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

Similarly to extremal Reissner-Nordstrom black holes (BHs), D3-branes and their intersecting bound states in lower dimensions enjoy a peculiar symmetry under conformal inversions that exchanges the horizon with infinity and keeps the photon sphere fixed. We explore the implications of this symmetry for the dynamics of massless and massive Bogomol'nyi-Prasad-Sommerfield (BPS) particles. In particular we find a remarkable identity between the scattering angle of a probe impinging from infinity and the inspiralling 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 of the two processes, when some cutoff regulator 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. We briefly discuss the fate of the symmetry at the quantum level as well as for nonspherically symmetric BHs and branes and sketch potential implications for the holographic correspondence.