In this paper, the null geodesic equations are computed in d = 4 space–time dimensions [Y. Younesizadeh, A. A. Ahmad, A. H. Ahmed, F. Younesizadeh, Ann. Phys. 420, 168246 (2020)] by using the concept of symmetries and Hamilton–Jacobi equation and Carter separable method. With these null geodesics in hand, we evaluate the celestial coordinates (x, y) and the radius Rs of the BH shadow and represent it graphically. In addition, we have shown that the peak of this energy slowly shifts to lower frequencies and its height decreases with the increase in the YM magnetic charge (Q) values and decrease in the f(R) parameter (χ) values. In addition, we have analyzed the concept of effective potential barrier by transforming the radial equation of motion into standard Schrodinger form. The most important result derived from this study is that the height of this potential increases with increase in the YM magnetic charge (Q) values. Then, we study the quasinormal modes (QNMs) of these 4D black holes. For this purpose, we use the WKB approximation method upto third-order corrections. We have shown the perturbation’s decay in corresponding diagrams when the YM magnetic charge (Q) values and the f(R) parameter (χ) values change.

中文翻译：

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受杨-米尔斯场启发的 f(R) 重力模型中的阴影投射和准正常模式

在本文中，零测地线方程计算为d = 4时空维度 [Y. Younesizadeh, AA 艾哈迈德, AH 艾哈迈德, F. Younesizadeh,安。物理。 420, 168246 (2020)] 通过使用对称性的概念和 Hamilton-Jacobi 方程和 Carter 可分离方法。有了这些零测地线，我们评估天体坐标（X,是的) 和半径Rs的 BH 阴影并以图形方式表示它。此外，我们已经表明，这种能量的峰值缓慢地转移到较低的频率，并且它的高度随着 YM 磁荷的增加而降低（问) 值和减少F(R)范围 （χ) 值。此外，我们通过将径向运动方程转化为标准薛定谔形式，分析了有效势垒的概念。这项研究得出的最重要结果是，该电位的高度随着 YM 磁荷的增加而增加（问) 值。然后，我们研究了这些 4D 黑洞的准常态模式 (QNM)。为此，我们使用 WKB 近似方法进行三阶校正。当 YM 磁荷 (问) 值和F(R)范围 （χ) 值变化。