List of Footnotes

1 More precisely, it follows from the Paley–Wiener theorem (see Theorem IX.11 in [346]) that ω f ∈ 𝒮 with ˆ f(k) = 0 for |k| ≥ R if and only if f possesses an analytic extension ¯ n f : ℂ → ℂ such that for each N = 0,1,2,... there exists a constant CN with
R |Im(ζ)| |f¯(ζ)| ≤ CN-e-------, ζ ∈ ℂn. (1+ |ζ|)N
2 In this regard we should mention the Cauchy–Kovalevskaya theorem (see, for example, [161]), which always provides unique local analytic solutions to the Cauchy problem for quasilinear partial differential equations with analytic coefficients and data on a non-characteristic surface. However, this theorem does not say anything about causal propagation and stability with respect to high-frequency perturbations.
3 In fact, we will see in Section 3.2 that in the variable coefficient case, smoothness of the symmetrizer in k′ is required.
4 Here, the factor 2 m could, in principle, be replaced by any positive number, which shows that the symmetrizer is not always unique. The choice here is such that the expression ∗ u Hu is proportional to the physical energy density of the system. Notice, however, that for the massless case m = 0 one must replace 2 m = 0 by a positive constant in order for the symmetrizer to be positive definite.
5 Notice that ¯Ei has only two degrees of freedom since it is orthogonal to k. Likewise, the quantities ¯Wi, V¯i have two degrees of freedom and ¯Wij has three since it its orthogonal to k and trace-free.
6 Here, the advection term ∑n Bj∂∂xjv j=1 is subtracted from vt for convenience only.
7 The Fourier transform of the function v(t,⋅) = P (t,⋅,∂∕∂x)u(t,⋅) is formally given by
⌊ ⌋ 1 ∑n ˆv(t,⋅) = ---n∕2⌈ ˆAj(t,⋅)∗ikjˆu(t,⋅)+ ˆB(t,⋅)∗ ˆu(t,⋅)⌉ , (2π) j=1
where Aˆj(t,⋅), ˆB (t,⋅) denote the Fourier transform of Aj (t,⋅) and B (t,⋅), respectively, and where the star denotes the convolution operator. Unless Aj and B are independent of x, the different k-modes couple to each other.
8 These smoothness requirements are sometimes omitted in the numerical-relativity literature.
9 In principle, the maximum propagation speed v(t0) could be infinite. Finiteness can be guaranteed, for instance, by requiring the principal symbol P0(t,x,s) to be independent of (t,x) for |x| > R outside a large ball of radius R > 0.
10 See Theorem 4.1.3 in [327Jump To The Next Citation Point].
11 Geometrically, this means that the identity map ϕ : (M,g) → (M,˚g), p ↦→ p, satisfies the inhomogeneous harmonic wave map equation
μ D ˚D ∂ϕA-∂ϕB- μν D ∇ ∇ μϕ + Γ AB ∂xμ ∂xν g = H , (4.2)
where xμ and xA are local coordinates on (M, g) and (M,˚g), respectively.
12 As indicated above, given initial data (0) gαβ and (0) kαβ it is always possible to adjust the background metric ˚gαβ such that the initial data for hαβ is trivial; just replace ˚gαβ(t,x) by ˚gαβ(t,x)+ h(α0β)(x)+ tk(0α)β(x).
13 Notice that the condition of Tαβ being divergence-free depends on the metric gαβ itself, which is not known before actually solving the nonlinear wave equation (4.5View Equation), and the latter, in turn, depends on T αβ. Therefore, one cannot specify Tαβ by hand, except in the vacuum case, Tαβ = 0, or in the case Tαβ = − Λg αβ with Λ the cosmological constant. In the more general case, the stress-energy tensor has to be computed from a diffeomorphism-independent action for the matter fields and one has to consistently solve the coupled Einstein-matter system.
14 Weak hyperbolicity of the system (4.20View Equation, 4.21View Equation) with given shift and densitized lapse has also been pointed out in [254Jump To The Next Citation Point] based on a reduction, which is first order in time and space. However, there are several inequivalent such reductions, and so it is not sufficient to show that a particular one is weakly hyperbolic in order to infer that the second-order-in-space system (4.20View Equation, 4.21View Equation) is weakly hyperbolic.
15 Notice that even when f = 1, the evolution system (4.27View Equation, 4.28View Equation, 4.20View Equation, 4.21View Equation) is not equivalent to the harmonic system discussed in Section 4.1. In the former case, the harmonic constraint is exactly enforced in order to obtain evolution equations for the lapse and shift; while in the latter case first derivatives of the harmonic constraint are combined with the Hamiltonian and momentum constraints; see Eq. (4.13View Equation).
16 If γ˚ ij is time-independent but not flat, additional curvature terms appear in the equations; see Appendix A in [82Jump To The Next Citation Point].
17 However, one should mention that this convergence is usually not sufficient for obtaining accurate solutions. If the constraint manifold is unstable, small departures from it may grow exponentially in time and even though the constraint errors converge to zero they remain large for finite resolutions; see [254Jump To The Next Citation Point] for an example of this effect.
18 However, it should be noted that these solutions are not square integrable, due to their harmonic dependency in y. This problem can be remedied by truncation of uα; see Section 3.2 in [241Jump To The Next Citation Point].
19 Alternatively, if k = − ∂x is taken to be the unit outward normal, then the incoming normal characteristic fields are the ones with positive characteristic speeds with respect to k. This more geometrical definition will be the one taken in Section 5.2.
20 ˆk is not well defined if k = 0; however, in this case the scalar block comprising (&tidle;E1, &tidle;U1) decouples from the vector block comprising (&tidle;EA, &tidle;UA ) and it is simple to verify that the resulting system does not possess nontrivial simple wave solutions.
21 One can always assume that u(0,x) = ut(0,x) = 0 for all x ∈ Σ by a suitable redefinition of u, F and g.
22 The restriction to homogeneous boundary conditions g = 0 in the IBVP (5.1View Equation, 5.2View Equation, 5.3View Equation) is not severe, since it can always be achieved by redefining u, f and F.
23 Instead of imposing the constraint itself on the boundary one might try to set some linear combination of its normal and time derivatives to zero, obtaining a constraint-preserving boundary condition that does not involve zero speed fields. Unfortunately, this trick only seems to work for reflecting boundaries; see [405Jump To The Next Citation Point] and [106Jump To The Next Citation Point] for the case of general relativity. In our example, such boundary conditions are given by ∂tAt = ∂xAx = ∂tAy = ∂tAz = 0, which imply ∂t(∂νA ν) = 0
24 This relation is explicitly given in terms of the Weyl tensor C, namely Ψ0 = C αβγδK αQ βKγQ δ = 2(Eαβ − 𝜀γδ(αsγBδβ))Q αQ β, where Eαβ = CαγβδTγT δ and Bαβ = ∗CαγβδTγTδ are the electric and magnetic parts of C with respect to the timelike vector field T = ∂t.
25 A systematic derivation of exact solutions including the correction terms in M ∕R of arbitrarily-high order in the Schwarzschild metric was given in [41] in a different context. However, a generic asymptotically-flat spacetime is not expected to be Schwarzschild beyond the order of M ∕R, since higher-correction terms involve, for example, the total angular momentum. See [135Jump To The Next Citation Point] for the behavior of linearized gravitational perturbations of the Kerr spacetime near future null infinity.
26 Notice that this is one of the requirements of Theorem 7, for instance, where the boundary matrix must have constant rank in order for the theorem to be applicable.
27 See, for instance, Example 6 with initial data f ∈ 𝒮 ω.
28 This refers to the truncation error at a fixed final time, as opposed to the local one after an iteration.
29 There are cases in which this is not true, at least for not-too-large penalty strengths.