Generalized, Energy-conserving Numerical Simulations of Particles in General Relativity. I. Time-like and Null Geodesics

Abstract

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 spacetimes (Schwarzschild and Kerr metrics) and non-standard spacetimes (e.g. Schwarzschild metric plus non-classical perturbations or multiple black hole metrics). 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. *

The Event Horizon General Relativistic Magnetohydrodynamic Code Comparison Project

“…More details on the various •  B preserving schemes and their implementation in BHAC can be found in Olivares et al (2018Olivares et al ( , 2019. Originally designed to study BH accretion in ideal MHD, BHAC has been extended to incorporate nuclear equations of state, neutrino leakage, charged and purely geodetic test particles (Bacchini et al 2018(Bacchini et al , 2019, and nonblack-hole fully numerical metrics. In addition, a nonideal resistive GRMHD module has been developed (e.g., Ripperda et al 2019aRipperda et al , 2019b).…”

General-relativistic Resistive Magnetohydrodynamics with Robust Primitive-variable Recovery for Accretion Disk Simulations

“…Combined with the AMR strategy in BHAC, the new GRRMHD algorithm allows for resolving both the global accretion features governed by the MRI-induced turbulence, and the dissipative reconnection physics that are conjectured to be responsible for non-thermal radiation. These non-thermal processes can be subsequently modeled in BHAC with first-principle approaches-for example, the newly implemented general-relativistic (charged) particle module (Bacchini et al 2018(Bacchini et al , 2019Ripperda et al 2018).…”

A Novel Energy-conserving Scheme for Eight-dimensional Hamiltonian Problems

“…The radius r = 40 remains invariant, but the orbit is a spherical orbit rather than a circular orbit because of the spin of body 1 leading to the precession of orbits. This spherical orbit with the angular frequency ω so = ω o + ∂H so /∂L z = 0.0038 is an orbit in a eight-dimensional phase space and so our new EC method rather than the energy-conserving method of Bacchini et al (2018a) becomes useful. The four algorithms show the preference of such a spherical orbit during a short integration time in Figure 5(c).…”

“…The discretization of each partial derivative of the Hamiltonian with respect to one of the position and momentum variables is the average of four Hamiltonian difference terms. Recently, an extension to a 6-dimensional Hamiltonian system with three degrees of freedom was given by Bacchini et al (2018a). Here, each component of the Hamiltonian gradient is replaced with the average of six Hamiltonian difference terms.…”

“…Panels (e) and (f): quasi-spherical orbit with the inclusion of spin-orbit couplings and spin-spin effects. EC in (a) and (b) denotes the energy-conserving method ofBacchini et al (2018a) because this circular orbit is considered in a six-dimensional phase space, but it is our new method in (c)-(f) because the spherical and quasi-spherical orbits are considered in a eight-dimensional phase space. The time step is h = 1. position errors for the four algorithms solving the eccentrical orbit 11 in…”