  The 3+1 Hamiltonian formulation in the gauge $D_tN=-K$ on the lapse function
fixes the direction of time associated with the trace $K$ of the extrinsic
curvature tensor. The Hamiltonian equations hereby become hyperbolic. We study
this new system for black hole spacetimes that are asymptotically quiescent,
which introduces analyticity properties that can be exploited for numerical
calculations by compactification in spherical coordinates with complex radius
following a M\"obius transformation. Conformal flat initial data of two black
holes are hereby invariant, and correspond to a turn point in a pendulum, up
for a pair of separated black holes and down for a single black hole. Here,
Newton's law appears in the relaxation of $l=2$ deformations of semi-infinite
poloidal surface elements, defined by the moment of inertia of the binary.
