New method for shadow calculations: Application to parametrized axisymmetric black holes

Collaborative international efforts under the name of the Event Horizon Telescope project, using sub-mm very long baseline interferometry, are soon expected to provide the first images of the shadow cast by the candidate supermassive black hole in our Galactic center, Sagittarius A*. Observations of this shadow would provide direct evidence of the existence of astrophysical black holes. Although it is expected that astrophysical black holes are described by the axisymmetric Kerr solution, there also exist many other black hole solutions, both in general relativity and in other theories of gravity, which cannot presently be ruled out. To this end, we present calculations of black hole shadow images from various metric theories of gravity as described by our recent work on a general parametrization of axisymmetric black holes [R. Konoplya, L. Rezzolla, and A. Zhidenko, Phys. Rev. D 93, 064015 (2016).]. An algorithm to perform general ray-tracing calculations for any metric theory of gravity is first outlined and then employed to demonstrate that even for extremal metric deformation parameters of various black hole spacetimes, this parametrization is both robust and rapidly convergent to the correct solution.

Figure 1

Shadows cast by a Kerr black hole. Left panel: As viewed by an observer at $i\equiv {\theta }_{\mathrm{obs}}=90°$, with black hole spin parameters varied as 0 (red, leftmost), $0.1,0.2,\dots ,0.9,0.95,0.998$ and $1-{10}^{-6}$ (purple, rightmost). Right panel: Spin parameter fixed as $a=0.998$ and $i$ varied as 0° (red, leftmost), $10°,20°,\dots ,90°$ (blue, rightmost).

Figure 2

Left column: Shadows from the radial expansion of the Kerr-Sen metric for $b=0.5$ as viewed by an observer in the equatorial plane of the black hole for $a=0.95$ (top) and $a=0.998$ (bottom). The black curve represents the exact analytic form of the metric. Blue (first order), red (second order), orange (third order) and green (fourth order) curves represent the radial metric expansion. For comparison, the dotted curve shows the shadow from a Kerr black hole with the same spin parameter. Right column: Corresponding percentage error plots of each expansion order with respect to the original Kerr-Sen metric as a function of $\cos \psi$ along the shadow boundary curve.

Figure 3

As in Fig. 2, but now the deformation parameter $b=1$.

Figure 4

Top row: Shadows from the polar coordinate expansion of the EDGB metric with spin parameter $a=0.5$ for deformation parameters $\zeta =0.1$ (left panel) and $\zeta =0.15$ (right panel). Middle row: A magnified view of the polar region. Bottom row: Percentage error plots of each expansion order relative to the original EDGB metric. Blue, red, orange and green curves denote the expansion at first, second, third and fourth order in ${\cos }^2\theta$, respectively. For comparison, the dotted curve shows the shadow from a Kerr black hole with the same spin parameter.

Figure 5

Left column: Shadows from the radial expansion of the Johannsen-Psaltis metric with spin parameter $a=0.9$ for ${ϵ}_3=0.24$ (top), ${ϵ}_3=-0.5$ (middle) and ${ϵ}_3=-1$ (bottom). Only the first four orders of the expansion are shown. Right column: Percentage errors of each expansion order relative to the original Johannsen-Psaltis metric. Blue, red, orange and green curves denote the expansion at first, second, third and fourth order in ${\cos }^2\theta$, respectively. For comparison, the dotted curve shows the shadow from a Kerr black hole with the same spin parameter.