The paper develops several variants of the suggestion of Unruh (1976), to define vacuum initial conditions on the horizon through an analytic property of normal-mode solutions which expresses 'positive frequency' with respect to null translations on the horizon. It is verified that Unruh's condition corresponds to the absence of a flux of energy through an horizon surface, although there may be a flux parallel to the surface. A region with time-like isometries typically is bounded by four such surfaces, two of which may be the usual null infinites, I^+or-. In general, Unruh's condition may be applied on two adjacent sides, forcing the appearance of a Hawking flux on the other two. In special cases, however, the opposite horizons can be in 'equilibrium', so that no radiation occurs. In particular, for two-dimensional de Sitter space the vacuum state thus obtained has a stress tensor proportional to the metric times the curvature scalar. If two horizons are not in equilibrium, then no state invariant under the isometries can yield a non-singular stress tensor.