19a)(19b)(19c)(Note that four-dimensional components are always used for T µν and Gµν . Their components are raised and lowered using the full spacetime metric.) Varying the ADM action4

with respect to the metric fields givesδSADM H 2α2 γ G00 ,δαδSADM Hi 2α γ γ ij G0 j ,δβi δSADM (3) ij 1 (3) ij ijR Rγ t ( γ P ) α γδγij2 γ ( i j γ ij 2 )α γ k β i P jk β j P ik β k P ij 11 2 ijkijkl α γ 2P k P P P Pkl P P γ ij22 iji j 00 α γ (G β β G ) .(20a)(20b)(20c)Combining equations (19) and (20) with δSADM δSM 0 yields the Einstein equations1Gµν T µν .2(21)RIn the mechanics of a system of finitely many degrees of freedom, S L(q, q̇, t) dtwhere q are generalized coordinates and q̇ dq/dt are coordinate velocities. The transition to a Hamiltonian formulation begins with the definition of canonical momenta,p L/ R q̇. In field theory, there are infinitely many degrees of freedom; the LagrangianL L d3 x sums over every field variable. The discrete variables q are, in effect, replaced by infinitely many variables α(x)d3 x, and so on. The field Lagrangian is nowregarded as a function of both the generalized coordinates (α, β, γij ) and their velocities(α̇, β̇, γ̇ij ), where a dot denotes t . Note that the coordinate time t must be singled outto define generalized momenta, and the Hamiltonian formulation regards time and spacederivatives in very different ways — time derivatives act on individual generalized coordinates (the field values at fixed spatial position) while space derivatives relate differentfield values. Using equations (5) and (15), one finds the momenta conjugate to α, βi ,and γij are, respectively, LADM 0, α̇ LADM 0, β̇i LADM γ (Kγ ij K ij ) γ P ij . γ̇ijπα (22a)πi(22b)π ij(22c)(The matter Lagrangian is assumed to be independent of the time derivative of the metricso it makes no contribution to the momenta.) In the classical theory, the momentaconjugate to α and βi vanish because the Lagrangian is independent of α̇ and β̇. In5

quantum field theory, πα and π i vanish “weakly,” i.e. on shell (denoted by 0). In thelanguage of Dirac [2], equations (22a) and (22b) are called primary constraints.The Hamiltonian follows from Legendre transformation of the action:Z hiH α̇πα β̇i π i γ̇ij π ij LADM LM d3 xZ hiiijij α̇πα β̇i π 2π (i βj) 2αKij π LADM LM d3 x Z "LADMiji 2Kij π α̇πα β̇i π αα#i βi ( j π ij j π ji 2γjkπ jk ) LM d3 x .In the second line we have used equation (5) and in the last line we have integrated byparts the (i βj) terms and dropped the irrelevant boundary terms. Writing Kij in termsof π ij using equations (13) and (22c), we obtain the ADM Hamiltonian,Z iijH(α, βi , γij , πα , π , π ) α̇πα β̇i π i αH βi Hi LM d3 x ,(23)whereHi 1 γik γjl γij γkl π ij π kl γ (3)R ,2 ij π ijjiijk ( j π j π 2γjk π ) 2 γ j .γ1H γ(24)(25)These are exactly the same quantities introduced in equations (12a)–(12b) except thatnow they are expressed in terms of the canonical fields and momenta. The threedimensional Ricci scalar is a function of the fields γij only (it contains no time derivatives)and its spatial integral should be integrated by parts to eliminate the spatial derivatives.Note that the Hamiltonian densities H and Hi must be regarded as functions of thecanonical variables γij and π ij and not, for example, π i j γjk π ik or π γij π ij .The ADM Hamiltonian includes terms α̇πα β̇i π i that would seem to depend on velocities. In fact, α̇ and β̇i are Lagrange multipliers which enforce the primary constraintsπα 0 and π i 0. These Lagrange multipliers are arbitrary and will be constrained laterby gauge-fixing. General covariance (diffeomorphism-invariance) allows us to replace α̇and β̇i by any functions of the metric variables (α, βi , γij ). This procedure amounts tomaking a gauge choice. For now we impose no gauge conditions.We can obtain the equations of motion using equal-time Poisson brackets, which aredefined by Z δAδBδAδB{A, B} d3 x .(26) δγij (x, t) δπ ij (x, t) δπ ij (x, t) δγij (x, t)6

The fundamental Poisson brackets are{α(x), πα (x0 )} δ 3 (x x0 ) ,{βj (x), π i (x0 )} δ i j δ 3 (x x0 ) ,{γkl (x), π ij (x0 )} δ i (k δ j l) δ 3 (x x0 ) .(27a)(27b)(27c)Using them, we may obtain the time evolution of the canonical variables:α̇ {α, H} α̇(α, βi , γij ) ,β̇i {βi , H} β̇i (α, βi , γij ) ,αγ̇ij {γij , H} i βj j βi (2πij πγij ) ,γδSM π̇α {πα , H} H H α2 γ T 00 0 ,δαδS M Hi α γ γ ij T0j 0 ,π̇ i {π i , H} Hi δβi (3) ij 1 (3) ijijijπ̇ {π , H} α γR Rγ2 (i j)k2β π β k π ij i jij 2 γ ( γ )α γ k γ 11αδSM. 2π i k π jk ππ ij πkl π kl π 2 γ ij γ22δγij(28a)(28b)(28c)(28d)(28e)(28f)Equations (28a)–(28b) contain no dynamical content whatsoever; they arise as Lagrangemultipliers. Equations (28d)–(28f) are equivalent to equations (19)–(21). Equation (28c)reproduces equation (22c). Equations (28d) and (28e) are called secondary constraintsor dynamical constraints, as they enforce the primary constraints πα 0 and π i 0.In the quantum theory, they imply that the wave function does not depend on α andβ [2]. The last equation, (28f), contains the actual dynamics of the gravitational field.From equations (28c) and (28f), we see that γij (unlike α and βi ) obeys a second-orderdifferential equation in time.We now wish to eliminate the non-dynamical degrees of freedom from the Hamiltonian. This is done following the prescription given by Dirac [2] and detailed by Weinberg[4]. The first step is to solve the secondary constraints for the non-dynamical variables αand β. The results depend on the matter fields. For a scalar field φ(x) with Lagrangiandensity 1 µνLM g g ( µ φ)( ν φ) V (φ) ,(29)27

varying the action yields δSM11 ij k2 γ(φ̇ β k φ) γ ( i φ)( j φ) V (φ) ,δα2α22 γδSM (φ̇ β k k φ)γ ij ( j φ) ,δβi2α α γ ik jlδSM γ γ ( k φ)( l φ) .δγij2Solving equations (28d) and (28e) for α and βi gives the following constraints: ij2 2HCα α γ ( i φ)( j φ) 2V (φ) (φ̇ β k k φ)2 0 ,γ2αHiCi (φ̇ β k k φ)( i φ) 0 ,γ(30a)(30b)(30c)(31a)(31b)where H and Hi are the functions of γij and π ij given by equations (24) and (25).4Eliminating non-dynamical degrees of freedomEquations (31) are insufficient to fix α and βi . They must be supplemented by gaugeconditions. For example, the following conditions are equivalent in the weak-field limitto the transverse gauge conditions:χ0 i β i 0 , χi j (γ 1/3 γ ij ) 0 .(32)(Yes, that really is γ 1/3 .) These constraints include what Dirac called second-class constraints, i.e. those whose Poisson brackets with the primary and secondary constraintsdo not vanish. We must follow the procedure outlined by [2] and [4], using the algebraof second-class constraints to modify the Poisson brackets. With the modified brackets,the commutators of constraints will lead to no new constraints and thereby provide aLie algebra. This section remains to be completed.5Perturbed Robertson-Walker SpacetimeTo clarify the treatment of constraints and gauge-fixing, it is useful to analyze a perturbative example. We choose the perturbed Robertson-Walker spacetime because of itscosmological relevance and because it has been studied extensively using the Lagrangianformulation.The perturbed Robertson-Walker metric may be written ds2 a2 (t) e2Φ dt2 2wj E j i dxi dt 0 γik (x)E kl E l j dxi dxj ,(33)8

where the time-independent background Robertson-Walker spatial metric 0 γij and itsinverse 0 γ ij are used to raise and lower all spatial indices unless otherwise noted. Thetime coordinate is called conformal time and the spatial coordinates are called comovingcoordinates. We have introduced a spatial triad matrix E i j , which may be writtenE i j e Ψ ij δi j ψi j 1 i kψ ψ ··· .2! k j(34)We require that Ψ be symmetric, i.e. ψij 0 γik ψ k j ψji . The determinant of thespatial metric is γ 0 γa6 exp( 2ψ i i ) where 0 γ is the determinant of 0 γ ij .The metric of equation (33) has been parameterized in a fully general form but wewill treat (Φ, wi , ψ k j ) as being small perturbations and will compute the Hamiltonian tosecond order in these variables. The translation of our new metric variables to those ofequation (1) is 1/2 1 22Φ0 ij2α a e γ wi wj a 1 Φ Φ w ··· ,2βi a2 wj E j i a2 (wi wj ψ j i · · · ) ,a 2 γij 0 γik E kl E l j 0 γij 2ψij 2ψik ψ k j · · · ,a2 γ ij γ Ẽ l k Ẽ j l 0 γ ij 2ψ ij 2ψ i k ψ kj · · · ,0 ik(35)where Ẽ exp(Ψ) is the matrix inverse of E exp( Ψ), i.e. Ẽ i k E kj E i k Ẽ kj δ i j .The connection coefficients with respect to γij areihllm0nl0 m nk0 kk 0(36)γij γij Ẽ l (i E j) Ẽ En(i j) E m Ẽ m En(i E j) ,where 0 γijk is the connection and 0 i is the covariant derivative, both taken with respectto 0 γij . Taylor expanding E to second order in Ψ, we getγijk 0 γijk 1 γijk 2 γijk · · · ,(37)where1 kγij2 kγij k ψij 0 i ψ k j 0 j ψ k i , 2ψ k l 0 l ψij 0 (i ψ l j) 2ψ l (i 0 j) ψ k l 2ψl(i 0 k ψ l j) . 0(38a)(38b)Notice that the perturbations to the connection are three-tensors on the constant-timehypersurfaces.The extrinsic curvature, to second order in the perturbations, is given by 01 21 1220a Kij η 1 Φ Φ wγij (i wj) 2 t (a ψij ) (1 Φ)2a 1 ψ k (i 0 j) wk 0 (i ψ k j) 0 k ψij wk 2 t (a2 ψik ψ k j ) ,(39)a9

where η ȧ/a. This gives the following extrinsic curvature contribution to the ADMaction: α γ (Kij K ij K 2 )p 6η 2 2η 2B i i η(3Φ ψ)a2 0 γhijiji 2iiji 00eeee Kij K (K i ) 4η ΦK i ψ ψ̇ij w ( i ψ j ψ i ) η 2 3(Φ2 w2 ) 2Φψ ψ 2 4ψij ψ ij ,(40)where ψ ψ i i ande ij 0 (i wj) 1 t (a2 ψij )(41)Ka2reduces to the extrinsic curvature when ȧ 0. Cosmic expansion (ȧ 6 0) introducesmany terms. The second and third lines of equation (40) give the second-order contributions.Expanding the intrinsic curvature to second order in the perturbations gives(3)R a2 6K 4K(ψ ψij ψ ij ) 2(0 2 ψ 0 i 0 j ψ ij )i (0 i ψ)(0 ψ) (0 k ψ ij )(1 γijk ) ii 0 i 2ψ ij j ψ 2ψ jk (1 γjk) 0 γ jk (2 γjk) ,(42)where K is the three-dimensional curvature of the background Robertson-Walker spacee ij . The intrinsic curvature contribution to the ADM actionand has nothing to do with Kis then: (3)Rα γp 6K[1 (Φ ψ)] 4Kψ 2(0 2 ψ 0 i 0 j ψ ij )20aγ 3K[(Φ ψ)2 w2 ] 4K(Φ ψ)ψ 4Kψij ψ ij 2(Φ ψ)(0 2 ψ 0 i 0 j ψ ij ) iii (0 i ψ)(0 ψ) (0 k ψ ij )(1 γijk ) 0 i 2ψ ij j ψ 2ψ jk (1 γjk) 0 γ jk (2 γjk) .(43)The second and third lines give the second-order contributions and a total derivativeterm that may be discarded.The ADM Lagrangian follows from combining equations (40) and (43). The zerothorder part is p ijll0 0 γijk 0 γkl.(44)LADM (a, 0 γ ij ) a2 0 γ 6η 2 0 γ 0Aij , 0Aij 0 γlik 0 γjkVarying the total action with respect to a(τ ) and 0 γ ij (x) gives the Friedmann and energyconservation equations for homogeneous matter at rest in comoving coordinates. Weassume henceforth that the zeroth-order metric functions are known, and examine thefirst-order and second-order Lagrangians.10

The first-order ADM Lagrangian ishiLADMiep 2η 2K i η(3Φ ψ) (0 γ ji j 0 γ ij j ) i (Φ ψ)20aγhi kij (Φ ψ) 0 γ ij 2ψ ij 0Aij 2 0 γ 0 γ l(i l 0 γ j)k 0 k ψij .1(45)Extremizing the first-order action with respect to Φ gives the Friedmann equationa2δ(1 S ADM )1 δSM 6(η 2 K) 2a4 G00 p a4 T 00 16πGa2 ρ0 . (46)0γδΦa2 0 γ δΦ1pHere, ρ0 is the unperturbed density. Extremizing the first-order action with respect towi gives the consistency condition 0 j 0 γ ij 0. Extremizing the first-order action withrespect to ψij givesa2hpδ(1 S ADM )20γ 2(2η̇ η 2 3K)0 γ ij 2 0Aij p k0γ0γδψij1p0 kijγ 0 γ l(i l 0 γ j)k 4 0 γ kl(i 0 γ j) kl 2 0 γ (ij)k 0 γ l kl 2 0 γ k(i k 0 γ j)l l 2(2η̇ η 2 K)0 γ ij 2a4 Gij1 δSM a4 T ij 16πGa2 p0 0 γ ij . p20aγ δψij i(47)Here, p0 is the unperturbed pressure. In summary, extremizing the first-order action givesthe unperturbed Einstein equations. This repeats what happened with the zeroth-orderaction. If we define a(τ ) and 0 γ ij (x) to be the classical solutions for the RobertsonWalker spacetime, then the zeroth-order and first-order action both vanish identically.In the quantum theory, a and 0 γ ij equal the classical functions multiplied by the identityoperator so that they commute with all observables.The dynamics of the perturbations (Φ, wi , ψij ) follow from the second-order Lagrangian density,ihLADMji 00iji 2iijeeeep Kij K (K i ) 4η ΦK i ψ ψ̇ij w ( i ψ j ψ i )a2 0 γ (K η 2 )(3Φ2 2Φψ ψ 2 4ψij ψ ij ) 3(η 2 K)w22i 2(Φ ψ)(0 2 ψ 0 i 0 j ψ ij ) (0 i ψ)(0 ψ) (0 k ψ ij )(1 γijk ) , (48)where we have discarded the boundary terms of equation (48). Varying this Lagrangian11

with respect to the metric fields gives2a2δ(2 S ADM ) (0 2 2K)ψ 0 i 0 j ψ ij 2η(ψ̇ 3ηΦ 0 i wi )0γδΦ1p 3(η 2 K)(Φ ψ) ,(49a)211δ( S ADM )1p (0 2 2K)wi 0 i (0 j wj ) 0 j ψ̇ ij 0 i (ψ̇ 2ηΦ)02δwi222aγ 3(η 2 K)wi ,(49b)i1δ( S ADM )20 2ij0 (i j)0k 0 ijp ( t 2η t 2K)ψ ( t 2η) w ( k w ) γ2a2 0 γ δψijhi200kl 0 ij ψ̈ 2η(Φ̇ ψ̇) 2(2η̇ η )Φ k l ψγh2 (0 i 0 j 0 γ ij 0 2 )(Φ ψ) 2 0 (i k ψ j)k (2η̇ η 2 K)(Φ ψ)0 γ ij .(49c)In deriving these we used the commutators(0 k 0 l 0 l 0 k )wi K(δ i k 0 γ nl δ i l 0 γ nk )wn ,(0 k 0 l 0 l 0 k )ψ ij K(δ i k 0 γ nl δ i l 0 γ nk )ψ nj K(δ j k 0 γ nl δ j l 0 γ nk )ψ in . (50)Equations (49) (with ψ ij φ 0 γ ij hij ) reproduces the Einstein tensor componentsgiven in Ref. [5]. As we will see, the last line of each of equations (49) arises from theunperturbed Einstein tensor and will disappear when we add the matter action terms tothe Lagrangian.To show this, we write the Lagrangian for scalar field matter (29) in a perturbedRobertson-Walker spacetime by letting φ φ0 (t) φ(x) and using the perturbed metricto second order. The result isLMpa20γ 111 2φ̇0 Ṽ (φ0 , a) φ̇0 φ̇ φ̇2 Ve (φ0 φ, a) Ve (φ0 , a) 0 γ ij ( i φ)( j φ)2221 (Φ ψ)(φ̇0 φ̇)2 (wi i φ)(φ̇0 φ̇) (Φ ψ)Ve (φ0 φ, a)2 1 (Φ ψ)0 γ ij 2ψ ij ( i φ)( j φ)2 1 (Φ ψ)2 w2 (φ̇0 φ̇)2 (Φ ψ)(φ̇0 φ̇)(wi i φ)4 1 i (w i φ)2 (φ̇0 φ̇)ψ ij (wi j φ) (Φ ψ)ψ ij ψ i k ψ jk ( i φ)( j φ)2 11 220ijVe (φ0 φ, a) γ ( i φ)( j φ) , (Φ ψ) w(51)2212

whereVe (φ, a) a2 V (φ) .(52)We have not linearized the scalar field; φ can be arbitrarily large. We have only droppedterms higher than quadratic in the metric perturbations. The terms in the first bracketgive the Lagrangian for a spatially homogeneous scalar field φ0 (t). The second bracketgives contributions that are independent of the metric perturbations. The second andthird lines give terms that are first order in the metric perturbations; the remaining linesgive terms that are second order. The zeroth-order Lagrangian gives the equation ofmotion Veφ̈0 2η φ̇0 0(53) φ0and1 2 eφ̇ V (φ0 , a) 6(η 2 K) ,2 01 2 e φ̇ V (φ0 , a) 2(2η̇ η 2 K) .2 0a2 ρ0 (54a)a2 p 0(54b)Together these imply1 2φ̇ η 2 η̇ K .(55)4 0Now we linearize the scalar field by treating φ as a first-order quantity, similarly tothe metric perturbations. The first-order scalar-field Lagrangian is VeLM1 2p (Φ ψ) φ̇0 Ṽ (φ0 , a) φ̇0 φ̇ φ φ̇20 Φ .022 φ0γa1(56)Varying this with respect to φ0 reproduces equation (53). Varying it with respect to themetric perturbations and comparing with equations (46) and (47) reproduces equations(54a) and (54b). As with the gravitational action, the first-order matter action yieldsnothing new. We have to go to second order in the perturbations to see the dynamics ofthe perturbations.The second-order scalar-field Lagrangian is"#hi2 2 Ve 2LM1 2 0 ijipφ̇ γ ( i φ)( j φ) φ (Φ ψ)φ̇ w φφ̇0i2 φ20a2 0 γ 1 Ve1 φ .(57) (Φ ψ)2 w2 φ̇20 (Φ ψ)2 w2 Ve (φ0 , a) (Φ ψ)42 φ013

Differentiating it gives a2δ(2 S M ) Ve a2 ρ0 (Φ ψ) φ̇0 φ̇ φ Φφ̇200γδΦ φ01p2 a2 [ρ0 (Φ ψ) δρ] , δ( S M ) a2 ρ0 wi φ̇0 (0 γ ij ) j φ a2 ρ0 wi (ρ0 p0 )v i ,a2 0 γ δwi#"e1 δ(2 S M ) V p a2 p0 (Φ ψ) φ̇0 φ̇ φ Φφ̇20 0 γ ij φ0a2 0 γ δψij 1p a2 [p0 (Φ ψ) δp] 0 γ ij .(58a)(58b)(58c)As expected, the terms proportional to a2 ρ0 and a2 p0 cancel the last terms in equations(49) when the matter and gravitational actions are combined. The perturbations ofenergy density, velocity, and pressure are δρ, v i , and δp.5.1Hamiltonian FormulationBefore proceeding further with the ADM Lagrangian in a perturbed Robertson-Walkerspacetime, we first compute the Hamiltonian for the scalar field, using the second-orderLagrangian. The canonical momentum of the scalar field is πφ (2 LM )/ φ̇, which givesa2πpφ0γ φ̇ (Φ ψ)φ̇0 .(59)Performing the Legendre transformation, we get#"2eπφ2πφVHM1 2ppp φ 0 γ ij ( i φ)( j φ) (Φ ψ)φ̇0 φ̇0 wi i φ222200202 (a φ0aaγγ)γ 1 1 2 e Ve(60)φ̇0 V (φ0 , a) (Φ ψ)2 w2 .φ Φψ φ̇20 (Φ ψ) φ02 2With a 1, the first set of terms (in square brackets) gives the Hamiltonian density ofa scalar field in flat spacetime. The other terms give gravitational couplings.Next we compute the ADM Hamiltonian by Legendre transformation of equation(48). The momentum conjugate to ψij is π ij 2 LADM / ψ̇ij , which givesπ ijp2a20γhi ψ̇ ij 0 (i wj) 0 γ ij ψ̇ 2η(Φ ψ) 0 k wk .14(61)

The ADM Hamiltonian density is given (up to irrelevant boundary terms) by πij π ij 21 (π k k )2HADMπ ij 0pp Φ̇πΦ ẇi π i p (i wj) η(Φ ψ)0 γ ija2 0 γ4(a2 0 γ)2a2 0 γ 4ηwi i ψ 12η 2 Φψ 4Kψij ψ ij 2(Φ ψ) (0 2 2K)ψ 0 i 0 j ψ ij (0 i ψ)(0 i ψ) (0 k ψ ij )(1 γijk ) 3(η 2 K) (Φ ψ)2 w2 .(62)The last term cancels the last term of equation (60).The net Hamiltonian for the fields is given by adding equations (60) and (62):Ziij(HADM HM ) d3 xH[Φ, πΦ , wi , π , φ, πφ , ψij , π ] Z Hφ Hψ Hint ΦHΦ wi Hi d3 x , (63) whereHφ Hψ Hint HΦ Hi #2eV (64a)φ2 , 0 γ ij ( i φ)( j φ) 2 φ20(a2 0 γ)2p πij π ij 21 (π k k )2p ηψπ k k a2 0 γ (0 k ψ ij ) 0 k ψij 2 0 (i ψ k j)4a2 0 γpp a2 0 γ (0 i ψ) 0 i ψ 20 j ψij 4K(ψ 2 ψij ψ ij )a2 0 γ ,(64b)!p dVeφ ψ,(64c)φ̇0 πφ a2 0 γdφ0#"ep Vφ , (64d)φ̇0 πφ ηπ k k a2 0 γ 2(0 2 2η̇ 4η 2 )ψ 20 i 0 j ψ ij φ0p j π ij 0 γ i jk π jk a2 0 γ (φ̇0 j φ 4η j ψ)0 γ ij#!"ijpπp(64e) (φ̇0 j φ 4η j ψ)0 γ ij .a2 0 γ 0 ja2 0 γa2p0γ"π2pφWe have ignored the Lagrange multiplier terms Φ̇πΦ and ẇi π i since they play no role inthe dynamics; Φ and wi will follow from the equations of motion combined with gaugeconstraints. In equations (64), Hφ depends only on φ and its conjugate momentum, Hψdepends only on ψij and its conjugate momentum, and Hint is a coupling between φ andψij . Because the Hamiltonian is independent of Φ and wi , the corresponding momentavanish weakly: πΦ 0 and π i 0. As we will see, HΦ and Hi are constraints on thedynamical fields φ and ψij and their momenta.15

The fundamental Poisson brackets are{Φ(x), πΦ (y)} δ 3 (x y) ,{wj (x), π i (y)} δ i j δ 3 (x y) ,{φ(x), πφ (y)} δ 3 (x y) ,{ψkl (x), π ij (y)} δ i (k δ j l) δ 3 (x y) .(65a)(65b)(65c)(65d)Using them, we obtain the classical time evolution of the canonical variables:φ̇ {φ, H} a2π̇φ (Φ ψ)φ̇0 ,ηπ k k 0 γ ijp 2(0 2 2K)ψ ij 2(0 i 0 j 0 γ ij 0 2 )(Φ ψ)20γ0aγ 2 2(η̇ 2η 2 )Φ 2Kψ 0 k 0 l ψ kl 2η(0 k wk ) 0 γ ij!eφ̇πdV0φ0 ijφ pγ . 4 0 (i k ψ j)k dφ0a2 0 γπ̇ ijpa20γπij 21 (π k k )0 γ ij 0p (i wj) η(Φ ψ)0 γ ij ,2a2 0 γ {πΦ , H} HΦ (φ, πφ , ψij , π ij ) 0 , {π i , H} Hi (φ, ψij , π ij ) 0 ,"#2eep V Vφ (Φ ψ) φ̇0 (0 i wi ) {πφ , H} a2 0 γ 0 2 φ φ20 φ0ψ̇ij {ψij , H} π̇Φπ̇ iπpφ(66a)(66b)(66c)(66d), (66e)(66f)A superscript 0 has been neglected on the k on the third line of equation (66f) fornotational clarity. Equations (66a) and (66b) reproduce equations (59) and (61), respectively. Equations (66c) and (66d) are secondary constraints which enforce the primaryconstraints πΦ 0 and π i 0. They are equivalent to equations (49a) and (49b)combined with equations (58a) and (58b).The last two equations, (66e) and (66f), contain the actual dynamics of the scalar andgravitational field. Combining equations (66a) and (66e) gives the equation of motionfor the scalar field:φ̈ 2η φ̇ 0 2 φ Ve 2 Veφ 2Φ φ̇0 (Φ̇ ψ̇) φ̇0 (0 i wi ) . φ20 φ0(67)Combining equations (66b) and (66f) yields the equation of motion for ψij . They aresimplest when separated into the trace and trace-free parts. We writeψij Ψ0 γ ij sij ,160 ijγ sij 0 .(68)

Note that ψ 3Ψ where Ψ is the usual notation for the gauge-invariant spatial curvatureperturbation. By combining the Hamilton equations (66), we get1Ψ̈ η(2Ψ̇ Φ̇) 0 2 (Φ Ψ) KΨ (2η̇ η 2 )Φ3! 111 Ve2 φ̇0 φ̇ φ Φφ̇0 2 t a2 (0 k wk ) 0 i 0 j sij ,4 φ03a6 11(69) 4πGa2 δp 2 t a2 (0 k wk ) 0 i 0 j sij .3a6The traceless parts of equations (66b) and (66f) give 1 0 ij 0 20020 20( t 2η t 2K)sij ( t 2η) (i wj) i j γ (Ψ Φ)31(70) 0 γ ij ( t 2η)(0 k wk ) 2 0 (i k sj)k .2Equations (69) and (70) simplify when we impose the transverse gauge conditions 1 0 jk0kj0jk0jk 0.(71)χ 0 k w 0 , χ k s kψ γ ψ3We have introduced the symbols χ0 and χj for gauge constraints to be used later.For the remainder of this subsection we assume these gauge conditions hold andexplore the classical equations of motion. Then the right-hand side of equation (70)vanishes. Using the scalar-vector-tensor decomposition, the left-hand side separates intoparts that are doubly transverse (sij ), semi-transverse [0 (i wj) ], and doubly longitudinal(Φ ψ). All three parts must vanish separately, yielding( t2 2η t 0 2 K)sij 0 ,(72a)0( t 2η) (i wj) 0 ,(72b) 10 i 0 j 0 γ ij 0 γ ij 0 2 (Ψ Φ) 0 .(72c)3The first of these is the evolution equation for gravitational waves. The second equationimplies wi 0: in linear theory, a scalar field cannot generate a vector mode. The thirdequation implies that Ψ Φ is spatially homogeneous. Any time-varying contributionto this contribution may be gauged away by modifying that background curvature andhence is unmeasurable. We may therefore conclude that Φ Ψ.Next we examine the secondary constraints. Using equations (59), (61), and (64d),HΦ 0 gives!e V11φ Φφ̇20 η(0 k wk ) 0 i 0 j sijφ̇0 φ̇ (0 2 3K)Ψ 3η(Ψ̇ ηΦ) 4 φ02 4πGa2 δρ η(0 k wk ) 1710 0 i j sij .2(73)

From equations (49a) and (58a), this is equivalent to δ(2 S)/δΦ 0. Similarly, usingequations (61) and (64e), Hi 0 implies1 0 2111( 2K)wi 0 i (Ψ̇ ηΦ) 0 i (φ̇0 φ) 0 i (0 k wk ) 0 k ṡk i444211 4πGa2 (ρ0 p0 )vi 0 i (0 k wk ) 0 k ṡk i . (74)42From equations (49b) and (58b), this is equivalent to δ(2 S)/δwi 0. With the gaugeconditions (71) imposed, equations (73) and (74) reduce to the standard equations forgauge-invariant perturbations. They are initial-value constraints; their time derivativescombined with equation (67) are redundant with equations (69) and (70).Combining the equations of motion yields a single second-order equation for Ψ:Ψ̈ 3(1 c2w )η Ψ̇ 3(c2w w)η 2 Ψ (5 3w)KΨ 2 Ψ 0 ,(75)where

and (4)Rk i0j (4)Rk iljβ l (4)R0 iljβ l (4)R0 i0j βk α kK ij iK k j. (9) These equations may be combined to give: (4) R ijkl (3) ijkl Kik Kjl il jk, (10a) (4)R 0jkl (4) R ijklβ i α( kKjl lKjk) (4) kl0j, (10b) (4)R 0i0j α t Kij α 2 k i Kjk α i j αβ k k

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