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Title: 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 needed 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. In this work 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 additionally prove the versatility of our framework in handling more exotic metrics,more » such as Morris–Thorne wormholes and quantum-perturbed Schwarzschild black holes. The generalized method 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.« less

Authors:
ORCiD logo [1]; ORCiD logo [1]; ORCiD logo [2];  [3]
  1. KU Leuven (Belgium)
  2. Princeton Univ., NJ (United States)
  3. Columbia Univ., New York, NY (United States)
Publication Date:
Research Org.:
Columbia Univ., New York, NY (United States)
Sponsoring Org.:
USDOE Office of Science (SC), Fusion Energy Sciences (FES)
OSTI Identifier:
1542035
Grant/Contract Number:  
SC0016542
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
The Astrophysical Journal. Supplement Series (Online)
Additional Journal Information:
Journal Volume: 237; Journal Issue: 1; Journal ID: ISSN 1538-4365
Publisher:
American Astronomical Society/IOP
Country of Publication:
United States
Language:
English
Subject:
79 ASTRONOMY AND ASTROPHYSICS; gravitation; methods: numerical; relativistic processes; stars: black holes; stars: imaging

Citation Formats

Bacchini, F., Ripperda, B., Chen, A. Y., and Sironi, L. Generalized, Energy-conserving Numerical Simulations of Particles in General Relativity. I. Time-like and Null Geodesics. United States: N. p., 2018. Web. doi:10.3847/1538-4365/aac9ca.
Bacchini, F., Ripperda, B., Chen, A. Y., & Sironi, L. Generalized, Energy-conserving Numerical Simulations of Particles in General Relativity. I. Time-like and Null Geodesics. United States. https://doi.org/10.3847/1538-4365/aac9ca
Bacchini, F., Ripperda, B., Chen, A. Y., and Sironi, L. Tue . "Generalized, Energy-conserving Numerical Simulations of Particles in General Relativity. I. Time-like and Null Geodesics". United States. https://doi.org/10.3847/1538-4365/aac9ca. https://www.osti.gov/servlets/purl/1542035.
@article{osti_1542035,
title = {Generalized, Energy-conserving Numerical Simulations of Particles in General Relativity. I. Time-like and Null Geodesics},
author = {Bacchini, F. and Ripperda, B. and Chen, A. Y. and Sironi, L.},
abstractNote = {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 needed 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. In this work 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 additionally prove the versatility of our framework in handling more exotic metrics, such as Morris–Thorne wormholes and quantum-perturbed Schwarzschild black holes. The generalized method 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.},
doi = {10.3847/1538-4365/aac9ca},
url = {https://www.osti.gov/biblio/1542035}, journal = {The Astrophysical Journal. Supplement Series (Online)},
issn = {1538-4365},
number = 1,
volume = 237,
place = {United States},
year = {2018},
month = {7}
}

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Cited by: 4 works
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Works referenced in this record:

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Studying null and time-like geodesics in the classroom
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A QUANTITATIVE TEST OF THE NO-HAIR THEOREM WITH Sgr A* USING STARS, PULSARS, AND THE EVENT HORIZON TELESCOPE
journal, February 2016


Observable Emission Features of Black Hole GRMHD Jets on Event Horizon Scales
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Testing the No-Hair Theorem with Observations in the Electromagnetic Spectrum. iii. Quasi-Periodic Variability
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Photon Rings Around kerr and Kerr-Like Black Holes
journal, October 2013


Escape of photons from two fixed extreme Reissner-Nordström black holes
journal, November 2008


A Ray-Tracing Algorithm for Spinning Compact Object Spacetimes with Arbitrary Quadrupole Moments. ii. Neutron Stars
journal, June 2012


An Improved Distance and mass Estimate for sgr a* from a Multistar Orbit Analysis
journal, October 2016


Photon Rings Around kerr and Kerr-Like Black Holes
journal, October 2013


Testing the No-Hair Theorem with Observations in the Electromagnetic Spectrum. iii. Quasi-Periodic Variability
journal, December 2010


A periodic table for black hole orbits
journal, May 2008


A QUANTITATIVE TEST OF THE NO-HAIR THEOREM WITH Sgr A* USING STARS, PULSARS, AND THE EVENT HORIZON TELESCOPE
journal, February 2016


ODYSSEY : A PUBLIC GPU-BASED CODE FOR GENERAL RELATIVISTIC RADIATIVE TRANSFER IN KERR SPACETIME
journal, March 2016


Stability of a Schwarzschild Singularity
journal, November 1957


Symmetric integrator for nonintegrable Hamiltonian relativistic systems
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Spherical Photon Orbits Around a Kerr Black Hole
journal, November 2003


Astrophysical imaging of Kerr black holes with scalar hair
journal, October 2016


Shadows of Bonnor black dihole by chaotic lensing
journal, March 2018


New method for shadow calculations: Application to parametrized axisymmetric black holes
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    Works referencing / citing this record:

    Generalized, Energy-conserving Numerical Simulations of Particles in General Relativity. II. Test Particles in Electromagnetic Fields and GRMHD
    journal, February 2019


    General-relativistic Resistive Magnetohydrodynamics with Robust Primitive-variable Recovery for Accretion Disk Simulations
    journal, September 2019


    The Event Horizon General Relativistic Magnetohydrodynamic Code Comparison Project
    journal, August 2019


    A Novel Energy-conserving Scheme for Eight-dimensional Hamiltonian Problems
    journal, December 2019