Turning black-holes and D-branes inside out their photon-spheres

Very much as extremal Reissner-Nordst\"om BHs, D3-branes and their intersecting bound-states in lower dimensions enjoy a remarkable symmetry under conformal inversions that exchange the horizon with infinity and keep the photon-sphere fixed. We explore the implications of this symmetry for the dynamics of massless and massive BPS particles. In particular we find a remarkable identity between the scattering angle of a probe impinging from infinity and the in-spiraling angle of a probe with the very same energy and angular momentum falling into the horizon from inside the photon-sphere. We argue for the identity of the radial actions and Shapiro time-delays for the two processes, when a cutoff regulator or an AdS ambient is adopted. We spell out the detailed conditions for the inversion symmetry to hold in the case of large BPS BHs in four dimensions. We address conformal inversions for other BHs and Dp-branes with photon-spheres in various dimensions with flat or AdS asymptotics. We briefly discuss the fate of the symmetry at the quantum level as well as for non spherically symmetric BHs and branes and sketch potential implications for the holographic correspondence.