1  More precisely, it follows from the Paley–Wiener theorem (see Theorem IX.11 in [346]) that with for
if and only if possesses an analytic extension such that for each there exists a
constant with


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 noncharacteristic surface. However, this theorem does not say anything about causal propagation and stability with respect to highfrequency perturbations.  
3  In fact, we will see in Section 3.2 that in the variable coefficient case, smoothness of the symmetrizer in is required.  
4  Here, the factor 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 is proportional to the physical energy density of the system. Notice, however, that for the massless case one must replace by a positive constant in order for the symmetrizer to be positive definite.  
5  Notice that has only two degrees of freedom since it is orthogonal to . Likewise, the quantities , have two degrees of freedom and has three since it its orthogonal to and tracefree.  
6  Here, the advection term is subtracted from for convenience only.  
7  The Fourier transform of the function is formally given by


8  These smoothness requirements are sometimes omitted in the numericalrelativity literature.  
9  In principle, the maximum propagation speed could be infinite. Finiteness can be guaranteed, for instance, by requiring the principal symbol to be independent of for outside a large ball of radius .  
10  See Theorem 4.1.3 in [327].  
11  Geometrically, this means that the identity map , , satisfies the inhomogeneous harmonic wave map equation where and are local coordinates on and , respectively.  
12  As indicated above, given initial data and it is always possible to adjust the background metric such that the initial data for is trivial; just replace by .  
13  Notice that the condition of being divergencefree depends on the metric itself, which is not known before actually solving the nonlinear wave equation (4.5), and the latter, in turn, depends on . Therefore, one cannot specify by hand, except in the vacuum case, , or in the case with the cosmological constant. In the more general case, the stressenergy tensor has to be computed from a diffeomorphismindependent action for the matter fields and one has to consistently solve the coupled Einsteinmatter system.  
14  Weak hyperbolicity of the system (4.20, 4.21) with given shift and densitized lapse has also been pointed out in [254] 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 secondorderinspace system (4.20, 4.21) is weakly hyperbolic.  
15  Notice that even when , the evolution system (4.27, 4.28, 4.20, 4.21) 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. (4.13).  
16  If is timeindependent but not flat, additional curvature terms appear in the equations; see Appendix A in [82].  
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 [254] 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 . This problem can be remedied by truncation of ; see Section 3.2 in [241].  
19  Alternatively, if is taken to be the unit outward normal, then the incoming normal characteristic fields are the ones with positive characteristic speeds with respect to . This more geometrical definition will be the one taken in Section 5.2.  
20  is not well defined if ; however, in this case the scalar block comprising decouples from the vector block comprising and it is simple to verify that the resulting system does not possess nontrivial simple wave solutions.  
21  One can always assume that for all by a suitable redefinition of , and .  
22  The restriction to homogeneous boundary conditions in the IBVP (5.1, 5.2, 5.3) is not severe, since it can always be achieved by redefining , and .  
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 constraintpreserving boundary condition that does not involve zero speed fields. Unfortunately, this trick only seems to work for reflecting boundaries; see [405] and [106] for the case of general relativity. In our example, such boundary conditions are given by , which imply  
24  This relation is explicitly given in terms of the Weyl tensor , namely , where and are the electric and magnetic parts of with respect to the timelike vector field .  
25  A systematic derivation of exact solutions including the correction terms in of arbitrarilyhigh order in the Schwarzschild metric was given in [41] in a different context. However, a generic asymptoticallyflat spacetime is not expected to be Schwarzschild beyond the order of , since highercorrection terms involve, for example, the total angular momentum. See [135] 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 .  
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 nottoolarge penalty strengths. 
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Living Rev. Relativity 15, (2012), 9
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