It should be noted that formulas (55View Equation), (56View Equation), (57View Equation), (58View Equation), and (59View Equation) are valid even if subleading terms (as in Eq. (51View Equation)) in the expansion of the metric near the singular surface Σ are functions of 𝜃 [111Jump To The Next Citation Point]. Such more general metrics describe what might be called a “local Killing horizon”.