The imaging of the M87 supermassive black hole (BH) by the Event Horizon Telescope collaboration has established the effectiveness of Very Large Baseline Interferometry to probe the strong gravity region around astrophysical BHs. In due time, one may expect more BHs, and with smaller masses, will be imaged, increasing the statistical significance of the data. It is then expected these data can be used to test the Kerr hypothesis and even General Relativity, by constraining alternative gravity models. It is therefore timely to investigate theoretical issues related to the propagation of light near BHs and, in particular, the properties of BH shadows.
Crispino and colleagues in their recent paper, considered the following question: may two different black holes (BHs) cast exactly the same shadow? In spherical symmetry, they showed the necessary and sufficient condition for a static BH (i.e. non-Schwarzschild) to be shadow-degenerate with Schwarzschild is that the dominant photonsphere of both has the same impact parameter, when corrected for the (potentially) different redshift of comparable observers in the different spacetimes. They further distinguished these two different classes of shadow degenerate spacetimes (class I and II), using the results they established.
The first shadow-equivalent class contains metrics whose constant (areal) radius hypersurfaces are isometric to those of the Schwarzschild geometry, which is illustrated by the Simpson and Visser (SV) metric. The second shadow-degenerate class contains spacetimes with different redshift profiles and an explicit family of metrics within this class is presented. In the stationary, axi-symmetric case, they determined a sufficient condition for the metric to be shadow degenerate with Kerr for far-away observers.
Again, they provided two classes of examples. The first class contains metrics whose constant (Boyer-Lindquist-like) radius hypersurfaces are isometric to those of the Kerr geometry, which is illustrated by a rotating generalization of the SV metric, obtained by a modified Newman-Janis algorithm. The second class of examples pertains BHs that fail to have the standard north-south Z2 symmetry, but nonetheless remain shadow degenerate with Kerr. The latter provides a sharp illustration that the shadow is not a probe of the horizon geometry.
“We remark that two non-isometric spacetimes are shadow-degenerate does not imply that the gravitational lensing is also degenerate. In fact, it is generically not.”— said Crispino
Through all these examples, they illustrated that non-isometric BH spacetimes can cast the same shadow, albeit the lensing is generically different.
Reference: Haroldo C. D. Lima Junior, Luís C. B. Crispino, Pedro V. P. Cunha, Carlos A. R. Herdeiro, “Mistaken identity: can different black holes cast the same shadow?”, ArXiv, pp. 1-16, 2021. https://arxiv.org/abs/2102.07034
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