Null Rigidity and Wave-Sector Reduction of Conformally Deformed Black Holes

Several minimal-length constructions deform the spacetime metric conformally at leading order, g˜μν=C(x)gμν. We ask which observables can register such a deformation. The two structural results rest on standard properties of conformal rescalings; what is worked out here is their consequence for this class of metrics, carried to closed form on a black-hole background. For an arbitrary conformal factor the unparametrized null geodesics are unchanged, so the critical impact parameter, the shadow angular radius seen from infinity, the deflection angles and the Shapiro delay keep their undeformed values at all orders, and the photon sphere keeps its coordinate location although its areal radius changes. A minimally coupled massless test scalar on g˜ obeys, exactly, a wave equation on g with the single potential Vω=□C/C=−16(CR˜−R), into which the Weyl invariant does not enter; a conformally coupled massless scalar does not see the deformation at all. For a Reissner–Nordström background and the power-law family η=ε0(M/r)p, with p as a model input rather than a prediction, we obtain the leading large-l scalar quasinormal shift p(p−3)ε0/[12·3pl(l+1)], suppressed as l−2, and closed forms for the ISCO radius, the disk radiative efficiency and the orbital frequency. Within this family the channels fall silent at the distinct exponents p=3, 2 and 5/2. All results are checked by computer algebra.