6 Boundary Conditions for Einstein’s Equations

The subject of this section is the discussion of the IBVP for Einstein’s field equations. There are at least three difficulties when formulating Einstein’s equations on a finite domain with artificial outer boundaries. First, as we have seen in Section 4, the evolution equations are subject to constraints, which, in general, propagate with nontrivial characteristic speeds. As a consequence, in general there are incoming constraint fields at the boundary that need to be controlled in order to make sure that the constraints propagate correctly, i.e., that constraint-satisfying initial data yields a solution of the evolution equations and the constraints on the complete computational domain, and not just on its domain of dependence. The control of these incoming constraint fields leads to constraint-preserving boundary conditions, and a nontrivial task is to fit these conditions into one of the admissible boundary conditions discussed in the previous Section 5, for which well-posedness can be shown.

A second issue is the construction of absorbing boundary conditions. Unlike the simple examples considered in Section 5.3, for which the fields evolve on a fixed background and in- and outgoing solutions can be represented explicitly, or at least characterized precisely, in general relativity it is not even clear how to define in- and outgoing gravitational radiation since there are no local expressions for the gravitational energy density and flux. Therefore, the best one can hope for is to construct boundary conditions, which approximately control the incoming gravitational radiation in certain regimes, like, for example, in the weak field limit where the field equations can be linearized around, say, a Schwarzschild or Minkowski spacetime.

Finally, the third issue is related to the diffeomorphism invariance of the theory. Ideally, one would like to formulate a geometric version of the IBVP, for which the data given on the initial and boundary surfaces Σ0 and 𝒯 can be characterized in terms of geometric quantities such as the first and second fundamental forms of these surfaces as embedded in the yet unknown spacetime (M, g). In particular, this means that one should be able to identify equivalent data sets, i.e., those which are related to each other by a diffeomorphism of M, leaving Σ0 and 𝒯 invariant, by local transformations on Σ 0 and 𝒯, without knowing the solution (M, g). It is currently not even clear if such a geometric uniqueness property does exist; see [186Jump To The Next Citation Point, 355Jump To The Next Citation Point] for further discussions on these points.

A well-posed IBVP for Einstein’s vacuum field equations was first formulated by Friedrich and Nagy [187Jump To The Next Citation Point] based on a tetrad formalism, which incorporates the Weyl curvature tensor as an independent field. This formulation exploits the freedom of choosing local coordinates and the tetrad orientation in order to impose very precise gauge conditions, which are adapted to the boundary surface 𝒯 and tailored to the IBVP. These gauge conditions, together with a suitable modification of the evolution equations for the Weyl curvature tensor using the constraints (cf. Example 32), lead to a first-order symmetric hyperbolic system in which all the constraint fields propagate tangentially to 𝒯 at the boundary. As a consequence, no constraint-preserving boundary conditions need to be specified, and the only incoming fields are related to the gravitational radiation, at least in the context of the approximations mentioned above. With this, the problem can be shown to be well posed using the techniques described in Section 5.2.

After the pioneering work of [187Jump To The Next Citation Point], there has been much effort in formulating a well-posed IBVP for metric formulations of general relativity, on which most numerical calculations are based. However, with the exception of particular cases in spherical symmetry [249Jump To The Next Citation Point], the linearized field equations [309] or the restriction to flat, totally reflecting boundaries [404Jump To The Next Citation Point, 405Jump To The Next Citation Point, 106, 98Jump To The Next Citation Point, 219, 220Jump To The Next Citation Point, 410, 29, 15Jump To The Next Citation Point], not much progress had been made towards obtaining a manifestly well-posed IBVP with nonreflecting, constraint-preserving boundary conditions. The difficulties encountered were similar to those described in Examples 31 and 32. Namely, controlling the incoming constraint fields usually resulted in boundary conditions for the main system involving either derivatives of its characteristic fields or fields propagating with zero speed, when it was written in first-order symmetric hyperbolic form. Therefore, the theory of maximal dissipative boundary conditions could not be applied in these attempts. Instead, boundary conditions controlling the incoming characteristic constraint fields were specified and combined with more or less ad hoc conditions controlling the gauge and gravitational degrees of freedom and verified to satisfy the Lopatinsky condition (5.27View Equation) using the Laplace method; see [395, 108, 378Jump To The Next Citation Point, 220Jump To The Next Citation Point, 363Jump To The Next Citation Point, 368Jump To The Next Citation Point].

The breakthrough in the metric case came with the work by Kreiss and Winicour [267Jump To The Next Citation Point] who formulated a well-posed IBVP for the linearized Einstein vacuum field equations with harmonic coordinates. Their method is based on the pseudo-differential first-order reduction of the wave equation described in Section 5.1.3, which, when combined with Sommerfeld boundary conditions, yields a problem, which is strongly well posed in the generalized sense and, when applied to systems of equations, allows a certain hierarchical coupling in the boundary conditions. This work was then generalized to shifted wave equations and higher-order absorbing boundary conditions in [369Jump To The Next Citation Point]. Later, it was recognized that the results in [267Jump To The Next Citation Point] could also be established based on the usual a priori energy estimates based on integration by parts [263Jump To The Next Citation Point]. Finally, it was found that the boundary conditions imposed were actually maximal dissipative for a specific nonstandard class of first-order symmetric hyperbolic reduction of the wave system; see Section 5.2.1. Unlike the reductions considered in earlier work, such nonstandard class has the property that the boundary surface is noncharacteristic, which implies that no zero speed fields are present, and yields a strong well-posed system. Based on this reduction and the theory of quasilinear symmetric hyperbolic formulations with maximal dissipative boundary conditions [218, 388], it was possible to extend the results in [267Jump To The Next Citation Point, 263Jump To The Next Citation Point] and formulate a well-posed IBVP for quasilinear systems of wave equations [264Jump To The Next Citation Point] with a certain class of boundary conditions (see Theorem 8), which was sufficiently flexible to treat the Einstein equations. Furthermore, the new reduction also offers the interesting possibility to extend the proof to the discretized case using finite difference operators satisfying the summation by parts property, discussed in Sections 8.3 and 9.4.

In order to parallel the presentation in Section 4, here we focus on the IBVP for Einstein’s equations in generalized harmonic coordinates and the IBVP for the BSSN system. The first case, which is discussed in Section 6.1, is an application of Theorem 8. In the BSSN case, only partial results have been obtained so far, but since the BSSN system is widely used, we nevertheless present some of these results in Section 6.2. In Section 6.3 we discuss some of the problems encountered when trying to formulate a geometric uniqueness theorem and, finally, in Section 6.4 we briefly mention alternative approaches to the IBVP, which do not require an artificial boundary.

For an alternative approach to treating the IBVP, which is based on the imposition of the Gauss–Codazzi equations at 𝒯; see [191, 192, 194, 193]. For numerical studies, see [249, 104, 40, 404, 405, 98, 287, 244Jump To The Next Citation Point, 378, 253, 61, 362, 35, 33Jump To The Next Citation Point, 368, 57, 56], especially [366Jump To The Next Citation Point] and [369Jump To The Next Citation Point] for a comparison between different boundary conditions used in numerical relativity and [365Jump To The Next Citation Point] for a numerical implementation of higher absorbing boundary conditions. For review articles on the IBVP in general relativity, see [372, 355Jump To The Next Citation Point, 435].

At present, there are no numerical simulations that are based directly on the well-posed IBVP for the tetrad formulation [187Jump To The Next Citation Point] or the well-posed IBVP for the harmonic formulation [267Jump To The Next Citation Point, 263, 264Jump To The Next Citation Point] described in Section 6.1, nor is there a numerical implementation of the constraint-preserving boundary conditions for the BSSN system presented in Section 6.2. The closest example is the harmonic approach described in [286Jump To The Next Citation Point, 363Jump To The Next Citation Point, 366Jump To The Next Citation Point], which has been shown to be well posed in the generalized sense in the high-frequency limit [369Jump To The Next Citation Point]. However, as mentioned above, the well posed IBVP in [264Jump To The Next Citation Point] opens the door for a numerical discretization based on the energy method, which can be proven to be stable, at least in the linearized case.

 6.1 The harmonic formulation
  6.1.1 Well-posedness of the IBVP
 6.2 Boundary conditions for BSSN
 6.3 Geometric existence and uniqueness
  6.3.1 Geometric existence and uniqueness in the linearized case
 6.4 Alternative approaches

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