Geometrically Significant Surfaces of Black Holes from a Single Scalar
This paper demonstrates that a single scalar function, derived from the analytically continued membrane-paradigm pressure of the Kerr-Newman black hole, serves as a unified global detector that simultaneously encodes the locations of all critical geometric surfaces—including horizons, stationary limits, singularities, and asymptotic infinity—while also admitting an interpretation as a generalized van der Waals equation of state.