numerical algorithm exists: one chooses some initial data whose late-time evolution asymptotes to our stationary or static vacuum solution of interest. The first point to make is that this will never allow one to find dynamically unstable solutions, which may be of interest for theoretical reasons. However, the second more serious obstruction to this being a practical method is that dynamical evolutions, particularly involving horizons, are very complicated, partly due to the constraints that must be initially solved—already a hard elliptic problem in general—and then ensuring that these constraints are properly preserved under evolution. For these reasons we believe that such a dynamical approach is not a viable one in practice.

The challenge then is to find a more direct but unified method to solve the Einstein equation in stationary, static, and Euclidean contexts. In this paper we provide a partial answer to this challenge. We present a general covariant framework in which to address the problem of finding static and Euclidean vacuum solutions in arbitrary dimensions, and with no particular symmetry properties (i.e. any cohomogenity). The geometry may contain no horizon, or one horizon component, in which case the algorithm only yields the spacetime exterior to the horizon. We 