  The results on the initial boundary value problem for Einstein's vacuum field
equation obtained in \cite{friedrich:nagy} rely on an unusual gauge. One of the
defining gauge source functions represents the mean extrinsic curvature of the
time-like leaves of a foliation that includes the boundary and covers a
neighbourhoood of it. The others steer the development of a frame field and
coordinates on the leaves. In general their combined action is needed to
control in the context of the reduced field equations the evolution of the
leaves. In this article are derived the hyperbolic equations implicit in that
gauge. It is shown that the latter are independent of the Einstein equations
and well defined on arbitrary space-times. The analysis simplifies if boundary
conditions with constant mean extrinsic curvature are stipulated. It simplifies
further if the boundary is required to be totally geodesic.
