AI 中文总结
研究自对偶圈量子引力黑洞中极端质量比旋进的轨道动力学和引力波特征，分析测试粒子运动，构建引力波波形，考虑旋转影响并进行费舍尔矩阵分析，发现EMRI观测可探测黑洞时空中的量子引力效应。
AI 中文摘要
圈量子引力（LQG）预测了对经典黑洞时空的量子修正，这可能在强场区域中致密天体的动力学和引力波信号上留下印记。本文研究自对偶圈量子引力黑洞时空中极端质量比旋进（EMRIs）的轨道动力学和引力波特征。分析了由聚合物参数\(P\)和最小面积参数\(a_0\)这两个量子参数表征的静态、球对称自对偶LQG几何中的测试粒子运动，系统研究了有效势和轨道结构，量化了量子修正对圆轨道稳定性和强场轨道行为的影响。与经典史瓦西时空相比，LQG修正改变了近视界轨道动力学。基于轨道演化构建了引力波波形，研究了量子修正对波形形态的影响。发现LQG效应在长时间旋进阶段积累，导致信号与经典情况有明显偏差。为纳入旋转，用纽曼 - 贾尼斯算法构建了自对偶时空的旋转扩展。所得的LQG修正克尔几何用于分析轨道运动，揭示了强场轨迹中自旋和量子修正的相互作用。最后进行了费舍尔矩阵分析，以估计未来天基引力波探测器对量子参数的潜在约束。结果表明，EMRI观测为探测黑洞时空中的量子引力效应提供了一条有前景的途径。
英文摘要
Loop quantum gravity (LQG) predicts quantum modifications to classical black-hole spacetimes, which may leave imprints on the dynamics and gravitational-wave signals of compact objects in the strong-field regime. In this work, we investigate the orbital dynamics and gravitational-wave signatures of extreme mass-ratio inspirals (EMRIs) in a self-dual loop quantum gravity black hole spacetime. We analyze test-particle motion in the static, spherically symmetric self-dual LQG geometry characterized by two quantum parameters: the polymeric parameter $P$ and the minimal area parameter $a_0$. The effective potential and orbital structure are systematically studied, and we quantify the influence of quantum corrections on circular-orbit stability and strong-field orbital behavior. Compared with the classical Schwarzschild spacetime, LQG corrections modify the near-horizon orbital dynamics. Based on the orbital evolution, we construct gravitational-wave waveforms and investigate the impact of quantum corrections on waveform morphology. We find that LQG effects accumulate during the long inspiral phase, leading to noticeable signal deviations from the classical case. To incorporate rotation, we construct a rotating extension of the self-dual spacetime using the Newman--Janis algorithm. The resulting LQG-corrected Kerr geometry is used to analyze orbital motion, revealing the interplay between spin and quantum corrections in strong-field trajectories. Finally, we perform a Fisher matrix analysis to estimate potential constraints on quantum parameters from future space-based gravitational-wave detectors. Our results indicate that EMRI observations provide a promising avenue to probe quantum gravitational effects in black-hole spacetimes.
