Quick review of the ADM formalism in General Relativity Jerome Quintin Grad Seminar, McGill, 2015 Jerome Quintin Quick review of the ADM formalism in General Relativity Foliation of spacetime Let us have a spacetime (M ;g). Let Σ ⊂ M be a spacelike 3-dimensional hypersurface. Let nm be a future-pointing m (timelike) unit normal vector, so n nm = −1. Then, the induced metric is gmn = gmn + nm nn : (1) gmn is purely spatial, i.e., it resides entirely in Σ with no piece along nm : m m m n gmn = n gmn + n nm nn = nn − nn = 0: (2) Jerome Quintin Quick review of the ADM formalism in General Relativity Foliation of spacetime Let the hypersurface (Σ;g) have a covariant derivative D such that Dm gnt = 0. For instance, for some scalar f , n Dm f ≡ gm ∇n f ; (3) m and for some tensor F n , n c n s x Dm F t ≡ gm gx gt ∇c T s : (4) Here, ∇m is the usual covariant derivative on (M ;g). Jerome Quintin Quick review of the ADM formalism in General Relativity The extrinsic curvature The extrinsic curvature K is defined as 1 K ≡ − L g = −g t g s ∇ n = −D n ; (5) mn 2 n^ mn m n t s n m where L is the Lie derivative. The trace is often called the mean curvature: mn K ≡ g Kmn : (6) Jerome Quintin Quick review of the ADM formalism in General Relativity Constraint equations The 3- and 4-dimensional Riemann tensors on (Σ;g) and (M ;g), respectively, can be related to one another with the extrinsic curvature in what are called the Codazzi equation and Gauss’ equation. More intuitively, by Gauss’ equation, Einstein’s 4-dimensional equations, 1 (4)R − g (4)R = 8pGT ; (7) mn 2 mn mn give the following Hamiltonian constraint: (3) 2 mn R + K − Kmn K = 16pGr; (8) m n where r ≡ Tmn n n . Jerome Quintin Quick review of the ADM formalism in General Relativity Constraint equations The same way, by the Codazzi equation, Einstein’s equations give the following momentum constraint: n Dn K m − Dm K = 8pGJm ; (9) t n where Jm ≡ −Tnt n g m . Jerome Quintin Quick review of the ADM formalism in General Relativity Evolution equations To describe the evolution, let us decompose the time vector as follows: tm = Nnm + b m : (10) We introduced the laspe function, m N = −nm t ; (11) and the shift vector, m m n b = g n t : (12) Jerome Quintin Quick review of the ADM formalism in General Relativity Evolution equations The resulting evolution equation for the extrinsic curvature is (3) t LtKmn = −Dm Dn N + N( R − 2Kmt K n + Kmn K) 1 −8pGN[S − g (S − r)] + L K ; (13) mn 2 mn b mn t s mn where Smn ≡ g m g n Tts and S ≡ g Smn , and the evolution equation for the induced metric is Ltgmn = −2NKmn + Lb gmn : (14) Jerome Quintin Quick review of the ADM formalism in General Relativity Choosing basis vectors Now, let us choose tm = (1;0;0;0): (15) Then, Lt = ¶t . Also, b m = (0;b i ); (16) and 1 b i nm = ;− : (17) N N Furthermore, the line element becomes 2 m n 2 2 i i j j ds = gmn dx dx = −N dt + gij (dx + b dt)(dx + b dt): (18) Jerome Quintin Quick review of the ADM formalism in General Relativity ADM equations With this choice of basis vectors, we obtain the following set of ADM equations, (3) 2 ij R + K − Kij K = 16pGr; (19) ij ij i Dj (K − g K) = 8pJ ; (20) ¶t gij = −2NKij + Di bj + Dj bi ; (21) 1 ¶ K = N((3)R −2K K k +K K)−D D N −8pGN[S − g (S −r)] t ij ij ik j ij i j ij 2 ij k k k +b ¶k Kij + Kik ¶j b + Kkj ¶i b ; (22) m n i m ij m n where r = Tmn n n , J = −Tmj n g , Sij = Tmn gi gj , and ij S = g Sij . Jerome Quintin Quick review of the ADM formalism in General Relativity Example: flat FRW i m m 2 Choose N = 1, b = 0 (so n = t ), and take gij = a (t)dij , i.e. 2 2 2 i j ds = −dt + a (t)dij dx dx ; (23) Then, homogeneity and isotropy imply that Ji = 0 and 1 Kij = − 3 gij K, so Eq. (20) is satisfied. Furthermore, K = −3a_=a, so Eq. (21) is satisfied, and Eq. (19) implies that a_ 2 8pG = r: (24) a 3 Similarly, Eq. (22) gives ¨a 4pG = − (r + 3p); (25) a 3 where p = S=3. Jerome Quintin Quick review of the ADM formalism in General Relativity Fluid picture Let us introduce a fluid with um corresponding to the 4-velocity of observers comoving with the fluid. The gradient of um can be decomposed as 1 ∇ u = −u a + Θg + s + w ; (26) m n m n 3 mn mn mn m where am is the acceleration, the trace Θ ≡ ∇m u is the expansion rate, the traceless symmetric part smn is the shear tensor, and the antisymmetric part wmn is the vorticity tensor. From here on, we assume wmn = 0. E.g., in FRW, Θ = 3H, smn = wmn = 0. Jerome Quintin Quick review of the ADM formalism in General Relativity Fluid picture Using um to foliate the spacetime (i.e. taking nm = um ), the extrinsic curvature becomes 1 K = −s − Θg : (27) mn mn 3 mn The Hamiltonian and momentum constraints become 2 (3)R − 2s 2 + Θ2 = 16pGr; (28) 3 2 D s n = D Θ; (29) n m 3 m 2 1 m n where s ≡ 2 s n s m is the rate of shear of a fluid element. Jerome Quintin Quick review of the ADM formalism in General Relativity Fluid picture The evolution equations become (setting b m = 0, but keeping N general) 1 ¶ g = 2N s + Θg ; (30) t mn mn 3 mn (3) 2 m ¶t Θ = −N[ R + Θ − 12pG(r − p)] + Dm D N; (31) 1 1 ¶ s m = −NΘs m −N (3)R m − g m (3)R +Dm D N − g m Dt D N: t n n n 3 n n 3 n t (32) Eqs. (28) and (31) lead to Raychaudhury’s equation: 1 1 Θ_ + Θ2 = −2s 2 − 4pG(r + 3p) + Dm D N: (33) 3 N m _ m Here, Θ ≡ u ∇m Θ, derivative with respect to the proper time measured by observers comoving with the fluid (with N = 1, Θ_ = ¶t Θ). Jerome Quintin Quick review of the ADM formalism in General Relativity Stress–energy tensor conservation For a perfect fluid with Tmn = (r + p)um un + pgmn , mn un ∇m T = 0 =) r_ + Θ(r + p) = 0; (34) the local energy conservation law, and D p g ∇ T mn = 0 =) a = − a : (35) na m a r + p Jerome Quintin Quick review of the ADM formalism in General Relativity.