In a spacetime \((\mathcal {M},g)\) , a horizon is a null hypersurface where the deformation tensor \(\mathcal {K}:=\pounds _{\eta }g\) of a null and tangent vector \(\eta \) satisfies certain restrictions. In this work, we develop a formalism to study the geometry of general horizons (i.e. characterized by any \(\mathcal {K}\) ), based on encoding the zeroth and first transverse derivatives of \(\mathcal {K}\) on null hypersurfaces detached from any ambient spacetime. We introduce the notions of \(\mathcal {K}\) -tuple and non-isolation tensor. The former encodes the order zero of \(\mathcal {K}\) , while the latter is a symmetric 2-covariant tensor that codifies the “degree of isolation" of a horizon. In particular, the non-isolation tensor vanishes for homothetic, Killing and isolated horizons. As an application we derive a generalized near-horizon equation, i.e., an identity that holds on any horizon (regardless of its topology or whether it contains fixed points), which relates the non-isolation tensor, a certain torsion one-form, and curvature terms. By restricting this equation to a cross-section one can recover the near-horizon equation of isolated horizons and the master equation of multiple Killing horizons. Our formalism allows us to prove two existence theorems for horizons. Specifically, we establish the necessary and sufficient conditions for a non-degenerate totally geodesic horizon with any prescribed non-isolation tensor to be embeddable in a spacetime satisfying any (non-necessarily \(\Lambda \) -vacuum) field equations. We treat first the case of arbitrary topology, and then show how the result can be strengthened when the horizon admits a cross-section.
