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 »
- Authors:
-
[1];
[1];
[2];
- KU Leuven (Belgium)
- Princeton Univ., NJ (United States)
- 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:
- Accepted Manuscript
- Journal Name:
- The Astrophysical Journal. Supplement Series (Online)
- Additional Journal Information:
- Journal Name: The Astrophysical Journal. Supplement Series (Online); 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},
journal = {The Astrophysical Journal. Supplement Series (Online)},
number = 1,
volume = 237,
place = {United States},
year = {Tue Jul 03 00:00:00 EDT 2018},
month = {Tue Jul 03 00:00:00 EDT 2018}
}
Free Publicly Available Full Text
Publisher's Version of Record
Other availability
Search WorldCat to find libraries that may hold this journal
Cited by: 29 works
Citation information provided by
Web of Science
Web of Science
Save to My Library
You must Sign In or Create an Account in order to save documents to your library.
Works referenced in this record:
Three-dimensional general relativistic radiation magnetohydrodynamical simulation of super-Eddington accretion, using a new code harmrad with M1 closure
journal, May 2014
- McKinney, Jonathan C.; Tchekhovskoy, Alexander; Sadowski, Aleksander
- Monthly Notices of the Royal Astronomical Society, Vol. 441, Issue 4
GeodesicViewer – A tool for exploring geodesics in the theory of relativity
journal, February 2010
- Müller, Thomas; Grave, Frank
- Computer Physics Communications, Vol. 181, Issue 2
Wormholes in spacetime and their use for interstellar travel: A tool for teaching general relativity
journal, May 1988
- Morris, Michael S.; Thorne, Kip S.
- American Journal of Physics, Vol. 56, Issue 5
A Comprehensive Comparison of Relativistic Particle Integrators
journal, March 2018
- Ripperda, B.; Bacchini, F.; Teunissen, J.
- The Astrophysical Journal Supplement Series, Vol. 235, Issue 1
Understanding Modern Molecular Dynamics: Techniques and Applications
journal, January 2000
- Tuckerman, Mark E.; Martyna, Glenn J.
- The Journal of Physical Chemistry B, Vol. 104, Issue 2
Spectra of black hole accretion models of ultraluminous X-ray sources
journal, April 2017
- Narayan, Ramesh; Sa̧dowski, Aleksander; Soria, Roberto
- Monthly Notices of the Royal Astronomical Society, Vol. 469, Issue 3
Simulations of relativistic collisionless shocks: shock structure and particle acceleration
conference, January 2005
- Spitkovsky, Anatoly
- ASTROPHYSICAL SOURCES OF HIGH ENERGY PARTICLES AND RADIATION, AIP Conference Proceedings
RAPTOR: I. Time-dependent radiative transfer in arbitrary spacetimes★
journal, May 2018
- Bronzwaer, T.; Davelaar, J.; Younsi, Z.
- Astronomy & Astrophysics, Vol. 613
ipole – semi-analytic scheme for relativistic polarized radiative transport
journal, December 2017
- Mościbrodzka, M.; Gammie, C. F.
- Monthly Notices of the Royal Astronomical Society, Vol. 475, Issue 1
Viewing the Shadow of the Black Hole at the Galactic Center
journal, January 2000
- Falcke, Heino; Melia, Fulvio; Agol, Eric
- The Astrophysical Journal, Vol. 528, Issue 1
General relativistic radiative transfer code in rotating black hole space–time: ARTIST
journal, October 2016
- Takahashi, Rohta; Umemura, Masayuki
- Monthly Notices of the Royal Astronomical Society, Vol. 464, Issue 4
Beyond the event horizon or altogether without it?
journal, March 2017
- Lobanov, Andrei
- Nature Astronomy, Vol. 1, Issue 3
GYOTO: a new general relativistic ray-tracing code
journal, October 2011
- Vincent, F. H.; Paumard, T.; Gourgoulhon, E.
- Classical and Quantum Gravity, Vol. 28, Issue 22
The Mathematical Theory of Black Holes
book, January 1984
- Chandrasekhar, S.
- General Relativity and Gravitation
Modelling the black hole silhouette in Sagittarius A* with ion tori
journal, July 2012
- Straub, O.; Vincent, F. H.; Abramowicz, M. A.
- Astronomy & Astrophysics, Vol. 543
Reconnection and particle acceleration in interacting flux ropes I. Magnetohydrodynamics and test particles in 2.5D
journal, February 2017
- Ripperda, B.; Porth, O.; Xia, C.
- Monthly Notices of the Royal Astronomical Society
Higher order explicit symmetric integrators for inseparable forms of coordinates and momenta
journal, April 2016
- Liu, Lei; Wu, Xin; Huang, Guoqing
- Monthly Notices of the Royal Astronomical Society, Vol. 459, Issue 2
The black hole accretion code
journal, May 2017
- Porth, Oliver; Olivares, Hector; Mizuno, Yosuke
- Computational Astrophysics and Cosmology, Vol. 4, Issue 1
An Extension of the Athena++ code Framework for Grmhd Based on Advanced Riemann Solvers and Staggered-Mesh Constrained Transport
journal, August 2016
- White, Christopher J.; Stone, James M.; Gammie, Charles F.
- The Astrophysical Journal Supplement Series, Vol. 225, Issue 2
Particle acceleration and energy conservation in particle in cell simulations
journal, July 2011
- Lapenta, Giovanni; Markidis, Stefano
- Physics of Plasmas, Vol. 18, Issue 7
Imaging the Schwarzschild-radius-scale Structure of M87 with the Event Horizon Telescope Using Sparse Modeling
journal, March 2017
- Akiyama, Kazunori; Kuramochi, Kazuki; Ikeda, Shiro
- The Astrophysical Journal, Vol. 838, Issue 1
Why is Boris algorithm so good?
journal, August 2013
- Qin, Hong; Zhang, Shuangxi; Xiao, Jianyuan
- Physics of Plasmas, Vol. 20, Issue 8
Jet-Launching Structure Resolved Near the Supermassive Black Hole in M87
journal, September 2012
- Doeleman, S. S.; Fish, V. L.; Schenck, D. E.
- Science, Vol. 338, Issue 6105
Energy conserving schemes for the simulation of musical instrument contact dynamics
journal, March 2015
- Chatziioannou, Vasileios; van Walstijn, Maarten
- Journal of Sound and Vibration, Vol. 339
Studying null and time-like geodesics in the classroom
journal, March 2011
- Müller, Thomas; Frauendiener, Jörg
- European Journal of Physics, Vol. 32, Issue 3
Structure-preserving second-order integration of relativistic charged particle trajectories in electromagnetic fields
journal, May 2017
- Higuera, A. V.; Cary, J. R.
- Physics of Plasmas, Vol. 24, Issue 5
Observable Emission Features of Black Hole GRMHD Jets on Event Horizon Scales
journal, August 2017
- Pu, Hung-Yi; Wu, Kinwah; Younsi, Ziri
- The Astrophysical Journal, Vol. 845, Issue 2
Escape of photons from two fixed extreme Reissner-Nordström black holes
journal, November 2008
- Alonso, Daniel; Ruiz, Antonia; Sánchez-Hernández, Manuel
- Physical Review D, Vol. 78, Issue 10
A Ray-Tracing Algorithm for Spinning Compact Object Spacetimes with Arbitrary Quadrupole Moments. ii. Neutron Stars
journal, June 2012
- Bauböck, Michi; Psaltis, Dimitrios; Özel, Feryal
- The Astrophysical Journal, Vol. 753, Issue 2
An Improved Distance and mass Estimate for sgr a* from a Multistar Orbit Analysis
journal, October 2016
- Boehle, A.; Ghez, A. M.; Schödel, R.
- The Astrophysical Journal, Vol. 830, Issue 1
Photon Rings Around kerr and Kerr-Like Black Holes
journal, October 2013
- Johannsen, Tim
- The Astrophysical Journal, Vol. 777, Issue 2
Testing the No-Hair Theorem with Observations in the Electromagnetic Spectrum. i. Properties of a Quasi-Kerr Spacetime
journal, May 2010
- Johannsen, Tim; Psaltis, Dimitrios
- The Astrophysical Journal, Vol. 716, Issue 1
Testing the No-Hair Theorem with Observations in the Electromagnetic Spectrum. iii. Quasi-Periodic Variability
journal, December 2010
- Johannsen, Tim; Psaltis, Dimitrios
- The Astrophysical Journal, Vol. 726, Issue 1
Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics
journal, September 1963
- Kerr, Roy P.
- Physical Review Letters, Vol. 11, Issue 5
A periodic table for black hole orbits
journal, May 2008
- Levin, Janna; Perez-Giz, Gabe
- Physical Review D, Vol. 77, Issue 10
Einstein rings as a tool for estimating distances and the mass of a Schwarzschild black hole
journal, June 2008
- Müller, Thomas
- Physical Review D, Vol. 77, Issue 12
A Ray-Tracing Algorithm for Spinning Compact Object Spacetimes with Arbitrary Quadrupole Moments. i. Quasi-Kerr Black Holes
journal, December 2011
- Psaltis, Dimitrios; Johannsen, Tim
- The Astrophysical Journal, Vol. 745, Issue 1
A QUANTITATIVE TEST OF THE NO-HAIR THEOREM WITH Sgr A* USING STARS, PULSARS, AND THE EVENT HORIZON TELESCOPE
journal, February 2016
- Psaltis, Dimitrios; Wex, Norbert; Kramer, Michael
- The Astrophysical Journal, Vol. 818, Issue 2
ODYSSEY : A PUBLIC GPU-BASED CODE FOR GENERAL RELATIVISTIC RADIATIVE TRANSFER IN KERR SPACETIME
journal, March 2016
- Pu, Hung-Yi; Yun, Kiyun; Younsi, Ziri
- The Astrophysical Journal, Vol. 820, Issue 2
Stability of a Schwarzschild Singularity
journal, November 1957
- Regge, Tullio; Wheeler, John A.
- Physical Review, Vol. 108, Issue 4
Symmetric integrator for nonintegrable Hamiltonian relativistic systems
journal, December 2012
- Seyrich, Jonathan; Lukes-Gerakopoulos, Georgios
- Physical Review D, Vol. 86, Issue 12
Spherical Photon Orbits Around a Kerr Black Hole
journal, November 2003
- Teo, Edward
- General Relativity and Gravitation, Vol. 35, Issue 11
Astrophysical imaging of Kerr black holes with scalar hair
journal, October 2016
- Vincent, F. H.; Gourgoulhon, E.; Herdeiro, C.
- Physical Review D, Vol. 94, Issue 8
Shadows of Bonnor black dihole by chaotic lensing
journal, March 2018
- Wang, Mingzhi; Chen, Songbai; Jing, Jiliang
- Physical Review D, Vol. 97, Issue 6
New method for shadow calculations: Application to parametrized axisymmetric black holes
journal, October 2016
- Younsi, Ziri; Zhidenko, Alexander; Rezzolla, Luciano
- Physical Review D, Vol. 94, Issue 8
Understanding Modern Molecular Dynamics: Techniques and Applications
journal, July 2001
- Tuckerman, Mark E.; Martyna, Glenn J.
- The Journal of Physical Chemistry B, Vol. 105, Issue 31
Studying null and time-like geodesics in the classroom
text, January 2011
- Müller, Thomas; Frauendiener, Jörg
- arXiv
GYOTO: a new general relativistic ray-tracing code
text, January 2011
- Vincent, Frederic H.; Paumard, Thibaut; Gourgoulhon, Eric
- arXiv
A Symmetric Integrator for non-integrable Hamiltonian Relativistic Systems
text, January 2012
- Seyrich, Jonathan; Lukes-Gerakopoulos, Georgios
- arXiv
Jet Launching Structure Resolved Near the Supermassive Black Hole in M87
text, January 2012
- Doeleman, Sheperd S.; Fish, Vincent L.; Schenck, David E.
- arXiv
Energy conserving schemes for the simulation of musical instrument contact dynamics
text, January 2015
- Chatziioannou, Vasileios; van Walstijn, Maarten
- arXiv
A Quantitative Test of the No-Hair Theorem with Sgr A* using stars, pulsars, and the Event Horizon Telescope
text, January 2015
- Psaltis, Dimitrios; Wex, Norbert; Kramer, Michael
- arXiv
An Extension of the Athena++ Code Framework for GRMHD Based on Advanced Riemann Solvers and Staggered-Mesh Constrained Transport
text, January 2015
- White, Christopher J.; Stone, James M.; Gammie, Charles F.
- arXiv
Odyssey: A Public GPU-Based Code for General-Relativistic Radiative Transfer in Kerr Spacetime
text, January 2016
- Pu, Hung-Yi; Yun, Kiyun; Younsi, Ziri
- arXiv
A new method for shadow calculations: application to parameterised axisymmetric black holes
text, January 2016
- Younsi, Ziri; Zhidenko, Alexander; Rezzolla, Luciano
- arXiv
Reconnection and particle acceleration in interacting flux ropes I. Magnetohydrodynamics and test particles in 2.5D
text, January 2016
- Ripperda, Bart; Porth, Oliver; Xia, Chun
- arXiv
Observable Emission Features of Black Hole GRMHD Jets on Event Horizon Scales
text, January 2017
- Pu, Hung-Yi; Wu, Kinwah; Younsi, Ziri
- arXiv
Shadows of Bonnor black dihole by chaotic lensing
text, January 2017
- Wang, Mingzhi; Chen, Songbai; Jing, Jiliang
- arXiv
RAPTOR I: Time-dependent radiative transfer in arbitrary spacetimes
text, January 2018
- Bronzwaer, Thomas; Davelaar, Jordy; Younsi, Ziri
- arXiv
Viewing the Shadow of the Black Hole at the Galactic Center
text, January 1999
- Falcke, Heino; Melia, Fulvio; Agol, Eric
- arXiv
Modelling the black hole silhouette in Sagittarius A* with ion tori
journal, July 2012
- Straub, O.; Vincent, F. H.; Abramowicz, M. A.
- Astronomy & Astrophysics, Vol. 543
Studying null and time-like geodesics in the classroom
journal, March 2011
- Müller, Thomas; Frauendiener, Jörg
- European Journal of Physics, Vol. 32, Issue 3
Observable Emission Features of Black Hole GRMHD Jets on Event Horizon Scales
journal, August 2017
- Pu, Hung-Yi; Wu, Kinwah; Younsi, Ziri
- The Astrophysical Journal, Vol. 845, Issue 2
Imaging the Schwarzschild-radius-scale Structure of M87 with the Event Horizon Telescope using Sparse Modeling
text, January 2017
- Akiyama, Kazunori; Kuramochi, Kazuki; Ikeda, Shiro
- arXiv
A comprehensive comparison of relativistic particle integrators
text, January 2017
- Ripperda, Bart; Bacchini, Fabio; Teunissen, Jannis
- arXiv
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
- Bacchini, F.; Ripperda, B.; Porth, O.
- The Astrophysical Journal Supplement Series, Vol. 240, Issue 2
General-relativistic Resistive Magnetohydrodynamics with Robust Primitive-variable Recovery for Accretion Disk Simulations
journal, September 2019
- Ripperda, B.; Bacchini, F.; Porth, O.
- The Astrophysical Journal Supplement Series, Vol. 244, Issue 1
The Event Horizon General Relativistic Magnetohydrodynamic Code Comparison Project
journal, August 2019
- Porth, Oliver; Chatterjee, Koushik; Narayan, Ramesh
- The Astrophysical Journal Supplement Series, Vol. 243, Issue 2
A Novel Energy-conserving Scheme for Eight-dimensional Hamiltonian Problems
journal, December 2019
- Hu, Shiyang; Wu, Xin; Huang, Guoqing
- The Astrophysical Journal, Vol. 887, Issue 2
The Event Horizon General Relativistic Magnetohydrodynamic Code Comparison Project
collection, January 2019
- Porth, Oliver; Chatterjee, Koushik; Narayan, Ramesh
- Maryland Shared Open Access Repository
Generalized, energy-conserving numerical simulations of particles in general relativity. II. Test particles in electromagnetic fields and GRMHD
text, January 2018
- Bacchini, Fabio; Ripperda, Bart; Porth, Oliver
- arXiv
The Event Horizon General Relativistic Magnetohydrodynamic Code Comparison Project
text, January 2019
- Porth, Oliver; Chatterjee, Koushik; Narayan, Ramesh
- arXiv
Generalized, Energy-conserving Numerical Simulations of Particles in General Relativity. II. Test Particles in Electromagnetic Fields and GRMHD
journal, February 2019
- Bacchini, F.; Ripperda, B.; Porth, O.
- The Astrophysical Journal Supplement Series, Vol. 240, Issue 2
Numerical methods for General Relativistic particles
text, January 2018
- Bacchini, Fabio; Ripperda, Bart; Sironi, Lorenzo
- arXiv
Structure preserving algorithms for simulation of linearly damped acoustic systems
text, January 2020
- Chatziioannou, Vasileios
- arXiv