6.1 The harmonic formulation

Here, we discuss the IBVP formulated in [264Jump To The Next Citation Point] for the Einstein vacuum equations in generalized harmonic coordinates. The starting point is a manifold of the form M = [0,T] × Σ, with Σ a three-dimensional compact manifold with ∞ C-boundary ∂Σ, and a given, fixed smooth background metric ˚gαβ with corresponding Levi-Civita connection ∇˚, as in Section 4.1. We assume that the time slices Σt := {t} × Σ are space-like and that the boundary surface 𝒯 := [0,T ] × ∂Σ is time-like with respect to ˚g αβ.

In order to formulate the boundary conditions, we first construct a null tetrad μ μ μ ¯μ {K ,L ,Q ,Q }, which is adapted to the boundary. This null tetrad is based on the choice of a future-directed time-like vector field Tμ tangent to 𝒯, which is normalized such that g T μT ν = − 1 μν. One possible choice is to tie μ T to the foliation Σt, and then define it in the direction orthogonal to the cross sections ∂ Σt = {t} × ∂Σ of the boundary surface. A more geometric choice has been proposed in [186Jump To The Next Citation Point], where instead T μ is chosen as a distinguished future-directed time-like eigenvector of the second fundamental form of 𝒯, as embedded in (M, g). Next, we denote by N μ the unit outward normal to 𝒯 with respect to the metric gμν and complete μ T and μ N to an orthonormal basis μ μ μ μ {T ,N ,V ,W } of TpM at each point p ∈ 𝒯. Then, we define the complex null tetrad by

μ μ μ μ μ μ μ μ μ μ μ μ K := T + N , L := T − N , Q := V + iW , ¯Q := V − iW , (6.1 )
where i = √ −-1. Notice that the construction of these vectors is implicit, since it depends on the dynamical metric gαβ, which is yet unknown. However, the dependency is algebraic, and does not involve any derivatives of gαβ. We also note that the complex null vector μ Q is not unique since it can be rotated by an angle φ ∈ ℝ, Qμ ↦→ eiφQμ. Finally, we define a radial function r on 𝒯 as the areal radius of the cross sections ∂Σt with respect to the background metric.

Then, the boundary conditions, which were proposed in [264Jump To The Next Citation Point] for the harmonic system (4.5View Equation), are:

| ˚ 2 | ∇ K hKK + r-hKK || = qK , (6.2 ) |𝒯 ˚ 1- || ∇ KhKL + r (hKL + hQ ¯Q)| = qL, (6.3 ) |𝒯 ∇˚ h + 2-h || = q , (6.4 ) K KQ r KQ |𝒯 Q ˚ ˚ || ∇ KhQQ − ∇ QhQK | = qQQ, (6.5 ) ||𝒯 ∇˚K hQ ¯Q + ∇˚LhKK − ∇˚QhK ¯Q − ∇˚Q¯hKQ | = 2HK |𝒯 , (6.6 ) |𝒯 ∇˚K hLQ + ∇˚LhKQ − ∇˚QhKL − ∇˚ ¯QhQQ || = 2HQ | , (6.7 ) |𝒯 𝒯 ∇˚ h + ∇˚ h ¯ − ∇˚ h ¯− ∇˚ ¯h || = 2H | , (6.8 ) K LL L QQ Q LQ Q LQ 𝒯 L 𝒯
where ∇˚K hLQ := K μL αQ β∇˚ μhαβ, hKL := K αLβh αβ, HK := K μH μ, etc., and where qK and qL are real-valued given smooth functions on 𝒯 and qQ and qQQ are complex-valued given smooth functions on 𝒯. Since Q is complex, these constitute ten real boundary conditions for the metric coefficients hαβ. The content of the boundary conditions (6.2View Equation, 6.3View Equation, 6.4View Equation, 6.5View Equation) can be clarified by considering linearized gravitational waves on a Minkowski background with a spherical boundary. The analysis in [264Jump To The Next Citation Point] shows that in this context the four real conditions (6.2View Equation),(6.3View Equation, 6.4View Equation) are related to the gauge freedom; and the two conditions (6.5View Equation) control the gravitational radiation. The remaining conditions (6.6View Equation, 6.7View Equation, 6.8View Equation) enforce the constraint μ C = 0 on the boundary, see Eq. (4.6View Equation), and so together with the constraint propagation system (4.14View Equation) and the initial constraints (4.15View Equation) they guarantee that the constraints are correctly propagated. Based on these observations, it is expected that these boundary conditions yield small spurious reflections in the case of a nearly-spherical boundary in the wave zone of an asymptotically-flat curved spacetime.

6.1.1 Well-posedness of the IBVP

The IBVP consisting of the harmonic Einstein equations (4.5View Equation), initial data (4.7View Equation) and the boundary conditions (6.2View Equation6.8View Equation) can be shown to be well posed as an application of Theorem 8. For this, we first notice that the evolution equations (4.5View Equation) have the required form of Eq. (5.113View Equation), where E is the vector bundle of symmetric, covariant tensor fields hμν on M. Next, the boundary conditions can be written in the form of Eq. (5.115View Equation) with α = 1. In order to compute the matrix coefficients cμAB, it is convenient to decompose hμν = hAeA μν in terms of the basis vectors

¯ ¯ e1αβ := K αK β, e2αβ := − 2K (α Qβ), e3αβ := − 2K (αQ β), e4αβ := 2Q (αQ β), e5αβ := Q¯αQ¯β, e6αβ := Q αQβ, ¯ e7αβ := − 2L (αQ β), e8αβ := − 2L (αQ β), e9αβ := 2K (αL β), e10αβ := LαL β,
with h1 = hLL∕4, h2 = ¯h3 = hLQ ∕4, h4 = hQ ¯Q∕4, h5 = ¯h6 = hQQ ∕4, h7 = ¯h8 = hKQ ∕4, h9 = hKL ∕4, h10 = hKK ∕4. With respect to this basis, the only nonzero matrix coefficients are
μ1 ¯μ μ1 μ μ1 μ c 2 = Q , c 3 = Q , c 4 = − L , cμ25 = ¯Qμ, cμ27 = − L μ, cμ29 = Q μ, μ3 μ μ3 μ μ3 ¯ μ c 6 = Q , c 8 = − L , c 9 = Q , cμ47 = ¯Qμ, cμ48 = Qμ, cμ410 = − L μ, cμ5 = Qμ, cμ6 = ¯Qμ, 7 8
which has the required upper triangular form with zeros in the diagonal. Therefore, the hypothesis of Theorem 8 are verified and one obtains a well-posed IBVP for Einstein’s equations in harmonic coordinates.

This result also applies the the modified system (4.16View Equation), since the constraint damping terms, which are added, do not modify the principal part of the main evolution system nor the one of the constraint propagation system.


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