Hamiltonian Formulation of General Relativity

Massachusetts Institute of Technology Department of Physics Physics 8.962 Spring 2002 Hamiltonian Formulation of General Relativity c 2005 Edmund Bertschinger. All rights reserved. Revision date June 5, 2005 1 Introduction The usual approach to treating general relativity as a field theory is based on the La- grangian formulation. For some purposes (e.g. numerical relativity and canonical quan- tization), a Hamiltonian formulation is preferred. The Hamiltonian formulation of a field theory, like the Hamiltonian formulation of particle mechanics, requires choosing a preferred time variable. For a single particle, proper time may be used, and the Hamiltonian formulation remains manifestly covariant. For a continuous medium, the Hamiltonian formulation requires that a time variable be defined everywhere, not just along the path of one particle. Thus, the Hamiltonian formulation of general relativity requires a sepa- ration of time and space coordinates, known as a 3+1 decomposition. Although the form of the equations is no longer manifestly covariant, they are valid for any choice of time coordinate, and for any coordinate system the results are equivalent to those obtained from the Lagrangian approach. It is convenient to decompose the metric as follows: g = α2 + γijβ β , g = β , g = γ , (1) 00 − i j 0i i ij ij ij ik i where γ is the inverse of γij, i.e. γ γjk = δ j. This 3+1 decomposition of the metric replaces the 10 independent metric components by the lapse function α(x), the shift vector βi(x), and the symmetric spatial metric γij(x). The inverse spacetime metric components are 1 βi βiβj g00 = , g0i = , gij = γij , (2) −α2 α2 − α2 1 i ij where β γ βj. From now on, except as noted otherwise, all Latin (spatial) indices are raised≡ and lowered using the spatial metric. The determinant of the four-metric is 2 g = α γ where γ is the determinant of γij. The− 3+1 decomposition separates the treatment of time and space coordinates. In place of four-dimensional gradients, we use time derivatives and three-dimensional gradients. In these notes, the symbol denotes the three-dimensional covariant ∇i derivative with respect to the metric γij. We will not use the four-dimensional covariant derivative. Thus, 1 Ai = ∂ Ai + γi Ak , γi γil (∂ γ + ∂ γ ∂ γ ) . (3) ∇j j jk jk ≡ 2 j kl k jl − l jk In these notes we choose units so that 16πG = 1. We assume a coordinate basis through- out. These notes first consider a general metric and then specialize to a perturbed Robertson- Walker spacetime. 2 Curvature and Gravitational Actions In the 3+1 approach, spacetime is described by a set of three-dimensional hypersurfaces of constant time t = x0 propagating forward in time. These hypersurfaces have intrinsic curvature given by the three-dimensional Riemann tensor, (3)Ri = ∂ γi ∂ γi + γi γn γi γn . (4) lkm k lm − m kl kn lm − mn kl (3) (3) k Contractions define the three-dimensional Ricci tensor Rij = R ikj and Ricci scalar, (3) ij(3) R = γ Rij. In addition to the intrinsic curvature, the hypersurface of constant time has an extrinsic curvature Kij arising from its embedding in four-dimensional spacetime: 1 K = ( β + β ∂ γ ) . (5) ij 2α ∇i j ∇j i − t ij The full spacetime curvature is related to the intrinsic and extrinsic curvature of the constant-time hypersurfaces by the Gauss-Codazzi equations 1 (4)R0 = ( K K ) (6) jkl −α ∇k jl − ∇l jk and (4)Ri = (3)Ri (4)R0 βi + Ki K Ki K . (7) jkl jkl − jkl k jl − l jk MTW and other sources give these relations assuming an orthonormal basis, for which βi = 0 and α = 1. Equations (6) and (7) are exact for any coordinate basis. The other components of the four-dimensional Riemann tensor follow from 1 1 1 (4)R0 = ∂ K K kK α + (βkK )+ K βk (8) i0j −α t ij − i jk − α∇i∇j α ∇j ik jk∇i 2 and (4)Rk = (4)Rk βl + (4)R0 βl (4)R0 βk + α kK Kk . (9) i0j ilj ilj − i0j ∇ ij − ∇i j These equations may be combined to give: (4)R = (3)R + K K K K , (10a) ijkl ijkl ik jl − il jk (4)R = (4)R βi + α( K K )= (4)R , (10b) 0jkl ijkl ∇k jl − ∇l jk kl0j (4)R = α∂ K + α2K kK + α α + αβk K 0i0j t ij i jk ∇i∇j ∇k ij α (βkK ) α (βkK )+ (4)R βkβl , (10c) − ∇i jk − ∇j ik kilj and 4 (4)Rij = (3)Rij + Ki Kj Ki Kj + β[i Kj] , (11a) kl kl k l − l k α ∇[k l] 1 (4)R0j = ( Kj Kj ) , (11b) kl −α ∇k l − ∇l k 1 1 1 1 (4)R0i = ∂ Ki + Ki Kk i α + (βkKi ) ( βi)Kk . (11c) 0j −α t j k j − α∇ ∇j α∇j k − α ∇k j From these one obtains the four-dimensional Einstein tensor components 00 ij 2 (3) G = H , = √γ KijK K R , (12a) −2α2√γ H − − i i 0i α + β i ij ij G = H H , = 2√γ j(K Kγ ) , (12b) 2α2√γ H ∇ − i j ij β β 1 ij (3) ij 1 (3) ij G = H + ∂t(√γ P )+ R Rγ −2α2√γ α√γ − 2 1 1 ( i j γij 2)α + βiP jk + βjP ik βkP ij −α ∇ ∇ − ∇ α∇k − 1 1 +2P i P jk P P ij P P kl P 2 γij , (12c) k − − 2 kl − 2 ij where K γijK and ≡ P ij Kγij Kij , P γ P ij . (13) ≡ − ≡ ij (Components of the four-dimensional Riemann and Einstein tensors are raised and lowered using the four-dimensional metric; components of all other quantities, including Kij and the three-dimensional Riemann tensor, are raised and lowered using γij.) The four-dimensional Ricci scalar obeys √ g (4)R = α√γ K Kij K2 + (3)R 2∂ (√γ K) + 2∂ √γ(Kβi iα) . (14) − ij − − t i − ∇ 3 Equation (14) provides an expression for the Einstein-Hilbert Lagrangian in the 3+1 decomposition. This expression includes two derivative terms that make no contribution to the equations of motion. We may therefore define a new action involving the intrinsic and extrinsic curvatures of the hypersurfaces of constant coordinate time. The result is the ADM action [1]: S [α, β ,γ ]= d4x (α, β ,γ ) , = α√γ K Kij K2 + (3)R . (15) ADM i ij LADM i ij LADM ij − Z The intrinsic curvature term may be integrated by parts to give (3)R α√γd3x = (γijγk γjkγi ) α + αγij γkγl γk γl √γd3x (16) jk − jk ∇i li kj − ij kl Z Z plus a surface term α(γjkγi γijγk )dS , where dS is the covariant surface element. jk − jk i i H 3 ADM Formulation In the Lagrangian approach, the classical equations of motion follow from extremizing the total 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 is defined by the integrand of a variation, neglecting any boundary terms arising from total derivatives, e.g. 4 δS δS[γij] lim S[γij(x)+ δγij(x)] S[γij(x)] d x δγij(x) , (17) ≡ δγij →0 − ≡ δγ Z ij where there variation is carried to first order in δγij. The four-dimensional stress-energy tensor is given by 2 δS T µν = M . (18) √ g δg − µν Using equation (1), this gives δS M = α2√γ T 00 , (19a) δα − δSM ij 0 = α√γγ T j , (19b) δβi δS 1 M = α√γ (T ij βiβjT 00) . (19c) δγij 2 − (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 action 4 with respect to the metric fields gives δS ADM = = 2α2√γ G00 , (20a) δα −H δSADM i ij 0 = = 2α√γγ G j , (20b) δβi −H − δS 1 ADM = ∂ (√γ P ij) α√γ (3)Rij (3)Rγij δγ − t − − 2 ij +√γ ( i j γij 2)α √γ βiP jk + βjP ik βkP ij ∇ ∇ − ∇ − ∇k − 1 1 α√γ 2P i P jk P P ij P P kl P 2 γij − k − − 2 kl − 2 = α√γ (Gij βiβjG00) . (20c) − − Combining equations (19) and (20) with δSADM + δSM = 0 yields the Einstein equations 1 Gµν = T µν . (21) 2 In the mechanics of a system of finitely many degrees of freedom, S = L(q, q,t˙ ) dt where q are generalized coordinates andq ˙ = dq/dt are coordinate velocities. The tran- R sition to a Hamiltonian formulation begins with the definition of canonical momenta, p ∂L/∂q˙. In field theory, there are infinitely many degrees of freedom; the Lagrangian L≡= d3x sums over every field variable. The discrete variables q are, in effect, re- placedL by infinitely many variables α(x)d3x, and so on. The field Lagrangian is now R regarded as a function of both the generalized coordinates (α,β,γij) and their velocities (α, ˙ β,˙ γ˙ ij), where a dot denotes ∂t. Note that the coordinate time t must be singled out to define generalized momenta, and the Hamiltonian formulation regards time and space derivatives in very different ways — time derivatives act on individual generalized coordinates (the field values at fixed spatial position) while space derivatives relate different field values. Using equations (5) and (15), one finds the momenta conjugate to α, βi, and γij are, respectively, ∂ π LADM 0 , (22a) α ≡ ∂α˙ ≈ ∂ πi LADM 0 , (22b) ≡ ∂β˙i ≈ ∂ πij LADM = √γ (Kγij Kij)= √γ P ij .

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