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 , 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
Then, the boundary conditions, which were proposed in  for the harmonic system (4.5), are: 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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