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I've been reading about Arago spots and the Aragoscope. Some fascinating concepts! Basically, due to wave diffraction, there is a bright spot in the centre of the shadow behind any circular object.

It strikes me that there are some similarities between a black hole and the disc used for the Arago spot experiments. Black holes (or their event horizons) are spherical, so should cast a circular shadow. Have there been any observations of bright spots in the centre of a black hole? If so, would these be useful for astronomy, like a supersized Aragoscope?

I have read that the shadowing object has to be very precisely circular, so I am not sure if a black hole is circular enough. Oblateness caused by rotation might distort the event horizon? There might also be issues with relativity around the fringes of the black hole. I don't have a clear concept of how gravitational lensing and Fresnel diffraction would work together.

A proper analysis of this should treat light as an EM wave propagating in the curved spacetime background of the black hole. The papers

(1) "Wave optics and image formation in gravitational lensing," https://arxiv.org/abs/1207.6846

(2) "Viewing Black Holes by Waves," https://arxiv.org/abs/1303.5520

(3) "Wave Optics in Black Hole Spacetimes: Schwarzschild Case," https://arxiv.org/abs/1502.05468

present analyses for the case of a non-rotating (Schwarzschild) black hole. The papers (1) and (2) use numerical techniques (solving the wave equation for the massless scalar field using a finite difference method) and the paper (3) uses analytic techniques. The abstract of (2) says

We study scattering of waves by black holes. Solving a massless scalar field with a point source in the Schwarzschild spacetime, waves scattered by the black hole is obtained numerically. We then reconstruct images of the black hole from scattered wave data for specified scattering angles. For the forward and the backward directions, obtained wave optical images of black holes show rings that correspond to the black hole glories associated with existence of the unstable circular photon orbit in the Schwarzschild spacetime.

Figure 7 in (2) shows "Images of black holes reconstructed from scattering waves...", and here is an excerpt from that figure:

Notice the relatively bright spot in the center. To help interpret the picture, page 12 says:

In the geometric optics limit, images of black holes can be obtained by solving null geodesics. For the observer at $theta_0=0$, the primary null rays, which are deflected by the black hole but do not go around it, result in the Einstein ring. The secondary and the higher degrees of null rays that go around the black hole many times also form ring images with smaller angular radius compared to the Einstein ring.

This is called the glory effect. The words "Arago spot" are not used in the paper, but the words "Airy disk" are used. The papers (2) and (3) both say that an extension of this analysis to rotating (Kerr) black holes is underway, but I don't know if it has been published yet.