Flexural modes at the photon sphere of a black hole

The spacetime curvature in the vicinity of massive black holes induces the bending of ray trajectories and the trapping of light in a specific region of space called the photon sphere. We mimic in the laboratory the behavior of waves near a black hole by investigating the modes of vibration on a 3D curved surface corresponding to a particular metric of the black hole. This surface can be transformed to a flat disk with non-uniform distribution of refractive index. Here, we consider elastic waves guided in a thin plate with non-uniform thickness, which corresponds to a varying velocity. Selective laser melting has been used to 3D-print our model with an aluminum alloy. A short pulse is propagated, and the spatiotemporal profile of the velocity-field is recorded by scanning a laser vibrometer. The quasimodes of the system are obtained by modal analysis in the Fourier domain. We find two different classes of modes: (1) Modes similar to the eigensolutions of a uniform disk, including whispering gallery modes; (2) Modes strongly confined within the plate, along a circle with radius corresponding to the photon sphere. Interestingly, this radius is frequency dependent, which is supported by our theory in the regime of wave optics. These observations are supported by full 3D numerical simulations. This work reveals a new type of high-Q modes scared on the stable or unstable orbits, which offers interesting perspective for laser design, inspired from celestial objects.