4.2 The ADM formulation

In the usual 3+1 decomposition of Einstein’s field equations (see, for example, [214], for a through discussion of it) one evolves the three metric and the extrinsic curvature (the first and second fundamental forms) relative to a foliation Σt of spacetime by spacelike hypersurfaces. The motivation for this formulation stems from the Hamiltonian description of general relativity (see, for instance, Appendix E in [429]) where the “q” variables are the three metric γij and the associated canonical momenta πij (the “p” variables) are related to the extrinsic curvature Kij according to
ij √ -( ij ij ) π = − γ K − γ K , (4.19 )
where γ = det(γij) denotes the determinant of the three-metric and K = γijKij the trace of the extrinsic curvature.

In York’s formulation [444] of the 3+1 decomposed Einstein equations, the evolution equations are

∂0γij = − 2Kij, (4.20 ) 1 [ 1 ] ∂0Kij = R(i3j)− --DiDj α + KKij − 2KilKlj − 8πGN σij + -γij(ρ − σ) . (4.21 ) α 2
Here, the operator ∂ 0 is defined as ∂ := α −1(∂ − £ ) 0 t β with α and βi denoting lapse and shift, respectively. It is equal to the Lie derivative along the future-directed unit normal n to the time slices when acting on covariant tensor fields orthogonal to n. Next, (3) Rij and Dj are the Ricci tensor and covariant derivative operator belonging to the three metric γij, and ρ := nαnβT αβ and σij := Tij are the energy density and the stress tensor as measured by observers moving along the future-directed unit normal n to the time slices. Finally, ij σ := γ Tij denotes the trace of the stress tensor. The evolution system (4.20View Equation, 4.21View Equation) is subject to the Hamiltonian and momentum constraints,
( ) H := 1- γijR (3) + K2 − KijKij = 8πGN ρ, (4.22 ) 2 ij Mi := DjKij − DiK = 8πGN ji, (4.23 )
where β ji := − n Tiβ is the flux density.

4.2.1 Algebraic gauge conditions

One issue with the evolution equations (4.20View Equation, 4.21View Equation) is the principle part of the Ricci tensor belonging to the three-metric,

(3) 1 kl Rij = 2γ (− ∂k∂lγij − ∂i∂jγkl + ∂i∂kγlj + ∂j∂kγli) + l.o., (4.24 )
which does not define a positive-definite operator. This is due to the fact that the linearized Ricci tensor is invariant with respect to infinitesimal coordinate transformations γ ↦→ γ + 2∂ ξ ij ij (i j) generated by a vector field i ξ = ξ ∂i. This has the following implications for the evolution equations (4.20View Equation, 4.21View Equation), assuming for the moment that lapse and shift are fixed, a priori specified functions, in which case the system is equivalent to the second-order system ∂20γij = − 2R(i3j)+ l.o. for the three metric. Linearizing and localizing as described in Section 3 one obtains a linear, constant coefficient problem of the form (3.56View Equation), which can be brought into first-order form via the reduction in Fourier space described in Section 3.1.5. The resulting first-order system has the form of Eq. (3.58View Equation) with the symbol
n ( ) ∑ ˚jˆ 0 I Q(ik) = i|k | β kj + |k| − α˚2R (ˆk) 0 , (4.25 ) j=1
where R(k ) is, up to a factor 2, the principal symbol of the Ricci operator,
( ) R(ˆk )γ = γ˚lm ˆkˆk γ + ˆk ˆk γ − ˆk ˆk γ − ˆk ˆk γ . (4.26 ) ij l m ij ij lm i l mj j l mi
Here, α˚, β˚i and γ˚ ij refer to the frozen lapse, shift and three-metric, respectively. According to Theorem 2, the problem is well posed if and only there is a uniformly positive and bounded symmetrizer h (ˆk ) such that h (ˆk)R(ˆk ) is symmetric and uniformly positive for 2 ˆk ∈ S. Although R(ˆk ) is diagonalizable and its eigenvalues are not negative, some of them are zero since R (ˆk)γij = 0 for γij of the form γij = 2ˆk (iξj) with an arbitrary one-form ξj, so h(k)R (k) cannot be positive.

These arguments were used in [308Jump To The Next Citation Point] to show that the evolution system (4.20View Equation, 4.21View Equation) with fixed lapse and shift is weakly but not strongly hyperbolic. The results in [308Jump To The Next Citation Point] also analyze modifications of the equations for which the lapse is densitized and the Hamiltonian constraint is used to modify the trace of Eq. (4.21View Equation). The conclusion is that such changes cannot make the evolution equations (4.20View Equation, 4.21View Equation) strongly hyperbolic. Therefore, these equations, with given shift and densitized lapse, are not suited for numerical evolutions.14

4.2.2 Dynamical gauge conditions leading to a well-posed formulation

The results obtained so far often lead to the popular statement “The ADM equations are not strongly hyperbolic.” However, consider the possibility of determining the lapse and shift through evolution equations. A natural choice, motivated by the discussion in Section 4.1, is to impose the harmonic gauge constraint (4.3View Equation). Assuming that the background metric ˚gαβ is Minkowski in Cartesian coordinates for simplicity, this yields the following equations for the 3+1 decomposed variables,

(∂t − βj∂j )α = − α2fK + α3Ht, (4.27 ) ( ) (∂t − βj∂j)βi = − αγij∂jα + α2γijγkl ∂kγjl − 1-∂jγkl + α2(Hi + βiHt), (4.28 ) 2
with f a constant, which is equal to one for the harmonic time coordinate t. Let us analyze the hyperbolicity of the evolution system (4.27View Equation, 4.28View Equation, 4.20View Equation, 4.21View Equation) for the fields i u = (α, β ,γij,Kij), where for generality and later use, we do not necessarily assume f = 1 in Eq. (4.27View Equation). Since this is a mixed first/second-order system, we base our analysis on the first-order pseudodifferential reduction discussed in Section 3.1.5. After linearizing and localizing, we obtain the constant coefficient linear problem
˚k ˚2 (∂t − β ∂k )α = − α fK, ( ) (4.29 ) ˚k i ˚ ˚ij ˚2 ˚ij ˚kl 1 (∂t − β ∂k)β = − α γ ∂jα + α γ γ ∂kγjl − -∂jγkl , (4.30 ) 2 (∂t − β˚k∂k )γij = 2γ˚k (i∂j)βk − 2α˚Kij, (4.31 ) ˚k α˚-˚kl (∂t − β ∂k)Kij = − ∂i∂jα + 2 γ (− ∂k∂lγij − ∂i∂jγkl + ∂i∂kγlj + ∂j∂k γli), (4.32 )
where ˚ α, ˚k β and ˚ γ ij refer to the quantities corresponding to α, k β, γij of the background metric when frozen at a given point. In order to rewrite this in first-order form, we perform a Fourier transformation in space and introduce the variables ˆU = (a, bi,lij,pij) with
˚ ˚ j ˚ a := |k|αˆ∕α , bi := |k |γ ij ˆβ ∕α , lij := |k|ˆγij, pij := 2i ˆKij, (4.33 )
where ∘ -------- ˚ij |k | := γ kikj and the hatted quantities refer to their Fourier transform. With this, we obtain the first-order system ˆUt = P (ik)ˆU where the symbol has the form P (ik) = iβ˚sksI + α˚Q (ik) with
( ) ( a ) fp | | | ˆ 2ˆj 1ˆ | Q (ik)| bi| = i|k||| − kia + k lij − 2kil || , (4.34 ) ( lij) ( 2ˆk(ibj) + pij ) pij 2 ˆkiˆkja + lij + ˆkiˆkjl − 2 ˆksˆk(ilj)s
where ˆki := ki∕|k|, ˆki := γ˚ijˆkj, l := γ˚ijlij, and p := γ˚ijpij. In order to determine the eigenfields S (k)−1ˆU such that S(k)−1P (ik)S(k) is diagonal, we decompose
bi = ¯bˆki + ¯bi, ¯ˆ ˆ ˆ ¯ ˆ 1-˚ ˆˆ ¯′ ˆˆ ˆ 1-˚ ˆ ˆ ′ lij = lkikj + 2k(ilj) + lij + 2 (γ ij − kikj)l, pij = p¯kikj + 2k(i¯pj) + ˆpij + 2(γ ij − kikj)¯p
into pieces parallel and orthogonal to ˆ ki, similar to Example 15. Then, the problem decouples into a tensor sector, involving (ˆlij,pˆij), into a vector sector, involving (¯bi,¯li, ¯pi) and a scalar sector involving (a,¯b,¯l, ¯p,¯l′,p¯′). In the tensor sector, we have
( ˆl ) ( pˆ ) Q(tensor)(ik) ij = i|k| ˆij , (4.35 ) ˆpij lij
which has the eigenvalues ±i |k| with corresponding eigenfields ˆlij ± ˆpij. In the vector sector, we have
( ¯ ) ( ¯ ) bj lj Q (vector)(ik )( ¯lj ) = i|k|( ¯bj + ¯pj) , (4.36 ) ¯pj 0
which is also diagonalizable with eigenvalues 0, ±i |k| and corresponding eigenfields ¯pj and ¯lj ± (¯bj + ¯pj). Finally, in the scalar sector we have
( ) ( f ′ ) | a¯ | | 2(¯p +1 p¯¯)¯′ | | b | | − a + 2(l − l )| (scalar) || ¯l || || 2¯b + ¯p || Q (ik) || ¯p || = i|k ||| 2a + ¯l′ || . (4.37 ) ( ¯l′) ( ¯p′ ) ′ ¯′ ¯p l
It turns out Q(scalar)(ik ) is diagonalizable with purely imaginary values if and only if f > 0 and f ⁄= 1. In this case, the eigenvalues and corresponding eigenfields are ±i |k |, ±i|k|, ±i √f-|k| and ¯′ ′ l ± ¯p, ¯ ¯ l ± (2b + ¯p), ¯′ √ -- ′ a + f l∕(f − 1) ± f[¯p + (f + 1 )∕ (f − 1)¯p]∕2, respectively. A symmetrizer for P (ik), which is smooth in k ∈ S2, α˚, β˚k and γ˚ij, can be constructed from the eigenfields as in Example 15.

Remarks:

We conclude that the evolution system (4.27View Equation, 4.28View Equation, 4.20View Equation, 4.21View Equation) is strongly hyperbolic if and only if f > 0 and f ⁄= 1. Although the full harmonic gauge condition (4.3View Equation) is excluded from these restrictions,15 there is still a large family of evolution equations for the lapse and shift that give rise to a strongly hyperbolic problem together with the standard evolution equations (4.20View Equation, 4.21View Equation) from the 3+1 decomposition.

4.2.3 Elliptic gauge conditions leading to a well-posed formulation

Rather than fixing the lapse and shift algebraically or dynamically, an alternative, which has been considered in the literature, is to fix them according to elliptic equations. A natural restriction on the extrinsic geometry of the time slices Σt is to require that their mean curvature, c = − K ∕3, vanishes or is constant [391Jump To The Next Citation Point]. Taking the trace of Eq. (4.21View Equation) and using the Hamiltonian constraint to eliminate the trace of (3) R ij yields the following equation for the lapse,

[− DjD + KijK + 4πG (ρ + σ)] α = ∂ K, (4.39 ) j ij N t
which is a second-order linear elliptic equation. The operator inside the square parenthesis is formally positive if the strong energy condition, ρ + σ ≥ 0, holds, and so it is invertible when defined on appropriate function spaces. See also [203Jump To The Next Citation Point] for generalizations of this condition. Concerning the shift, one choice, which is motivated by eliminating the “bad” terms in the expression for the Ricci tensor, Eq. (4.24View Equation), is the spatial harmonic gauge [25Jump To The Next Citation Point]. In terms of a fixed (possibly time-dependent) background metric γ˚ij on Σt, this gauge is defined as (cf. Eq. (4.3View Equation))
( ) ( ) k ij k ˚k ij kl ˚ 1-˚ 0 = V := γ Γ ij − Γ ij = γ γ Dk γlj − 2 Djγkl , (4.40 )
where ˚D is the Levi-Civita connection with respect to ˚ γ and ˚k Γ ij denote the corresponding Christoffel symbols. The main importance of this gauge is that it permits one to rewrite the Ricci tensor belonging to the three metric in the form
(3) 1- kl˚ ˚ Rij = − 2 γ DkDl γij + D (iVj) + l.o., (4.41 )
where ˚ Dk denotes the covariant derivative with respect to the background metric ˚ γ and where the lower-order terms “l.o.” depend only on γij and its first derivatives ˚Dk γij. When Vk = 0 the operator on the right-hand side is second-order quasilinear elliptic, and with this, the evolution system (4.20View Equation, 4.21View Equation) has the form of a nonlinear wave equation for the three-metric γij. However, the coefficients and source terms in this equation still depend on the lapse and shift. For constant mean curvature slices the lapse satisfies the elliptic scalar equation (4.39View Equation), and with the spatial harmonic gauge the shift is determined by the requirement that Eq. (4.40View Equation) is preserved throughout evolution, which yields an elliptic vector equation for it. In [25Jump To The Next Citation Point] it was shown that the coupled hyperbolic-elliptic system consisting of the evolution equations (4.20View Equation, 4.21View Equation) with the Ricci tensor R (3) ij rewritten in elliptic form using the condition k V = 0, the constant mean curvature condition (4.39View Equation), and this elliptic equation for i β, gives rise to a well-posed Cauchy problem in vacuum. Besides eliminating the “bad” terms in the Ricci tensor, the spatial harmonic gauge also has other nice properties, which were exploited in the well-posed formulation of [25Jump To The Next Citation Point]. For example, the covariant Laplacian of a function f is
DkDkf = γij˚Di˚Djf − V k˚Dkf, (4.42 )
which does not contain any derivatives of the three metric ij γ if k V = 0. For applications of the hyperbolic-elliptic formulation in [25] to the global existence of expanding vacuum cosmologies; see [26, 27].

Other methods for specifying the shift have been proposed in [391], with the idea of minimizing a functional of the type

∫ √ -- I [β ] = ΘijΘij γd3x, (4.43 ) Σt
where Θij := ∂tγij∕2 = − αKij + D (iβj) is the strain tensor. Therefore, the functional I[β] minimizes time changes in the three metric in an averaged sense. In particular, I[β ] attains its absolute minimum (zero) if ∂t is a Killing vector field. Therefore, one expects the resulting gauge condition to minimize the time dependence of the coordinate components of the three metric. An alternative is to replace the strain by its trace-free part on the right-hand side of Eq. (4.43View Equation), giving rise to the minimal distortion gauge. Both conditions yield a second-order elliptic equation for the shift vector, which has unique solutions provided suitable boundary conditions are specified. For generalizations and further results on these type of gauge conditions; see [73, 203, 204]. However, it seems to be currently unknown whether or not these elliptic shift conditions, together with the evolution system (4.20View Equation, 4.21View Equation) and an appropriate condition on the lapse, lead to a well-posed Cauchy problem.

4.2.4 Constraint propagation

The evolution equations (4.20View Equation, 4.21View Equation) are equivalent to the components of the Einstein equations corresponding to the spatial part of the Ricci tensor,

( ) 1- μν Rij = 8πGN Tij − 2 γijg T μν , (4.44 )
and in order to obtain a solution of the full Einstein equations one also needs to solve the constraints H = 8πGN ρ and Mi = 8 πGN ji. As in Section 4.2.3, the constraint propagation system can be obtained from the twice contracted Bianchi identities, which, in the 3+1 decomposition, read
∂ H + 1-Dj (α2M ) − 2KH − (Kij − K γij)R = 0, (4.45 ) 0 α2 j ij 1 ( 2 ) 1 j ( kl ) ∂0Mi + α2-Di α H − KMi + α-D αRij − α γijγ Rkl = 0. (4.46 )
The condition of the stress-energy tensor being divergence-free leads to similar evolution equations for ρ and ji. Therefore, the equations (4.44View Equation) lead to the following symmetric hyperbolic system [190Jump To The Next Citation Point, 445] for the constraint variables ℋ := H − 8πGN ρ and ℳi := Mi − 8πGN ji,
1 ( ) ∂0ℋ = − -2Dj α2 ℳj + 2K ℋ, (4.47 ) α ( ) ∂0ℳi = − 1-Di α2ℋ + K ℳi. (4.48 ) α2
As has also been observed in [190], the constraint propagation system associated with the standard ADM equations, where Eq. (4.44View Equation) is replaced by its trace-reversed version R − γ gμνR ∕2 = 8πG T ij ij μν N ij is
-1- j ( 2 ) ∂0ℋ = − α2 D α ℳj + K ℋ, D α ∂0ℳi = − --i-ℋ + K ℳi, α
which is only weakly hyperbolic. Therefore, it is much more difficult to control the constraint fields in the standard ADM case than in York’s formulation of the 3+1 equations.
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