Black hole boundary conditions and coordinate conditions

This paper treats boundary conditions on black hole horizons for the full (3+1)D Einstein equations. Following a number of authors, the apparent horizon is employed as the inner boundary on a space slice. It is emphasized that a further condition is necessary for the system to be well posed; the ``prescribed curvature conditions'' are therefore proposed to complete the coordinate conditions at the black hole. These conditions lead to a system of two 2D elliptic differential equations on the inner boundary surface, which coexist nicely to the 3D equation for maximal slicing (or related slicing conditions). The overall 2D-3D system is argued to be well posed and globally well behaved. The importance of ``boundary conditions without boundary values'' is emphasized. This paper is the first of a series.