{"title":"量子 Schwarzschild 黑洞光学方面","authors":"Anas El Balali","doi":"10.1134/S0202289324010043","DOIUrl":null,"url":null,"abstract":"<p>We investigate the optical behavior of a quantum Schwarzschild black hole with a space-time solution including a parameter <span>\\(\\lambda\\)</span> that encodes its discretization. Specifically, we derive the effective potential of this solution. In particular, we study circular orbits around the quantum black hole. Indeed, we find that the effective potential is characterized by a minimum and a maximum yielding double photon spheres denoted by <span>\\(r_{p_{1}}\\)</span> and <span>\\(r_{p_{2}}\\)</span>. Then, we analyze the double shadow behavior as a function of the parameter <span>\\(\\lambda\\)</span>, where we show that it controls the shadow circular size. An inspection of the Innermost Stable Circular Orbits (ISCO) shows that the radius <span>\\(r_{\\textrm{ISCO}}\\)</span> increases as a function of <span>\\(\\lambda\\)</span>. Besides, we find that this radius is equal to <span>\\(6M\\)</span> for an angular momentum <span>\\(L=2\\sqrt{3}\\)</span> independently of <span>\\(\\lambda\\)</span>. A numerical analysis shows that the photon sphere of radius <span>\\(r_{p_{1}}\\)</span> generates a shadow with a radius larger than <span>\\(r_{\\textrm{ISCO}}\\)</span>. Thus, a truncation of the effective potential is imposed to exclude such behavior. Finally, the <span>\\(\\lambda\\)</span>-effect is inspected depending on the deflection angle of such a black hole, showing that it increases when higher values of the parameter <span>\\(\\lambda\\)</span> are considered. 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引用次数: 0
摘要
Abstract We investigate the optical behavior of a quantum Schwarzschild black hole with a space-time solution including a parameter \(\lambda\) that encodes its discretization.具体地说,我们推导了这个解的有效势。我们特别研究了围绕量子黑洞的圆形轨道。事实上,我们发现有效势的特征是一个最小值和一个最大值,产生双光子球,分别用 \(r_{p_{1}}\) 和 \(r_{p_{2}}\) 表示。然后,我们分析了作为参数 \(\lambda\)函数的双影行为,结果表明它控制着影子的圆形大小。对最内层稳定圆形轨道(ISCO)的考察表明,半径 \(r_{textrm{ISCO}}\)随着 \(\lambda\)的函数而增加。此外,我们还发现,在角动量(L=2(sqrt{3}))与(\(lambda))无关的情况下,这个半径等于(6M)。数值分析表明,半径为(r_{p_{1}})的光子球产生的阴影半径大于(r_{textrm{ISCO}})。因此,我们对有效势能进行了截断,以排除这种行为。最后,根据这种黑洞的偏转角对\(\lambda\)效应进行了检验,结果表明当考虑到参数\(\lambda\)的较高值时,这种效应会增加。然而,这种增加受到一个由 \({6M}/{b}\) 给出的上限的限制。
We investigate the optical behavior of a quantum Schwarzschild black hole with a space-time solution including a parameter \(\lambda\) that encodes its discretization. Specifically, we derive the effective potential of this solution. In particular, we study circular orbits around the quantum black hole. Indeed, we find that the effective potential is characterized by a minimum and a maximum yielding double photon spheres denoted by \(r_{p_{1}}\) and \(r_{p_{2}}\). Then, we analyze the double shadow behavior as a function of the parameter \(\lambda\), where we show that it controls the shadow circular size. An inspection of the Innermost Stable Circular Orbits (ISCO) shows that the radius \(r_{\textrm{ISCO}}\) increases as a function of \(\lambda\). Besides, we find that this radius is equal to \(6M\) for an angular momentum \(L=2\sqrt{3}\) independently of \(\lambda\). A numerical analysis shows that the photon sphere of radius \(r_{p_{1}}\) generates a shadow with a radius larger than \(r_{\textrm{ISCO}}\). Thus, a truncation of the effective potential is imposed to exclude such behavior. Finally, the \(\lambda\)-effect is inspected depending on the deflection angle of such a black hole, showing that it increases when higher values of the parameter \(\lambda\) are considered. However, such an increase is limited by an upper bound given by \({6M}/{b}\).
期刊介绍:
Gravitation and Cosmology is a peer-reviewed periodical, dealing with the full range of topics of gravitational physics and relativistic cosmology and published under the auspices of the Russian Gravitation Society and Peoples’ Friendship University of Russia. The journal publishes research papers, review articles and brief communications on the following fields: theoretical (classical and quantum) gravitation; relativistic astrophysics and cosmology, exact solutions and modern mathematical methods in gravitation and cosmology, including Lie groups, geometry and topology; unification theories including gravitation; fundamental physical constants and their possible variations; fundamental gravity experiments on Earth and in space; related topics. It also publishes selected old papers which have not lost their topicality but were previously published only in Russian and were not available to the worldwide research community