On boundary conditions for the Einstein equations

The use of the projection of the Einstein tensor normally to a timelike boundary as a set of boundary conditions for the initial-value problem of the vacuum Einstein equations is investigated within the setting of a particular first-order strongly hyperbolic formulation. It is found that the components of such a projection give rise to boundary conditions that are appropriate, in a certain sense, for the initial-value problem of the evolution equations and for the initial-value problem of the auxiliary system of propagation of the constraints, at the same time. It can be concluded that imposing such boundary conditions on the values of the fundamental variables of the initial-value problem guarantees the propagation of the constraints. This contribution presents a unified account of results that have recently appeared separately in the literature. The presentation is meant to be accessible to a broader readership.