In order to formulate the boundary conditions, we first construct a null tetrad , which is adapted to the boundary. This null tetrad is based on the choice of a future-directed time-like vector field tangent to , which is normalized such that . One possible choice is to tie to the foliation , and then define it in the direction orthogonal to the cross sections of the boundary surface. A more geometric choice has been proposed in [186], where instead is chosen as a distinguished future-directed time-like eigenvector of the second fundamental form of , as embedded in . Next, we denote by the unit outward normal to with respect to the metric and complete and to an orthonormal basis of at each point . Then, we define the complex null tetrad by

where . Notice that the construction of these vectors is implicit, since it depends on the dynamical metric , which is yet unknown. However, the dependency is algebraic, and does not involve any derivatives of . We also note that the complex null vector is not unique since it can be rotated by an angle , . Finally, we define a radial function on as the areal radius of the cross sections with respect to the background metric.Then, the boundary conditions, which were proposed in [264] for the harmonic system (4.5), are:

where , , , etc., and where and are real-valued given smooth functions on and and are complex-valued given smooth functions on . Since is complex, these constitute ten real boundary conditions for the metric coefficients . The content of the boundary conditions (6.2, 6.3, 6.4, 6.5) can be clarified by considering linearized gravitational waves on a Minkowski background with a spherical boundary. The analysis in [264] shows that in this context the four real conditions (6.2),(6.3, 6.4) are related to the gauge freedom; and the two conditions (6.5) control the gravitational radiation. The remaining conditions (6.6, 6.7, 6.8) enforce the constraint on the boundary, see Eq. (4.6), and so together with the constraint propagation system (4.14) and the initial constraints (4.15) 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.

The IBVP consisting of the harmonic Einstein equations (4.5), initial data (4.7) and the boundary conditions (6.2 – 6.8) can be shown to be well posed as an application of Theorem 8. For this, we first notice that the evolution equations (4.5) have the required form of Eq. (5.113), where is the vector bundle of symmetric, covariant tensor fields on . Next, the boundary conditions can be written in the form of Eq. (5.115) with . In order to compute the matrix coefficients , it is convenient to decompose in terms of the basis vectors

This result also applies the the modified system (4.16), 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.

Living Rev. Relativity 15, (2012), 9
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