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Quick review of the ADM formalism in

Jerome Quintin

Grad Seminar, McGill, 2015

Jerome Quintin Quick review of the ADM formalism in General Relativity Foliation of

Let us have a spacetime (M ,g). Let Σ ⊂ M be a spacelike 3-dimensional hypersurface. Let nµ be a future-pointing µ (timelike) unit normal vector, so n nµ = −1. Then, the induced metric is

γµν = gµν + nµ nν . (1)

γµν is purely spatial, i.e., it resides entirely in Σ with no piece along nµ :

µ µ µ n γµν = n gµν + n nµ nν = nν − nν = 0. (2)

Jerome Quintin Quick review of the ADM formalism in General Relativity Foliation of spacetime

Let the hypersurface (Σ,γ) have a D such that Dµ γντ = 0. For instance, for some scalar f ,

ν Dµ f ≡ γµ ∇ν f , (3)

µ and for some tensor F ν ,

ν χ ν σ ξ Dµ F τ ≡ γµ γξ γτ ∇χ T σ . (4)

Here, ∇µ 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 γ = −γ τ γ σ ∇ n = −D n , (5) µν 2 nˆ µν µ ν τ σ ν µ where L is the Lie derivative. The trace is often called the mean curvature: µν K ≡ γ Kµν . (6)

Jerome Quintin Quick review of the ADM formalism in General Relativity Constraint equations

The 3- and 4-dimensional Riemann tensors on (Σ,γ) 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 = 8πGT , (7) µν 2 µν µν give the following Hamiltonian constraint:

(3) 2 µν R + K − Kµν K = 16πGρ, (8) µ ν where ρ ≡ Tµν 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:

ν Dν K µ − Dµ K = 8πGJµ , (9) τ ν where Jµ ≡ −Tντ n γ µ .

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:

tµ = Nnµ + β µ . (10) We introduced the laspe function,

µ N = −nµ t , (11) and the shift vector,

µ µ ν β = γ ν 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) τ LtKµν = −Dµ Dν N + N( R − 2Kµτ K ν + Kµν K)

1 −8πGN[S − γ (S − ρ)] + L K , (13) µν 2 µν β µν τ σ µν where Sµν ≡ γ µ γ ν Tτσ and S ≡ γ Sµν , and the evolution equation for the induced metric is

Ltγµν = −2NKµν + Lβ γµν . (14)

Jerome Quintin Quick review of the ADM formalism in General Relativity Choosing basis vectors

Now, let us choose tµ = (1,0,0,0). (15)

Then, Lt = ∂t . Also,

β µ = (0,β i ), (16) and

 1 β i  nµ = ,− . (17) N N Furthermore, the line element becomes

2 µ ν 2 2 i i j j ds = gµν dx dx = −N dt + γij (dx + β dt)(dx + β 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 = 16πGρ, (19)

ij ij i Dj (K − γ K) = 8πJ , (20)

∂t γij = −2NKij + Di βj + Dj βi , (21)

1 ∂ K = N((3)R −2K K k +K K)−D D N −8πGN[S − γ (S −ρ)] t ij ij ik j ij i j ij 2 ij

k k k +β ∂k Kij + Kik ∂j β + Kkj ∂i β , (22) µ ν i µ ij µ ν where ρ = Tµν n n , J = −Tµj n γ , Sij = Tµν γi γj , and ij S = γ Sij . Jerome Quintin Quick review of the ADM formalism in General Relativity Example: flat FRW

i µ µ 2 Choose N = 1, β = 0 (so n = t ), and take γij = a (t)δij , i.e.

2 2 2 i j ds = −dt + a (t)δij dx dx , (23)

Then, homogeneity and isotropy imply that Ji = 0 and 1 Kij = − 3 γij K, so Eq. (20) is satisfied. Furthermore, K = −3a˙/a, so Eq. (21) is satisfied, and Eq. (19) implies that a˙ 2 8πG = ρ. (24) a 3 Similarly, Eq. (22) gives

¨a 4πG = − (ρ + 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 uµ corresponding to the 4-velocity of observers comoving with the fluid. The gradient of uµ can be decomposed as 1 ∇ u = −u a + Θγ + σ + ω , (26) µ ν µ ν 3 µν µν µν µ where aµ is the acceleration, the trace Θ ≡ ∇µ u is the expansion rate, the traceless symmetric part σµν is the shear tensor, and the antisymmetric part ωµν is the vorticity tensor. From here on, we assume ωµν = 0.

E.g., in FRW, Θ = 3H, σµν = ωµν = 0.

Jerome Quintin Quick review of the ADM formalism in General Relativity Fluid picture

Using uµ to foliate the spacetime (i.e. taking nµ = uµ ), the extrinsic curvature becomes 1 K = −σ − Θγ . (27) µν µν 3 µν The Hamiltonian and momentum constraints become 2 (3)R − 2σ 2 + Θ2 = 16πGρ, (28) 3 2 D σ ν = D Θ, (29) ν µ 3 µ 2 1 µ ν where σ ≡ 2 σ ν σ µ 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 β µ = 0, but keeping N general)  1  ∂ γ = 2N σ + Θγ , (30) t µν µν 3 µν

(3) 2 µ ∂t Θ = −N[ R + Θ − 12πG(ρ − p)] + Dµ D N, (31)

 1  1 ∂ σ µ = −NΘσ µ −N (3)R µ − γ µ (3)R +Dµ D N − γ µ Dτ D N. t ν ν ν 3 ν ν 3 ν τ (32) Eqs. (28) and (31) lead to Raychaudhury’s equation: 1 1 Θ˙ + Θ2 = −2σ 2 − 4πG(ρ + 3p) + Dµ D N. (33) 3 N µ ˙ µ Here, Θ ≡ u ∇µ Θ, derivative with respect to the 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– tensor conservation

For a perfect fluid with Tµν = (ρ + p)uµ uν + pgµν ,

µν uν ∇µ T = 0 =⇒ ρ˙ + Θ(ρ + p) = 0, (34) the local energy conservation law, and

D p γ ∇ T µν = 0 =⇒ a = − α . (35) να µ α ρ + p

Jerome Quintin Quick review of the ADM formalism in General Relativity