Generalized, Energy-conserving Numerical Simulations of Particles in General Relativity. I. Time-like and Null Geodesics
Generalized, Energy-conserving Numerical Simulations of Particles in General Relativity. I. Time-like and Null Geodesics
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DOI：
10.3847/1538-4365/aac9ca
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发表时间：
2018-01
期刊：
影响因子：
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通讯作者：
F. Bacchini;B. Ripperda;Alexander Y. Chen;L. Sironi
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文献类型：
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作者：
F. Bacchini;B. Ripperda;Alexander Y. Chen;L. Sironi
弯曲时空中粒子轨迹的数值积分是获得大质量致密物体（如黑洞和中子星）周围粒子动力学的真实模型的基础。对于标准时空（史瓦西度规和克尔度规）和非标准时空（例如史瓦西度规加非经典扰动或多重黑洞度规）的研究，都需要能够处理一般度量的广义算法。最常用的显式数值方案（例如，龙格-库塔）无法在临界点产生高度精确的结果，例如，在接近事件视界的区域，重力导致时空的极端曲率，在可接受的计算成本。本文描述了在任意给定的3 + 1分裂时空中对类时（大质量粒子）和零（光子）测地线进行数值积分的一种广义算法。基于底层哈密顿量的守恒，我们引入了一种新的完全节能的隐式积分格式，并将其与标准的四阶龙格-库塔显式格式和隐式中点格式的性质进行了比较。我们测试了三种格式在Schwarzschild和Kerr时空中粒子和光子轨道解析解的数值性能。我们还证明了我们的框架在处理更多奇异度量方面的通用性，例如莫里斯-索恩虫洞和量子摄动史瓦西黑洞。本文还从未来扩展到更复杂的粒子动力学的角度讨论了广义方法，例如，添加作用于带电粒子的洛伦兹力，这允许在GRMHD模拟中进行测试粒子诊断。
The numerical integration of particle trajectories in curved spacetimes is fundamental for obtaining realistic models of the particle dynamics around massive compact objects such as black holes and neutron stars. Generalized algorithms capable of handling generic metrics are required for studies of both standard (Schwarzschild and Kerr metrics) and nonstandard (e.g., Schwarzschild metric plus nonclassical perturbations or multiple black hole metrics) spacetimes. The most commonly employed explicit numerical schemes (e.g., Runge–Kutta) are incapable of producing highly accurate results at critical points, e.g., in the regions close to the event horizon where gravity causes extreme curvature of the spacetime, at an acceptable computational cost. Here we describe a generalized algorithm for the numerical integration of time-like (massive particles) and null (photons) geodesics in any given 3 + 1 split spacetime. We introduce a new, exactly energy-conserving implicit integration scheme based on the preservation of the underlying Hamiltonian, and we compare its properties with a standard fourth-order Runge–Kutta explicit scheme and an implicit midpoint scheme. We test the numerical performance of the three schemes against analytic solutions of particle and photon orbits in Schwarzschild and Kerr spacetimes. We also prove the versatility of our framework in handling more exotic metrics, such as Morris–Thorne wormholes and quantum-perturbed Schwarzschild black holes. The generalized approach is also discussed in the perspective of future extensions to more complex particle dynamics, e.g., the addition of the Lorentz force acting on charged particles, which allows for test particle diagnostics in GRMHD simulations.