Geometrically Significant Surfaces of Black Holes from a Single Scalar

Black hole spacetimes contain several geometrically distinguished hypersurfaces, including event and Cauchy horizons, stationary-limit surfaces, curvature singularities, and asymptotic infinity. These structures are usually identified by different geometric or causal criteria. Here, we show that for the Kerr-Newman black hole, a single scalar function encodes all of them at once. The function arises by analytically continuing the membrane-paradigm pressure of the stretched horizon into the full spacetime. In fully factorized form, its zeros locate the outer and inner horizons, its poles locate the outer and inner stationary-limit surfaces, its higher-order divergence identifies the ring singularity, and its decay at large $r$ captures the asymptotic region. Thus, the analytically continued membrane pressure serves as a unified global detector of the critical surfaces in the Kerr-Newman geometry. We further note that the same analytic structure admits a secondary interpretation as an effective generalized multi-component van der Waals-type equation of state, whose intrinsic scales are fixed by the distinguished radii of the spacetime itself.