Geometric horizon for a distorted static spherically symmetric black hole
Recently, an alternative definition of a black hole horizon was proposed; such geometric definition is afforded by curvature invariants-scalars formed from the contractions involving the curvature tensor and its derivatives. It has been shown that for any stationary horizon, one can construct certain combinations of the gradient of curvature invariants such that they have vanishing norm on the horizon; this allows us to define a geometric horizon. However, its role in dynamical spacetimes is still not yet fully understood. In this work, we study the geometric horizon of a static spherically symmetric spacetime under first-order metric perturbations. Specifically, we derive an equation for a perturbed geometric horizon–defined by the vanishing squared norm of the gradient of the Kretschmann scalar; this equation is valid for any static spherically symmetric background spacetime in dimension d ≥ 4, as long as the gradient of the Kretschmann scalar is well-defined.