Observational signatures of near-extremal Kerr-like black holes in a modified gravity theory at the Event Horizon Telescope

Minyong Guo, Niels A. Obers, and Haopeng Yan

Phys. Rev. D 98, 084063 – Published 31 October 2018

Abstract

We study the shadows cast by near-extremal Kerr-modified gravity (MOG) black holes for different values of the parameter in modified gravity. In particular, we consider an isotropic emitter orbiting near such black holes and analytically compute the positions, fluxes and redshift factors of their images. The size of the shadow decreases when the modified parameter is increased. For each shadow, the images of the emitter appear on a special part of the shadow which has a rich structure. The primary image and secondary images are similar to those produced for the near-extremal (high-spin) Kerr black hole, but the near-extremal Kerr-MOG black hole can have a spin ( $\hat{J} / M_{α}^{2}$ ) which is finitely lower than 1. When the modified parameter is varied, the typical positions of the corresponding images do not change, nor does the typical redshift factor associated with the primary image. However, another typical redshift factor associated with the secondary image increases when the modified parameter is increased. We also find that the fluxes increase in that case. These images appear periodically with periods greater than that of Kerr. This provides an alternative signature away from the Kerr case which may be tested by the Event Horizon Telescope.

Received 5 August 2018

DOI:https://doi.org/10.1103/PhysRevD.98.084063

Physics Subject Headings (PhySH)

Alternative gravity theories Astrophysical studies of gravity Classical black holes

Astronomical black holes

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Minyong Guo

Department of Physics, Beijing Normal University, Beijing, 100875, China and Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada

Niels A. Obers and Haopeng Yan

The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen Ø, Denmark

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Issue

Vol. 98, Iss. 8 — 15 October 2018

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Images

Figure 1
Edges of near-extremal Kerr-MOG black hole shadows, where the dashed lines are the open parts (49) and the vertical solid lines are the NHEK-MOG lines (53). The green, magenta, blue and red curves have $a = 1$ , 0.9, 0.8, 0.717, respectively.
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Figure 2
Left to right: plots of $G (q)$ , $q ({\hat{t}}_{o})$ and $F_{o} / F_{N} (q)$ for $a = 1$ (light curves) and $a = 0.8$ (bright curves) with $m = 2$ , $b = 0$ , $s = + 1$ and the parameters choices of (54b). For $a = 1$ , the light yellow, light magenta and light cyan curves have $N = - 6$ , $- 7$ , $- 8$ , respectively, and no extra label $i$ , while the light red, light green and light blue curves have $N = - 9$ and $i = 1$ , 2, 3, respectively. For $a = 0.8$ , the yellow and magenta curves have $N = - 6, - 7$ , respectively, and no extra label $i$ , while the cyan, red and green curves have $N = - 8$ and $i = 1$ , 2, 3, respectively. Note that for $a = 1$ , the Kerr-MOG case reduce to the Kerr case so that the light curves agree with those in Ref. [15] exactly. (Although the condition (29) allows the whole range of $q$ , we have imposed a small $q$ cutoff since the corresponding image fluxes are negligibly small.)
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Figure 3
Observables of the most important few images for three different values of near-extremal spin of Kerr-MOG black holes with the parameter choices of (54b). From top to bottom, we plot positions, fluxes and redshift factors. Form left to right, we have $a = 1$ (Kerr case [15]), $a = 0.8$ and $a = 0.717$ (critical case for there exist ISCO for a near-extremal Kerr-MOG black hole). The color-coding is the same as that of Ref. [15]: each of these colored lines maybe a composed of continuous multiple track segments. For example, the green line (denoting the primary image) is composed of 4, 3, and 3 segments in the $a = 1$ , 0.8, 0.717 cases, respectively.
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Figure 4
Left to right: plots of $F_{o} / F_{N}$ for $a = 1$ (Kerr case [15]), $a = 0.8$ and $a = 0.717$ (critical case for there exist ISCO for a near-extremal Kerr-MOG black hole) with parameter choices of (54b). We set $q = 1.5$ , 1.38, and 1.35 for $a = 1$ , 0.8, and 0.717, respectively, and we let $m$ vary in each case. We denote the direct/reflect ( $b = 0 / b = 1$ ) images by blue/red dots. For $m = 0$ , we have only one image corresponding to $s = - 1$ , for each other value of $m$ , we have two images corresponding to $s = \pm 1$ .
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Figure 5
Left to right: plots of flux $F_{o} / F_{N}$ (green) and winding number $Δ ϕ / 2 π$ (gray) of the primary image for $a = 1$ (Kerr case [15]), $a = 0.8$ and $a = 0.717$ (critical case for there exist ISCO for a near-extremal Kerr-MOG black hole) with parameter choices of (54b). Note that as mentioned below Fig. 3, the single primary image for each case is a composed of multiple track segments.
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