diff --git "a/3tFAT4oBgHgl3EQfERy2/content/tmp_files/load_file.txt" "b/3tFAT4oBgHgl3EQfERy2/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/3tFAT4oBgHgl3EQfERy2/content/tmp_files/load_file.txt" @@ -0,0 +1,1405 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf,len=1404 +page_content='Peculiar properties in quasi-normal spectra from loop quantum gravity effect Guoyang Fu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='∗ Dan Zhang 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='† Peng Liu 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='‡ Xiao-Mei Kuang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='§ and Jian-Pin Wu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='4¶ 1 Center for Gravitation and Cosmology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' College of Physical Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Yangzhou University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Yangzhou 225009,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' China 2 Key Laboratory of Low Dimensional Quantum Structures and Quantum Control of Ministry of Education,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Synergetic Innovation Center for Quantum Effects and Applications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' and Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Hunan Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Changsha,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Hunan 410081,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' China 3 Department of Physics and Siyuan Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Jinan University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Guangzhou 510632,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' China and 4 Shanghai Frontier Science Center for Gravitational Wave Detection, Shanghai Jiao Tong University, Shanghai 200240, China Abstract We investigate the quasi-normal mode (QNM) spectra for scalar and electromagnetic fields over a covairant loop quantum gravity black hole (LQG-BH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' For the fundamental modes, the LQG effect reduces the oscillations in the scalar field, however it induces stronger oscillations in the elec- tromagnetic field, comparing to the classical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Under the scalar field perturbation, the system enjoys faster decaying modes with more oscillations than the electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Some peculiar phenomena emerge in the scalar field’s QNM spectra with high overtones for the angular quantum numbers l > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' It is that the LQG-BH has a larger real part of QNM with high overtones than the Schwarzschild black hole (SS-BH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Such an anomalous phenomenon results in the oscillation of the scalar field in the LQG-BH to be nearly identical to that in the SS-BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Therefore, the high overtone modes of the scalar field in LQG-BH play an important role in the modes with l > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' This anomalous phenomenon, however, does not occur in the electromagnetic field’s QNM spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' ∗FuguoyangEDU@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='com †danzhanglnk@163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='com ‡phylp@email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='jnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='cn §xmeikuang@yzu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='cn ¶jianpinwu@yzu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='cn 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='08421v1 [gr-qc] 20 Jan 2023 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' INTRODUCTION A non-perturbative and background-independent technique, loop quantum gravity (LQG) [1–4], provides a scenario for quantizing space-time structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' This approach has been suc- cessfully applied to quantize symmetry reduced cosmological space-times, known as loop quantum cosmology (LQC) [5–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Effective LQC theory can be constructed by incorporat- ing two key quantum gravity effects, namely the inverse volume correction and the holonomy correction, which can be achieved using both the canonical approach [13–18] and the path integral perspectives [19–25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' The quantum gravity effects in LQC can be connected to low-energy physics, resulting in a solvable cosmological model for studying quantum gravity effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' In particular, the big bang singularity in classical general relativity (GR) is suc- cessfully avoided by the quantum gravity effects [5–12, 26–31], which instead result in a non-singular big bounce even at the semi-classical level [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Following the same idea in LQC [5–12], several effective black holes (BH) models with LQG corrections have been constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Up to date, most of effective LQG-BHs are im- plemented through the input of the holonomy correction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' see, for example, [34–44] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' A common feature of LQG-BHs is that the singularity is replaced by a transition surface between a trapped and an anti-trapped region, which can be understood as the interior region of black hole and white hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' The heart of the holonomy correction is the phase space regularisation technique called polymerisation [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Because of this, the effective LQG-BH with holonomy correction is also known as the polymer BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' The basic idea behind polymerisation is the replacement of the conjugate momentum p with their regularised counterpart sin(¯λp)/¯λ, where ¯λ is a quantity known as polymerisation scale, which is linked to the area-gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Depending on whether the polymerization scale is constant or phase space dependent function, the polymer BHs are classified into two basic types: µ0-type scheme In this scheme, the polymerization scale is assumed to remain constant over the whole phase space [34–38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' This approach has the drawback that the final result is reliant on the fiducial structures, which are introduced in the construction of the classical phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' In addition, even in the low-curvature regimes, significant quantum effects may manifest, making these models unphysical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' 2 ¯µ-type scheme The polymerization scale in ¯µ-type scheme is chosen to be a function of the phase space [39–42] such that the dependency on fiducial structures is removed, though in this scheme significant quantum corrections near the horizon may still survive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Further, to fix the problems mentioned above, the authors in [46, 47] came up with an generalized version of the µ0-scheme in which the polymerization scale relies on the black hole mass such that it stays the same only along the dynamical trajectories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' This effective model does remove the problems described above, namely, the dependence on fiducial structures and the large quantum effects that emerge in the low-curvature regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' This model, however, also suffers from mass amplifications when crossing the transition surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' More recently, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Ashtekar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Olmedo, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Singh proposed an improved version of the generalized µ0-scheme [48, 49], which is now known as the AOS model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' The polymerization scale in this model is thought to depend on the effective Hamiltonian itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' This model is quite interesting since it removes not only the drawbacks of the µ0-scheme, but also the mass amplifications in the primitive version of the generalized µ0-scheme [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' However, the covariance in the aforementioned models is usually broken [50–53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Follow- ing the idea of the anomaly-free polymerization in [54], the authors in [55, 56] construct a covariant model of a spherically symmetric BH with holonomy correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' The polymeriza- tion scale ¯λ is a constant in this model, and it is related to a fundamental length scal r0 by a constant of motion m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' The resulting geometry corresponds to a singularity-free inte- rior region and two asymptotically flat exterior regions of equal mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' It is expected that this covariant model can alleviate certain long-standing concerns in the LQG community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Therefore, it certainly deserves further exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' In this paper, we will mainly study the properties of the quasi-normal modes (QNMs) of a probe scalar field and a probe Maxwell field over this covariant polymer BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' As we all know, during the ringdown phase of binary system coalescence, the BH emits the gravita- tional waves (GWs) with typical discrete frequencies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=', quasi-normal frequencies (QNFs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' According to [57], QNFs encode decaying scales and damped oscillating frequencies .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Cer- tainly, quantum effects have the imprints in the QNM spectra, which are expected to be detected in GW observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Also, conversely, GW detection will serve as an important criterion for the correctness of candidate quantum gravity theories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' 3 Our paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' In section II, we present a brief discussion on the effective potentials of scalar and Maxwell fields over the covariant LQG-BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Section III is dedicated to the properties of the QNM spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Then, we further study the ringdown waveform in section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' We present the conclusions and discussions in section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Appendixes A and B present the detailed derivation of the wave equations and the QNMs in the eikonal limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' SCALAR FIELD AND MAXWELL FIELD OVER A COVARIANT LQG-BH In [55, 56], the authors proposed a novel effective LQG black hole model with holonomy correction that is covariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' The exterior geometry of this covariant LQG black hole is given by ds2 = −f(r)dt2 + 1 g(r)f(r)dr2 + r2dΩ2 , f(r) = 1 − 2m r , g(r) = 1 − r0 r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (1) Thanks to the quantum gravity effects, a new length scale r0 is introduced r0 = 2m ¯λ2 1 + ¯λ2 , (2) where the parameter ¯λ is a dimensionless constant related to the fiducial length of the holonomies and m is a constant of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' The length scale r0 define a minimum area r2 0 of this model [55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' In the ¯λ → ∞ limit, this new length scale vanishes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=', r0 = 0, restoring the classical Schwarzschild black hole is recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' In addition, as we move to the low curvature regions, the quantum gravity effects die off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Without loss of generality, we shall set m = 1/2 through this paper, which leads to the horizon located at rh = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' We focus on the perturbations of the massless scalar field Φ and electromagnetic field Aµ over this LQG black hole and study their response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' We write down the covariant equations for the test scalar field and electromagnetic field as follows: 1 √−g(gµν√−gΦµ),ν = 0 , (3) 1 √−g(gαµgσν√−gFασ),ν = 0 , (4) where Fασ = ∂αAσ − ∂σAα is the field strength of the Maxwell field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' After the separation of variables, the aforementioned equations can be packaged into the Schr¨odinger-like form (for 4 more details, see Appendix A) ∂2Ψ ∂r2 ∗ + (ω2 − Veff)Ψ = 0 , (5) where r∗ is the tortoise coordinate and Veff is the effective potentials: Veff = f(r)l(l + 1) r2 + 1 − s r d dr∗ f(r) � g(r) , (6) with l being the angular quantum numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' s = 0 and s = 1 correspond to the scalar field and electromagnetic field, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='1 and 2 demonstrate the effective potentials as a function of r∗ for scalar and electromagnetic fields with different l and r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' It is found that both effective potentials are positive, indicating the LQG black hole is stable under scalar and electromagnetic perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Furthermore, we would like to compare the differences in effective potentials between scalar and electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' It is easy to find that for the electromagnetic field (s = 1), the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (6) vanishes, such that all the peaks of the effective potentials Vel have the same height for different r0 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' However, for the scalar field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=', s = 0, the second term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (6) survives and the height of the effective potential Vs depends on r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' In particular, with increasing r0, the height of Vs decreases (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' The shape of the effective potentials shall definitely results in different properties of the QNMs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' r0=0 r0=1/4 r0=1/2 r0=3/4 10 10 20 r* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='10 Vs(r) l=0 r0=0 r0=1/4 r0=1/2 r0=3/4 10 10 20 r* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='4 Vs(r) l=1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' 1: The effective potentials Vs(r∗) of the scalar field for different r0 with fixed l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' 5 r0=0 r0=1/4 r0=1/2 r0=3/4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='280 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='285 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='290 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='295 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='300 10 10 20 r* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='30 Vel(r) l=1 r0=0 r0=1/4 r0=1/2 r0=3/4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='87 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='89 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='90 10 10 20 r* 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='8 Vel(r) l=2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' 2: The effective potentials Vel(r∗) of the electromagnetic field for different r0 with fixed l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' QUASI-NORMAL MODES In this section, we investigate the QNMs spectra and specially focus on the effects from quantum gravity corrections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' The nature of determining the QNMs is to solve the eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' To this end, we will impose a purely outgoing wave at infinity and purely ingoing wave at the horizon: horizon: ∂tΨ − ∂r∗Ψ = 0, infinity: ∂tΨ + ∂r∗Ψ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (7) By solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (5) with the aforementioned boundary conditions, numerous techniques have been developed to determining the QNMs spectra, such as the WKB method [58–63], Horowitz-Hubeny method [64], continued fraction method [65], asymptotic iteration method [66–68], pseudo-spectral method [69, 70], and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' In this paper, we will solve the eigen- value problem using the pseudo-spectral method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' For more applications of pseudo-spectral method in determining the QNMs in the black hole physics, we can refer to [71–80] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' It is convenient to work in the Eddington-Finkelstein coordinate, which makes the wave equation (5) is linear in the frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' To achieve this goal, it is direct to make a transformation as r → 1/u and Ψ = e−iωr∗(u)ψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (8) 6 Then, the wave equation (5) turns into the following form: ψ′′(u) + � f ′(u) f(u) + g′(u) 2g(u) + 2iω u2f(u) � g(u) � ψ′(u) −1 u � 2iω u2f(u) � g(u) + Veff(u) u3f(u)2g(u) + f ′(u) f(u) + g′(u) 2g(u) � ψ(u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (9) Combining with the boundary conditions (7), one can solve Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (9) by the pseudo-spectral method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='0r0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='21 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='22 Reω l=0, n=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='8 r0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='14 Imω l=0, n=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='0r0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='570 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='575 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='580 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='585 Reω l=1, n=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='8 r0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='14 Imω l=1, n=0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' 3: QNFs as a function of r0 for the scalar field perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Now, we evaluate the QNM spectra for various values of the free parameter r0 to explore the LQG effects on these spectra, as well as the differences between them and those of the Schwarzschild black hole (r0 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' First, we focus on the fundamental modes and show the QNF as a function of r0 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='3 for the scalar field and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='4 for the electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' The main properties are presented as what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' For scalar field, the real parts of the QNF, Reω decreases with increasing r0 (left plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' It means that the LQG effect reduces the oscillations in comparison to 7 the Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' By contrast, Reω of the electromagnetic field exhibits an inverse tendency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' That is, when r0 increases, so does Reω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' As a result, the LQG effect produces stronger oscillations in the electromagnetic field than that of the Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Whether for scalar or electromagnetic field, the imaginary part of QNF Imω al- ways lives in the lower half-plane, and their absolute values are less than that of the Schwarzschild black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Therefore, the system is stable in the presence of scalar or electromagnetic field perturbations, and the LQG effect results in faster decaying modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' When we fix r0, the scalar field has larger absolute values of Reω or Imω than the electromagnetic field (see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='3 and 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' It indicates that in comparison to the elec- tromagnetic field, the system under scalar field perturbation enjoys faster decaying modes with greater oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='8 r0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='505 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='510 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='515 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='520 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='525 Reω l=1, n=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='8 r0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='12 Imω l=1, n=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='8 r0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='920 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='930 Reω l=2, n=0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='8 r0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='12 Imω l=2, n=0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' 4: QNFs as a function of r0 for the electromagnetic field perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' We also work out the QNM spectra with high overtones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Table I shows the results for 8 scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' For l = 0, we see that the LQG-BH has a lower Reω than the Schwarzschild black hole (SS-BH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' This is consistent with the fundamental mode’s behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' The most striking difference occurs at the case with l > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' To demonstrate this, we define the difference in QNFs between SS-BH (r0 = 0) and LQG-BH (r0 ̸= 0) as follows: δω = ωLQG − ωSS .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (10) Observing the left plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='5 (also see the second column in Table I), Reω with high overtones is larger for the LQG-BH than the SS-BH for l > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' It indicates that the overtones may play an important role in the modes with l > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' We shall testify this point by the time evolution of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' We also briefly address the properties of the QNMs for the electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Table II shows the QNM spectra for the electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' It is found that for all modes, Reω for the LQG-BH is always smaller than that of the SS-BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' It differs from the scalar field, where Reω with high overtones is larger for the LQG-BH than the SS-BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' This discrepancy also results in the corresponding difference in the evolutions of the scalar field and the electromagnetic field, which will be addressed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' l = 0 l = 1 n ω (r0 = 0) ω (r0 = 1/2) ω (r0 = 0) ω (r0 = 1/2) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='220910-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='209792i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='200799-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='164527i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='585872-0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='588236i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='395550-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='981462i TABLE I: The QNM spectra for the scalar field perturbation with different n, l, and r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' 9 r0=1/100 r0=1/10 r0=1/2 1 2 3 4 n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='02 0.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='5 δImω l=1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' 5: The difference of QNFs of scalar field for the mode with l = 1 between the LQG-BH and SS-BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' l = 1 l = 2 n ω (r0 = 0) ω (r0 = 1/2) ω (r0 = 0) ω (r0 = 1/2) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='496527-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='184975i 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='322640-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='956797i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='602986-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='439431i 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='730157-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='888692i TABLE II: The QNM spectra for the electromagnetic field perturbation with different system parameters n, l and r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Finally, we will discuss the properties of the QNMs in the eikonal limit (l → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' In [81], Cardoso et al have demonstrated that, in the eikonal limit, QNMs may be connected with the behavior of null particle trapped on the unstable circular geodesic of the spacetime, which have been validated in most static, spherically symmetric, asymptotically flat spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' The Reω is determined by the angular velocity Ωc at the unstable null geodesic [82–86], whereas the Imω is connected to the Lyapunov exponent λ [87, 88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' In the LQG-BH background, we can calculate the QNMs in the eikonal limit, which is given by ω = Ωcl − i � n + 1 2 � |λ| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (11) 10 For the detailed calculation, we can refer to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' It is found that as SS-BH, the angular velocity Ωc is completely determined by the black hole mass: Ωc = 1 3 √ 3m .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (12) Therefore, the Reω is independent of the LQG parameter r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' While the Lyapunov exponent λ is given by λ = � − r2 c f(rc) � d2 dr2 ∗ f(r) r2 � ��� r=rc , (13) where rc is the radius of the photon sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Obviously, the Lyapunov exponent is affected by the LQG correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Left plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='6 shows the Lyapunov exponent λ as a function of r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' We see that the Lyapunov exponent decreases with r0 increasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Correspondingly, the absolution value of Imω is suppressed by the the LQG effect (see the right plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='0r0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='12 |λ|/2 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='60 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='65 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='70 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='12 Re ω Im ω l=100,n=0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' 6: Left plot: The Lyapunov exponent λ as a function of the LQG corrected parameter r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Right plot: The QNFs for different r0 for large l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' We notice that since the real part of QNF is independent of the LQG parameter r0 in the eikonal limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Therefore, we expect that as l increases, the difference in Reω between LQG-BH and SS-BH will be suppressed and vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='7 validates this argument that as l increases, the difference rapidly decreases and goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' 11 90 92 94 96 98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='000100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='000098 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='000096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='000094 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='000092 20 40 60 80 100l 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='005 δReω n=0 90 92 94 96 98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='00040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='00041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='00042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='00043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='00044 20 40 60 80 100 l 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='015 δReω n=1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' 7: The difference of QNFs of scalar field between the LQG-BH and SS-BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Left plot is for n = 0 and right plot for n = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' RINGDOWN WAVEFORM In this section, we will study the time evolution of the scalar and electromagnetic per- turbations, which help us to further know the total contributions from overtones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Here, we will use the finite difference method (FDM) technics to implement the dynamical evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' For more details on the FDM, we can refer to Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' [73, 89–92] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' To this end, we write the wave equation in difference form as −(Ψi+1,j − 2Ψi,j + Ψi−1,j) △t2 + (Ψi,j+1 − 2Ψi,j + Ψi,j−1) △r2 ∗ − VjΨi,j + O(△t2) + O(△r2 ∗) = 0 , (14) where △t and △r∗ are the time and radial intervals, respectively, wihch are defined by t = i△t and r∗ = j△r∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' The Vj is the discrete form of the effective potential (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Then, the iterate formula is derived as: Ψi+1,j = −Ψi−1,j + △t2 △r2 ∗ (Ψi,j+1 + Ψi,j−1) + (2 − 2 △t2 △r2 ∗ − △t2Vj)Ψi,j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (15) Notice that the Courant-Friedrichs-Lewy (CFL) condition for instability requires that △t/△r∗ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Using the iterate formula (15) with the initial Gaussian distribution Ψ(r∗, t < 0) = 0 and Ψ(r∗, t = 0) = exp − (r∗−a)2 2b2 , one can obtain the ringdown profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' In general, there are three different stages in time-evolution profile: initial outburst, quasinormal ringing, which depends only on the black hole’s characteristics and is very important for GW observations [57, 93–95], and the late tail, which exhibits the power-law 12 behavior for the asymptotically flat spacetimes or exponential behavior for asymptotically de-Sitter spacetimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' We will focus on the properties of the latter two stages in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Schwarschild BH r0=1/4 r0=1/2 0 50 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='100 1 10 100 t |Ψs| l=0 Schwarschild BH r0=1/4 r0=1/2 0 50 100 150 200 10-9 10-7 10-5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='100 10 t |Ψs| l=1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' 8: The time evolution of the scalar field |Ψs(r)| for different r0 with fixed l (left plot for l = 0 and right plot for l = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Left plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='8 shows the time-domain profile for the scalar field perturbation with l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' In comparison to the SS-BH, the oscillation is slightly weaker and the decay becomes slower during the intermediate time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Recalling that both the real and imaginary parts of the fundamental QNF all reduce as the LQG corrected parameter r0 enhances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' It suggests that in the scalar evolution of LQG-BH with l = 0 the fundamental QNMs dominate over the ones with high overtones, which is consistent with the case of SS-BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' At asymptotically late-times, quasinormal ringing is suppressed, and it follows the same power-law tail as Ψ(t) ∼ t−(2l+3) for both LQG-BH and SS-BH [96–98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' However, for multipoles l > 0, we observe some peculiar behavior that differs from that of l = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Carefully observing the right plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='8, we find that the slope of quasinormal ringing is smaller for the LQG-BH than for the SS-BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' This observation is consistent with the QNFs, which show that the absolute value of Imω for all discrete overtones is smaller for the LQG-BH than for the SS-BH (see the right column in Table I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Nevertheless, the oscillation for the LQG-BH is nearly coincide with that for the SS-BH (the right plot in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='8 and also see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Recalling that for l > 0, the LQG-BH has a lower Reω for the fundamental mode than the SS-BH, whereas the case is reversed for high overtones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' It means that the contribution from the high overtone modes reduces the difference between the LQG-BH and SS-BH time-domain profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' That is, the high overtone modes in the Reω 13 of the LQG-BH play an important role in determining the oscillation of the time evolution of the scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' r0=0 r0=1/100 r0=1/10 r0=1/4 0 10 20 30 40 50 30 20 10 0 10 20 30 40 t |Ψs| l=1 r0=0 r0=1/100 r0=1/10 r0=1/4 10 20 30 40 50 60 70 10-4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='100 1 10 t |Ψs| l=1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' 9: The time evolution of the scalar field |Ψs(r)| for different r0 with fixed l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Notice that the right plot is the semi-log plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Schwarschild BH r0=1/2 0 50 100 150 200 10-6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='001 1 t |Ψel| l=1 Schwarschild BH r0=1/2 0 50 100 150 200 10-9 10-6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='001 1 t |Ψel| l=2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' 10: The semi-log plot of the time evolution of the electromagnetic field |Ψel(r)| for different r0 with fixed l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' We also study the time evolution of the electromagnetic field, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' We observe that the slop of quasinormal ringing is smaller for the LQG-BH than the SS-BH, which is similar to the scalar field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' However, unlike in the case of scalar field, the oscillations of the LQG-BH and the SS-BH are not the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' It is expected because for l > 0, the anomalous phenomena seen in the QNMs of scalar field that Reω with high overtones is larger for the LQG-BH than the SS-BH does not happen for electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' 14 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' CONCLUSION AND DISCUSSION As the rapid development of the GW detection technics, it is expected to detect the quantum gravity effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' To extract substantial information from GW detectors, one must thoroughly know the main features and behaviors of QNM for LQG-BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' As the first step, we investigate the QNM for scalar and electromagnetic fields over the covariant LQG-BH proposed in [55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' The QNM spectra for scalar and electromagnetic fields share some common features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' But they also exhibit many different features and behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' First, we focus on the fundamental modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' It is found that the system is always stable under scalar field or electromagnetic field perturbations, and the LQG effect results in faster decaying modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' The difference is that the LQG effect reduces the oscillations in the scalar field, however it enhances oscillations in the electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' In addition, we find that the system under the scalar field perturbation enjoys faster decaying modes with more oscillations than the electromagnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Some peculiar phenomena emerge in the scalar field QNM spectra with high overtones for l > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' It is that the LQG-BH has a larger ωR with high overtones than the SS-BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Such an anomalous phenomenon results in the oscillation of the scalar field in the LQG-BH to be nearly identical to that in the SS-BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Therefore, the high overtone modes of the scalar field in LQG-BH play an important role in the modes with l > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' This anomalous phenomenon, however, cannot be observed in the electromagnetic field’s QNM spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Finally, we comment some open questions deserving further exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' It would be interesting to extend our investigation to the Dirac field and see if the peculiar property still emerges in the QNM spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' It is definitely interesting and valuable to further study the QNM spectrum of the gravity perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' It provides us a platform for detecting quantum gravity effects using the GW detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' In addition, we can examinate if the isospectrality still holds in this LQG-BH model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' In [99], the anomalous decay rate of QNMs of a massive scalar field is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' De- pending on how large the mass of the scalar field is, the decay timescales of the QNMs either grow or decay with increasing angular harmonic numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' This anomalous be- havior is seen in much larger class models beyond a simple massive scalar field, see 15 [100–104] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' It will interesting to see how the LQG effect affects this anomalous behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' We can also construct an effective rotating LQG-BH solution using the Newman-Janis algorithm, starting with this spherical sysmetric LQG-BH, and study the LQG effects on its QNM spectrum and shadow, allowing us to constrain the LQG parameters using the GW detector and the Event Horizon Telescope (EHT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' We plan to investigate these questions and publish our results in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Acknowledgments This work is supported by National Key R&D Program of China (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' 2020YFC2201400), the Natural Science Foundation of China under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' 11905083, the Postgraduate Re- search & Practice Innovation Program of Jiangsu Province under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' KYCX20 2973, the Postgraduate Scientific Research Innovation Project of Hunan Province, the Science and Technology Planning Project of Guangzhou (202201010655), the Fok Ying Tung Education Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' 171006, the Natural Science Foundation of Jiangsu Province under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='BK20211601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' is also supported by Top Talent Support Program from Yangzhou University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Appendix A: Wave equations In this appendix, we will derive the wave equations for the scalar and electromagnetic fields in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' First, we shall provide a generic version of the wave equation in a static spherically symmetric spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' The cases of scalar field and electromagnetic field are then discussed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Because the spacetime is static spherically symmetric, we can separate variables using the spherical function and write the radial equation in the form (K(r)S(r)ˆΨ′(r))′ + � ΛF(r) + K(r) ω2 S(r) � ˆΨ(r) = 0 , (A1) where ˆΨ is the radial part of the wave function, the coefficient functions {K , F , S} only depend on the radial coordinate r, and Λ is the separation constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' After introducing the 16 tortoise coordinate r∗ and redefining the wave function as dr∗ dr = 1 S(r) , ˆΨ(r) = Ψ � K(r) , (A2) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (A1) can be recasted into the following form d2Ψ(r∗) dr2 ∗ + (ω2 − Veff(r∗))Ψ(r∗) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (A3) The above formula provides a general transformation from the usual wave equation to its Schr¨odinger-like counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' In the following, we will go over the specific form of the wave equations for scalar and electromagnetic fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' For the scalar field equation, we perform the separation as Φ(t, r, θ, φ) = � l,m ˆΨ(r)e−iωtYlm(θ, φ), where Ylm(θ, φ) is the spherical harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' When the particular form of the LQG-BH background (1) is substituted into the wave equation (A1), one obtains � r2f(r) � g(r)ˆΨ′(r) �′ + � r2ω2 f(r) � g(r) − l(l + 1) � g(r) � ˆΨ(r) = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (A4) We can read off the coefficient functions by comparing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (A4) to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (A1) K(r) = r2 , S = f(r) � g(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (A5) The Schr¨odinger-like version of the wave equation is then easily given as ∂2Ψ ∂r2 ∗ + (ω2 − Vs)Ψ = 0 , (A6) Vs = f(r)l(l + 1) r2 + 1 2r d drf(r)2g(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (A7) For the electromagnetic field, we can expand the gauge field Aµ in vector spherical har- monics [105, 106], Aµ(t, r, θ, φ) = � l,m � � � � � � � � ������ 0 0 alm(r) sin θ ∂φYlm −alm(r) sin θ∂θYlm � ������ + � ������ plm(r)Ylm hlm(r)Ylm klm(r)∂θYlm klm(r)∂φYlm � ������ � � � � � � � e−iωt , (A8) where the first term is the odd (axial) perturbation and second term is even (polar) pertur- bation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Then, in the following, we will show how to derive the odd perturbation equation and even perturbation equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' 17 When we switch on the odd electromagnetic field perturbation, we can explicitly write down the radial equation as � f(r) � g(r)a′ lm(r) �′ + � ω2 f(r) � g(r) − l(l + 1) r2� g(r) � alm(r) = 0 , (A9) It is easy to find that K = 1 and S = f(r) � g(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Thus, we have Vodd = f(r)l(l + 1) r2 , (A10) where Ψ = alm(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' For the even perturbation of the electromagnetic field, the radial equation becomes p′′ lm(r) + q(r)p′ lm(r) + iω (h′ lm(r) + q(r)hlm(r)) + l(l + 1) r2f(r)g(r)(plm(r) + iωklm(r)) = 0 , −iωp′ lm(r) + ω2hlm(r) + l(l + 1) r2 f(r)(−hlm(r) + k′ lm(r)) = 0 , (A11) where q(r) = 2 r + g′(r) 2g(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' After introducing a new variable ˆΨ(r) = −p′ lm(r) − iωhlm(r) , (A12) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (A11) can be reduced to (r4f(r)g(r)3/2 ˆΨ′(r))′ + � r4ω2� g(r) f(r) − l(l + 1)r2 � g(r) + 1 2J(r) � ˆΨ(r) = 0 (A13) where J(r) = r2� g(r)(rf ′(r)(4g(r) + rg′(r)) + f(r)(4g(r) + r(6g′(r) + rg′′(r)))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Thus, the coefficient functions are K = r4� g(r) and S = f(r) � g(r) and then we have Veven = f(r)l(l + 1) r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (A14) We find that the effective potentials for odd and even electromagnetic field perturbations are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Therefore, we will use Vel to signify the effective potential of the electromagnetic field rather than Vodd and Veven.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Appendix B: QNMs in the eikonal limit In this appendix, we will show the connection between the QNMs in the eikonal limit and the behavior of null particle trapped on the unstable circular geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' For a null particle, 18 the Lagrange is1 L(x, ˙x) = 1/2gµν ˙xµ ˙xν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (B1) We start with the spherically symmetric geometry (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Thanks to the symmetry, one can only consider the geodesics in the equatorial plane: θ = π/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Then the Lagrangian (B1) becomes 2L = −f(r)˙t2 + ˙r2 f(r)g(r) + r2 ˙φ2 , (B2) where the dot represents the derivative with respect to the affine parameter τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' In this system, there are two constants of the motion, which are Pt = −f(r)˙t = −E , Pφ = r2 ˙φ = L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (B3) Using the canonical transform and combining the above equations (B2) and (B3), we have the following reduced Hamiltonian system: 2H = E ˙t + ˙r2 f(r)g(r) + L ˙φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (B4) Since the Hamiltonian H satisfies the constraint H = 0, we have ˙r2 + Veff = 0 , (B5) where the effective potential is Veff = g(r) �L2 r2 f(r) − E2 � , (B6) Because ˙r2 > 0, the photon can only emerge in the area of negative potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' When the angular momentum is small, the photon will fall from infinity into the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' However, for the large angular momentum, the photon will escape the bondage of the black hole and go back to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Therefore, the critical circular orbit for the photon can be derived from the unstable conditions Veff = 0 , ∂Veff ∂r = 0 , ∂2Veff ∂r2 < 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (B7) 1 For the calculation details of the geodesic of a null particle, please refer to [81, 82, 107, 108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' 19 From the above conditions, we can obtain the equation for the critical radius rc 2fc(r) = rcf ′ c(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (B8) Correspondingly, we have the critical impact parameters bc: bc = L E = rc � fc(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (B9) Then, the shadow radius Rs and Lyapunov exponents λ can be calculated as follows: Rs = � ζ2 + η2 = bc = 3 √ 3m , (B10) λ = � V ′′ eff 2˙t2 = � − r2 c f(rc) � d2 dr2 ∗ f(r) r2 � ��� r=rc , (B11) where {ζ , η} are the celestial coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' We find that the shadow radius reduces to the one of SS-BH [109, 110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' It means that the LQG effect doesn’t change the shape of the shadow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' However, the LQG correction affects the Lyapunov exponent λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' On the other hand, we shall use the first order WKB approximation to obtain the analytic form of the QNMs in the eikonal limit (l → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' In this limit, the last term of the effective potential (6) can be ignored, resulting in the following form of the effective potential V∞(r) = f(r) l2 r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' (B12) Reminding that the potential (B6) and (B12) are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} +page_content=' Therefore, in the eikonal limit, the QNMs may be obtained by the multiples of the frequency and the instability timescale of the unstable circular null geodesic [81]: ω = Ωcl − i(n + 1 2)|λ| , (B13) where Ωc is the angular velocity and can be worked out as Ωc = ˙φ ˙t = 1 bc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3tFAT4oBgHgl3EQfERy2/content/2301.08421v1.pdf'} 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