Shadow of a noncommutative geometry inspired Ayón Beato García black hole

Abstract

We introduce the noncommutative geometry inspired Ayón Beato García black hole metric and study various properties of this metric by which we try to probe the allowed values of the noncommutative parameter $\vartheta$ under certain conditions. We then construct the shadow (apparent shape) cast by this black hole. We derive the corresponding photon orbits and explore the effects of noncommutative spacetime on them. We then study the effects of noncommutative parameter $\vartheta$, smeared mass m(r), smeared charge q(r) on the silhouette of the shadow analytically and present the results graphically. We then discuss the deformation which arises in the shape of the shadow under various conditions. Finally, we introduce a plasma background and observe how the shadow behaves in this scenario.

Appendix

In this “Appendix”, we present the explicit expressions for $\xi$ and $\eta$. These can be obtained by substituting Eq. 31 in Eqs. (28) and (29). The expressions read

$$\begin{aligned} \xi = \frac{\mathcal {X}}{a\mathcal {D}} \qquad \eta = \frac{\mathcal {G}}{\mathcal {D}} \end{aligned}$$

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where

$$\begin{aligned} \mathcal {X}= & {} (r^2+a^2)\left( -\frac{2m^\prime (r)r^4}{(r^2+q(r)^2)^{3/2}} - \frac{4m(r)r^3}{(r^2+q(r)^2)^{3/2}} + \frac{6m(r)r^4}{(r^2+q(r)^2)^{5/2}}(r+q(r)q^\prime (r))\right. \nonumber \\&+\, \frac{2q(r)q^\prime (r)r^4}{(q(r)^2+r^2)^2} + \frac{2q(r)^2r^3}{(q(r)^2+r^2)^2} - (q(r)q^\prime (r)+r)\frac{4q(r)^2r^4}{(q(r)^2+r^2)^3}+2r-\frac{4m(r)r^3}{(r^2+q(r)^2)^\frac{3}{2}}\nonumber \\&\left. +\,\frac{2q(r)^2r^3}{(r^2+q(r)^2)^2}\right) -4r^3 +\frac{8m(r)r^5}{(r^2+q(r)^2)^\frac{3}{2}} -\frac{4q(r)^2r^5}{(r^2+q(r)^2)^2} \end{aligned}$$

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$$\begin{aligned} \mathcal {D}= & {} -\frac{2m^\prime (r)r^4}{(r^2+q(r)^2)^{3/2}} - \frac{4m(r)r^3}{(r^2+q(r)^2)^{3/2}}+\, \frac{6m(r)r^4}{(r^2+q(r)^2)^{5/2}}(r+q(r)q^\prime (r)) + \frac{2q(r)q^\prime (r)r^4}{(q(r)^2+r^2)^2}\nonumber \\&+\, \frac{2q(r)^2r^3}{(q(r)^2+r^2)^2} - (q(r)q^\prime (r)+r)\frac{4q(r)^2r^4}{(q(r)^2+r^2)^3}+\,2r-\frac{4m(r)r^3}{(r^2+q(r)^2)^\frac{3}{2}} +\frac{2q(r)^2r^3}{(r^2+q(r)^2)^2} \nonumber \\ \end{aligned}$$

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$$\begin{aligned} \mathcal {G}= & {} 4r^3+4a^2r+2a\xi \left( -\frac{2m^\prime (r)r^4}{(r^2+q(r)^2)^{3/2}} - \frac{4m(r)r^3}{(r^2+q(r)^2)^{3/2}}\right. +\frac{6m(r)r^4}{(r^2+q(r)^2)^{5/2}}(r+q(r)q^\prime (r)) \nonumber \\&+\, \frac{2q(r)q^\prime (r)r^4}{(q(r)^2+r^2)^2}+\frac{2q(r)^2r^3}{(q(r)^2+r^2)^2} - (q(r)q^\prime (r)+r)\frac{4q(r)^2r^4}{(q(r)^2+r^2)^3}+2r-\frac{4m(r)r^3}{(r^2+q(r)^2)^\frac{3}{2}}\nonumber \\&\left. +\,\frac{2q(r)^2r^3}{(r^2+q(r)^2)^2}-r\right) -(a^2+\xi ^2)\left( -\frac{2m^\prime (r)r^4}{(r^2+q(r)^2)^{3/2}} - \frac{4m(r)r^3}{(r^2+q(r)^2)^{3/2}} \right. \nonumber \\&+\, \frac{6m(r)r^4}{(r^2+q(r)^2)^{5/2}}(r+q(r)q^\prime (r)) + \frac{2q(r)q^\prime (r)r^4}{(q(r)^2+r^2)^2} + \frac{2q(r)^2r^3}{(q(r)^2+r^2)^2}\nonumber \\&\left. -\, (q(r)q^\prime (r)+r)\frac{4q(r)^2r^4}{(q(r)^2+r^2)^3}+2r-\frac{4m(r)r^3}{(r^2+q(r)^2)^\frac{3}{2}}+\frac{2q(r)^2r^3}{(r^2+q(r)^2)^2}\right) .\nonumber \\ \end{aligned}$$

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