Numerical Modeling of High-energy Transients from Black Holes and Neutron Stars Citation Kim, Yoonsoo (2025) Numerical Modeling of High-energy Transients from Black Holes and Neutron Stars. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/8skm-3x17. https://resolver.caltech.edu/CaltechTHESIS:04162025-220303394 Abstract Along with recent breakthroughs in relativistic astrophysics and multi-messenger astronomy, theoretical studies on compact objects and the dynamics of relativistic matter surrounding them have a growing significance. General relativistic approaches are required to properly describe astrophysical phenomena taking place in a strong gravity regime, yet the high complexity and nonlinearity of the equations governing those systems compel numerical approaches. In this thesis, we develop a computational method for and present global numerical simulations of relativistic plasma around compact objects, particularly focusing on high-energy electromagnetic transients originating from black holes and neutron stars. Our works include a new hybrid numerical scheme for modeling force-free magnetospheres of compact objects, large-scale simulations of a spinning black hole immersed in a magnetized wind, and magnetospheric transients from a merging black hole--neutron star binary. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: computational astrophysics; numerical relativity; relativistic astrophysics; black holes; neutron stars; plasma astrophysics Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Physics Thesis Availability: Public (worldwide access) Research Advisor(s): Most, Elias R. (advisor) Teukolsky, Saul A. (co-advisor) Thesis Committee: Teukolsky, Saul A. (chair) Phinney, E. Sterl Graham, Matthew J. Most, Elias R. Defense Date: 8 April 2025 Funders: Funding Agency Grant Number Sherman Fairchild Foundation UNSPECIFIED National Science Foundation PHY-2309211 National Science Foundation PHY-2309231 National Science Foundation OAC-2209656 Record Number: CaltechTHESIS:04162025-220303394 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:04162025-220303394 DOI: 10.7907/8skm-3x17 Related URLs: URL URL Type Description https://doi.org/10.1103/PhysRevD.109.123019 DOI Article adapted for Ch 2 https://doi.org/10.1103/PhysRevD.111.083025 DOI Article adapted for Ch 3 https://doi.org/10.3847/2041-8213/adbff9 DOI Article adapted for Ch 4 https://doi.org/10.5281/zenodo.4290404 DOI SpECTRE code ORCID: Author ORCID Kim, Yoonsoo 0000-0002-4305-602 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17159 Collection: CaltechTHESIS Deposited By: Yoonsoo Kim Deposited On: 02 May 2025 21:16 Last Modified: 29 Jul 2025 19:14 Thesis Files PDF - Final Version See Usage Policy. 29MB Repository Staff Only: item control page

Numerical Modeling of High-energy Transients from Black Holes and Neutron Stars Thesis by Yoonsoo Kim In Partial Fulfillment of the Requirements for the degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2025 Defended Apr 8 2025 ii © 2025 Yoonsoo Kim ORCID: 0000-0002-4305-6026 All rights reserved iii ACKNOWLEDGEMENTS I would like to thank my advisors Elias Most and Saul Teukolsky for their guidance and support throughout my PhD studies. Elias was always willing to offer unstinting help ranging from elementary physics questions to practical research skills, and I am taking a leaf out of his book for an untiring enthusiasm towards newly emerging science. I have also greatly benefited from Saul’s concise but always incisive comments and advice for many years. One of the most valuable lessons from Saul was the easiest way to be an expert: to fail in every possible way. I cannot overstate how fortunate and grateful I have been for having Elias and Saul, passionate scholars and considerate mentors, as my doctoral advisors. My early years in grad school were enriched by the mentorship from Nils Deppe, who taught me how to rigorously assess numerical methodologies and properly apply them to problems in computational physics. I would also like to express my gratitude to Mark Scheel, who instructed me how to split a long-term technical project into smaller logical units that can be tested and tackled individually. Thank you Nils and Mark. My special thanks go to JoAnn Boyd, the world’s best admin, for her help, support, and generosity in office candies, matched only by her satiric humor. It was my genuine pleasure to share spacetime trajectories at Caltech with Sarah Habib, Isaac Legred, Kyle Nelli, Nicholas Rui, Kiran Shila, and Rhiannon Udall, not only as colleagues at the workplace but also as friends who have livened up my personal life in Pasadena. To our postdocs Mike Pajkos and Nils Vu, thank you for all your help, encouraging words, and office companionship; I wish you all the best. I also thank the members of Caltech TAPIR and the SXS Collaboration for helpful discussions, collaborative efforts, and, most of all, supportive atmosphere. To my family, this could not have been done without your unconditional support and encouragement. Thank you. Lastly, I would like to thank everyone—within which this margin is too narrow to contain—who made my years at Caltech happy and memorable. Y.K. acknowledges support from National Science Foundation, Sherman Fairchild Foundation, and Korea Foundation for Advanced Studies. Figures in this article were produced using matplotlib (The Matplotlib Development Team, 2025), numpy (Harris et al., 2020), and scipy (Virtanen et al., 2020) packages. v ABSTRACT Along with recent breakthroughs in relativistic astrophysics and multi-messenger astronomy, theoretical studies on compact objects and the dynamics of relativistic matter surrounding them have a growing significance. General relativistic approaches are required to properly describe astrophysical phenomena taking place in a strong gravity regime, yet the high complexity and nonlinearity of the equations governing those systems compel numerical approaches. In this thesis, we develop a computational method for and present global numerical simulations of relativistic plasma around compact objects, particularly focusing on high-energy electromagnetic transients originating from black holes and neutron stars. Our works include a new hybrid numerical scheme for modeling force-free magnetospheres of compact objects, large-scale simulations of a spinning black hole immersed in a magnetized wind, and magnetospheric transients from a merging black hole–neutron star binary. vi PUBLISHED CONTENT AND CONTRIBUTIONS Kim, Yoonsoo, Elias R. Most, William Throwe, et al. (2024). “General relativistic force-free electrodynamics with a discontinuous Galerkin-finite difference hybrid method”. In: Physical Review D 109.12, p. 123019. doi: 10.1103/PhysRevD. 109.123019. arXiv: 2404.01531. url: https://journals.aps.org/prd/ abstract/10.1103/PhysRevD.109.123019. Y.K. is a lead developer of the code module, ran test problems, and wrote the majority of the manuscript. Kim, Yoonsoo and Elias R. Most (2025). “General relativistic magnetized BondiHoyle-Lyttleton accretion with a spin-field misalignment: Jet nutation, polarity reversals, and Magnus drag”. In: Physical Review D 111 (8), p. 083025. doi: 10.1103/PhysRevD.111.083025. url: https://link.aps.org/doi/10. 1103/PhysRevD.111.083025. Y.K. developed several code functionalities required for the study, performed simulations, analyzed the data, and wrote the majority of the manuscript. Kim, Yoonsoo, Elias R. Most, Andrei M. Beloborodov, et al. (2024). “Black Hole Pulsars and Monster Shocks as Outcomes of Black Hole–Neutron Star Mergers”. In: Astrophysical Journal Letters 982.2, p. L54. doi: 10.3847/2041- 8213/ adbff9. url: https://iopscience.iop.org/article/10.3847/20418213/adbff9. Y.K. performed simulations, analyzed the data, and wrote the majority of the manuscript. Deppe, Nils et al. (Mar. 2025). SpECTRE v2025.03.17. Version 2025.03.17. doi: 10.5281/zenodo.15040490. url: https://spectre-code.org. Y.K. has been a developer of the code since 2021. vii TABLE OF CONTENTS Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Published Content and Contributions . . . . . . . . . . . . . . . . . . . . . . vi Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi List of Illustrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x Chapter I: Prologue: Astrophysical plasmas in strong gravity . . . . . . . . . 1 Chapter II: A hybrid numerical method for general relativistic force-free electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 General relativistic force-free electrodynamics . . . . . . . . . . . . 7 2.3 Numerical implementation . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Chapter III: General relativistic magnetized Bondi-Hoyle-Lyttleton accretion . 43 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 3.3 Fiducial model: Jet launching from horizontally magnetized wind . . 53 3.4 Dependence on accretion parameters . . . . . . . . . . . . . . . . . 68 3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.A Extraction radius of density-related integral quantities . . . . . . . . 84 Chapter IV: Magnetospheric transients from black hole–neutron star merger . 86 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 4.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.3 Monster shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 4.4 Transient black hole pulsar . . . . . . . . . . . . . . . . . . . . . . . 93 4.5 Electromagnetic transient . . . . . . . . . . . . . . . . . . . . . . . 108 4.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Appendix A: Spherical Kerr-Schild coordinates . . . . . . . . . . . . . . . . 141 Appendix B: The Wald magnetosphere solution . . . . . . . . . . . . . . . . 143 viii LIST OF ILLUSTRATIONS Number Page 2.1 Fast wave test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.2 Stationary Alfvén wave test . . . . . . . . . . . . . . . . . . . . . . 22 2.3 FFE breakdown test . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2.4 A half-cut illustration of the spherical grid used for black hole magnetosphere tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.5 Vacuum Wald problem: simulation snapshots . . . . . . . . . . . . . 27 2.6 Vacuum Wald problem: total magnetic flux through the upper hemisphere of the outer horizon . . . . . . . . . . . . . . . . . . . . . . . 28 2.7 Magnetospheric Wald problem . . . . . . . . . . . . . . . . . . . . . 29 2.8 A zoom-in view of the computational domain used for the pulsar magnetosphere tests . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.9 Aligned rotator test . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.10 Oblique rotator test: simulation snapshots . . . . . . . . . . . . . . . 35 2.11 Oblique rotator test: inclination angle dependence of the spin-down luminosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.12 Accuracy comparison between DG and FD methods for a periodic wave problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.13 Single-core wall-clock speedup of the 1D Alfvén wave problem for different fractions of elements using the FD grid . . . . . . . . . . . 39 3.1 Initial jet launching process in the 𝛽10 -𝜃 90 -𝑅200 simulation. . . . . . . 54 3.2 Jets and magnetic flux eruptions in GRMHD Bondi-Hoyle-Lyttleton accretion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.3 Time evolution of physical quantities for the representative model . . 59 3.4 A magnetic flux eruption event at 𝑡 = 16500𝑟 𝑔 /𝑐 . . . . . . . . . . . 61 3.5 A three-dimensional rendering of the simulation . . . . . . . . . . . 63 3.6 Distribution of the radial Poynting flux . . . . . . . . . . . . . . . . 64 3.7 Magnetic field reversal of the jets between three eruption epochs in the fiducial model . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.8 Time evolution of physical quantities for three models with 𝜃 𝐵 = 23.5◦ , 45◦ and 67.5◦ , all with 𝛽 = 10 and 𝑅𝑎 = 200 . . . . . . . . . . 69 ix 3.9 3.10 3.11 3.12 3.13 3.14 3.15 3.16 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 Duration of a magnetically arrested accretion epoch in units of the accretion timescale 𝜏𝑎 = 2000𝑟 𝑔 /𝑐 for the four models with 𝑅𝑎 = 200𝑟 𝑔 and 𝛽∞ = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . 70 Time series data for the 𝛽100 -𝜃 90 -𝑅200 , 𝛽10 -𝜃 90 -𝑅50 , and 𝛽10 -𝜃 90 -𝑅400 models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Distribution of the mass density 𝜌 in the meridional plane for the simulations 𝛽10 -𝜃 90 -𝑅50 and 𝛽10 -𝜃 90 -𝑅400 . . . . . . . . . . . . . . . 73 ¤ 2 − 𝐸¤ from all simulations . . . . . . . 75 Energy outflow power 𝑃 = 𝑀𝑐 ¤ 𝑀¤ BHL from all simulations . . . . 76 Energy conversion efficiency 𝜂 𝑀/ Magnus force in all simulations . . . . . . . . . . . . . . . . . . . . 78 Mass accretion rate extracted at different radii for the 𝛽10 -𝜃 90 -𝑅200 model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 The 𝑥ˆ component of the momentum drag and gravitational drag computed with different radii from the 𝛽10 -𝜃 90 -𝑅200 model . . . . . . . . 84 Non-disrupting merger of a BH–NS binary . . . . . . . . . . . . . . 91 Structure of the perturbed magnetosphere of the BH–NS binary 0.9 ms before merger for the aligned (𝜃 𝐵 = 0◦ ) model . . . . . . . . . . . 92 Monster shocks launched from BH–NS mergers . . . . . . . . . . . . 93 Post-merger magnetosphere of the remnant black hole having settled down to a rotating split monopole . . . . . . . . . . . . . . . . . . . 95 Angular velocity of magnetic field lines threading the apparent horizon for 𝜃 𝐵 = 0◦ simulation . . . . . . . . . . . . . . . . . . . . . . . 96 A spacetime diagram of the latitude of the post-merger magnetospheric BH current sheet . . . . . . . . . . . . . . . . . . . . . . . . 97 Time evolution of the current sheet inclination angle . . . . . . . . . 98 Balding and ring-down of the remnant BH . . . . . . . . . . . . . . 100 Imaginary part of 𝜙2(𝑙=1,𝑚=1) normalized with the magnitude of magnetic field for the aligned case 𝜃 𝐵 = 0◦ at 𝑡 − 𝑡 merger = 1.05 ms. . . . . 101 Pulsar-like striped wind from the remnant black hole . . . . . . . . . 103 Toroidal magnetic field of the striped wind |𝐵 𝜙 (𝑟)| . . . . . . . . . . 104 The dissipation luminosity from a BH pulsar 𝐿 𝐷 (𝑡) computed with an analytic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 x LIST OF TABLES Number Page 2.1 Simulation setup for 1D tests in Sec. 2.4.1 . . . . . . . . . . . . . . . 20 2.2 Convergence tests of different DG-𝑃 𝑁 schemes on the exact Wald solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.1 Models and parameters of the GRMHD simulations presented in Chatper 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 1 Chapter 1 PROLOGUE: ASTROPHYSICAL PLASMAS IN STRONG GRAVITY Compact objects such as black holes and neutron stars can harbor the strongest magnetic field in the Universe. When a compact object interacts with astrophysical plasma surrounding it, this extreme magnetic field can trigger a variety of observable electromagnetic bursts. Deciphering these electromagnetic transient signals provide a unique window for probing an unknown realm of physics in strong gravity and of dense matter, which cannot be reproduced in any terrestrial laboratories. Following the first detection of gravitational waves from a merging binary black hole (B. P. Abbott et al., 2016), the past decade has witnessed unprecedented advances in relativistic astrophysics, including the first multi-messenger observation of a binary neutron star merger (B. P. Abbott et al., 2017) and the first direct imaging of a black hole (Akiyama et al., 2019). However, many aspects of the high-energy astrophysical phenomena, such as fast radio bursts and gamma ray bursts, which are thought to be associated with compact objects and their interactions with the surrounding plasma via hydrodynamic or magnetic processes, still remain unclear. As we expect a larger number of and more precise observations of such electromagnetic bursts from upcoming next-generation detectors and space missions, improved theoretical models are required in order to properly interpret the observed features and to understand physical properties of compact objects and their host environment. Since the physics underlying these systems — general relativity, electromagnetism, hydrodynamics — are described with highly complex and nonlinear governing equations, numerical simulations on supercomputers are often drawn to study a particular scenario of our interest. Numerical relativity and computational relativistic astrophysics, products of scientific efforts to accurately and efficiently immitate those physical systems on computers, provides an ideal set of toolkits for this purpose. This thesis collects three independent studies on numerical modeling of astrophysical plasmas around compact objects. We develop a new computational method for and present two global, large-scale numerical simulations particularly focusing on highenergy electromagnetic transients originating from black holes and neutron stars. In Chapter 2, we present a new numerical method for relativistic electrodynamics. 2 The key idea is hybridizing the discontinuous Galerkin method with the shockcapturing finite difference method to selectively combine the merits of each approaches. This hybrid method achieves spectral convergence in smooth regions while robustly resolving sharp dissipative features such as current sheets and reconnection points. Our new approach can perform two to three times better efficiently than conventional simulation techniques, potentially up to ten times in an ideal setting. A black hole flying through a magnetized medium can be related to many astrophysical contexts, including common envelope phase, wind-fed X-ray binaries, or a recoiled remnant of a binary black hole merger in a gaseous environment. In Chapter 3, we present 3D general relativistic magnetohydrodynamics simulations of the Bondi-Hoyle-Lyttleton accretion onto a spinning black hole when the magnetic field of the incoming wind is inclined relative to the spin of the black hole. This study brings a detailed 3D picture of an accreting black hole in a magnetized wind, and sheds light on electromagnetic transients from compact binary merger remnants in a gaseous environment. When a compressive mode is excited within a compact object magnetosphere, the resulting outgoing waves can evolve into strongly radiative shockwaves. This particular mechanism is thought to produce monster shocks, the strongest shockwaves in the Universe, which can power electromagnetic bursts in various wavelengths. In Chapter 4, we show that strong disturbances in the circumbinary magnetosphere during a neutron star–black hole merger develop into monster shocks within milliseconds following the merger. The remnant black hole dissipates the post-merger magnetosphere via magnetic reconnection and launches a magnetized wind resembling that of pulsars. This study is the first self-contained demonstration of the monster shock formation as well as the emergence of a transient ‘black hole pulsar’ state from a compact binary merger, unveiling a novel type of shock-powered and reconnection-driven transient associated with merging compact objects. 3 Chapter 2 A HYBRID NUMERICAL METHOD FOR GENERAL RELATIVISTIC FORCE-FREE ELECTRODYNAMICS Kim, Yoonsoo et al. (2024). “General relativistic force-free electrodynamics with a discontinuous Galerkin-finite difference hybrid method”. In: Physical Review D 109.12, p. 123019. doi: 10.1103/PhysRevD.109.123019. arXiv: 2404.01531. url: https://journals.aps.org/prd/abstract/10.1103/PhysRevD. 109.123019. Relativistic plasmas around compact objects can sometimes be approximated as being force-free. In this limit, the plasma inertia is negligible and the overall dynamics is governed by global electric currents. We present a novel numerical approach for simulating such force-free plasmas, which allows for high accuracy in smooth regions as well as capturing dissipation in current sheets. Using a highorder accurate discontinuous Galerkin method augmented with a conservative finitedifference method, we demonstrate efficient global simulations of black hole and neutron star magnetospheres. In addition to a series of challenging test problems, we show that our approach can — depending on the physical properties of the system and the numerical implementation — be up to 10× more efficient than conventional simulations, with a speedup of 2–3× for most problems we consider in practice. 2.1 Introduction Compact objects such as neutron stars and black holes can feature some of the strongest magnetic fields in the universe. Under these conditions, the environments surrounding them can be filled with a highly conducting plasma. The plasma dynamics of these magnetospheres are thought to be responsible for several observable transients in the radio (Bochenek et al., 2020; Lyubarsky, 2020; Mahlmann, Philippov, Levinson, et al., 2022) and X-ray (Thompson and Duncan, 1995; Kaspi et al., 2003; Beloborodov, 2013; Archibald et al., 2017; Tavani et al., 2021) bands. While the description of emission processes fundamentally necessitates modeling the relevant kinetic scales (Philippov, Cerutti, et al., 2015; A. Y. Chen and Beloborodov, 2017), the available energy budget as well as the presence of any dissipative or emitting region inside the magnetosphere is a result of the bulk dynamics. It is this 4 latter aspect that our present work aims to advance. Since these scenarios are highly nonlinear, their effective description requires numerical approaches. The global dynamics of the plasma is usually modeled under several simplifying assumptions. In a very strongly magnetized magnetosphere, the inertia of the plasma can approximately be neglected (Uchida, 1997; Gruzinov, 1999). In this force-free electrodynamics (FFE) state, the evolution of the system is governed largely by bulk currents, obtained via an effective closure of the Maxwell equations. It is important to point out that the main assumption—neglecting plasma inertia—can break down, e.g., during shock formation, as well as the absence of physically meaningful dissipation in reconnection regions. In this regime, the closest extension of force-free electrodynamics is magnetohydrodynamical (MHD) models, retaining a single-component plasma rest-mass density. MHD studies of relativistic magnetospheres are not commonly employed.1 Instead, most studies adopt an FFE approach. Recent examples include applications to magnetar quakes (Bransgrove, Beloborodov, and Y. Levin, 2020), nonlinear steepening of Alfvén waves (Yuan, Beloborodov, A. Y. Chen, and Y. Levin, 2020; Yuan, Beloborodov, A. Y. Chen, Y. Levin, et al., 2022), magnetar giant flares (Parfrey, Beloborodov, and Hui, 2013; Mahlmann, Akgün, et al., 2019; Carrasco, Viganò, et al., 2019; Mahlmann, Philippov, Mewes, et al., 2023), outbursts from gravitational collapse of a neutron star (Lehner et al., 2012),2 and black hole and neutron star magnetospheres (Komissarov, 2004a; Spitkovsky, 2006; McKinney, 2006b; A. Y. Chen, Yuan, and Vasilopoulos, 2020; Carrasco and Shibata, 2020). Apart from isolated compact objects, force-free electrodynamics has also been employed in the context of jets from massive black hole mergers (Palenzuela, Garrett, et al., 2010; Palenzuela, Lehner, and Liebling, 2010) and potential electromagnetic precursor to gravitational wave events involving merger of compact objects (Alic et al., 2012; Palenzuela, Lehner, and Yoshida, 2010; Carrasco, Shibata, and Reula, 2021; Most and Philippov, 2020; Most and Philippov, 2022; Most and Philippov, 2023a; Most and Philippov, 2023b). Several approaches have been adopted in the literature for numerically solving the FFE equations. Most commonly, either unlimited finite-difference (Spitkovsky, 2006; Kalapotharakos and Contopoulos, 2009; Palenzuela, Garrett, et al., 2010; 1 See, e.g. Tchekhovskoy and Spitkovsky (2013) for a notable exception in neutron star magne- tospheres. 2 See also Nathanail, Most, and Rezzolla (2017) and Most, Nathanail, and Rezzolla (2018) for related studies in electrovacuum. 5 A. Y. Chen, Yuan, and Vasilopoulos, 2020; Carrasco and Reula, 2017) or conservative finite-volume schemes (Komissarov, 2002; Cho, 2005; Asano, Uchida, and Matsumoto, 2005; McKinney, 2006a; Yu, 2011; Etienne, Wan, et al., 2017; Most and Philippov, 2020; Mahlmann, Aloy, et al., 2021a) have been employed. These methods are robust, can easily capture strong gradients inside the magnetosphere, and work well with commonly employed mesh refinement techniques (Dubey et al., 2014). However, they come with a major drawback. Properly capturing wave solutions over long integration times, e.g., Alfvén waves (Yuan, Beloborodov, A. Y. Chen, and Y. Levin, 2020), requires a large number of grid points, especially when less accurate versions of finite-difference/volume schemes are being used. This prohibitively increases computational costs, especially for applications such as compact binary magnetospheres in which scale separations can span two orders of magnitude. On the other hand, spectral-type methods such as the pseudospectral method offer exponential convergence for smooth solutions, providing a maximum in accuracy over computational cost. Several studies have made use of spectral schemes to solve the FFE equations (Parfrey, Beloborodov, and Hui, 2012; Petri, 2012; Cao, L. Zhang, and Sun, 2016). One serious limitation of spectral methods is the appearance of unphysical oscillations (Gibbs phenomenon) near a discontinuity or a large gradient e.g. current sheets, which are naturally present in compact object magnetospheres. Remedying these numerical instabilities requires special treatments such as filtering or a limiting procedure (Hesthaven and Warburton, 2007). In addition, globally spectral methods are not easily parallelizable, making it difficult to simulate physical scenarios with large scale separations. To counteract this shortcoming, popular approaches in the literature focus on spectral element methods in which the computational domain is divided into non-overlapping spectral elements, communicating only with the directly neighboring elements through the element boundaries. This approach allows for highly parallelizable implementations while retaining the exponential convergence property for smooth solutions. A concrete example of this approach, a discontinuous Galerkin (DG) method, is gaining its popularity in computational fluid dynamics and astrophysics (e.g. Bugner et al., 2016; Zanotti, Fambri, Dumbser, and Hidalgo, 2015; Zanotti, Fambri, and Dumbser, 2015; Fambri et al., 2018; Reinarz et al., 2020; Deppe et al., 2022; Tichy et al., 2023; Dumbser, Zanotti, Gaburro, et al., 2024; Cernetic et al., 2024), as well as in FFE (Petri, 2015; Petri, 2016). While a DG scheme naturally permits a discontinuity at the element boundary, 6 without special care to suppress unphysical oscillations, it suffers from the same fate as globally spectral methods described above. Several strategies have been proposed in the DG literature, which are frequently referred to as limiters. Common types of DG limiters are implemented as direct manipulations on spectral coefficients, addition of artificial viscosity, or a flattening correction of the solution with respect to its average value within an element. We refer the reader to Zanotti, Fambri, Dumbser, and Hidalgo (2015) and Deppe, Hébert, et al. (2022) and references therein for the available types of DG limiters and related discussions. DG limiters currently are not particularly accurate or reliable compared with corresponding finite-volume or finite-difference techniques, especially for curved meshes or relativistic applications (Deppe, Hébert, et al., 2022). A recently developed alternative strategy is to supplement the DG evolution with a more robust sub-element discretization, which has been mostly chosen to be finite volume (e.g. Dumbser, Zanotti, Loubère, et al., 2014; Zanotti, Fambri, Dumbser, and Hidalgo, 2015; Zanotti, Fambri, and Dumbser, 2015; Fambri et al., 2018; Vilar, 2019; Núñez-de la Rosa and Munz, 2018; Rueda-Ramírez, Pazner, and Gassner, 2022; Maltsev et al., 2023). Motivated by the idea of the a posteriori finite-volume limiting approach of Dumbser, Zanotti, Loubère, et al. (2014), the discontinuous Galerkin-finite difference (DG-FD) hybrid method was introduced by Deppe, Hébert, et al. (2022); see also Deppe et al. (2022) and Legred et al. (2023) for applications to relativistic fluid dynamics simulations. Here we present a new numerical scheme and code for general-relativistic FFE simulations based on a discontinuous Galerkin discretization. Our motivation is twofold. First, we explore the suitability of the DG-FD hybrid approach to enable large-scale, parallel yet accurate numerical simulations, especially of compact binary magnetospheres. Second, since FFE on physical grounds has very localized regions of non-smoothness such as current sheets, these simulations serve as an ideal testbed to calibrate and assess the usefulness of the DG-FD hybrid approach. Our hybrid scheme also incorporates previously developed implicit-explicit time integration schemes by Pareschi and Russo (2005), which allows us to enforce a set of algebraic constraints present in the FFE system. This joint approach achieves high-order convergence in smooth regions while capturing discontinuous features such as magnetic reconnection points and current sheets. This chapter is organized as follows. In Sec. 2.2, we briefly review Maxwell’s equations in general relativity and introduce the formulation we adopt in this work. 7 We also discuss our strategy for maintaining the force-free conditions in simulations. In Sec. 2.3, we describe the numerical implementation of spatial discretization, time stepping, and the discontinuous Galerkin-finite difference hybrid solver. We present results from a set of test problems in Sec. 2.4, and conclude with a discussion of result in Sec. 2.5. In this chapter, we adopt geometrized (𝑐 = 𝐺 = 1) Heaviside-Lorentz units, where √ electric and magnetic fields have been rescaled by 1/ 4𝜋 compared to Gaussian units. We use the abstract index notation using latin indices (𝑎, 𝑏, · · · ) for spacetime tensors, but reserve {𝑖, 𝑗, 𝑘, . . .} for spatial tensors. We follow the sign convention of the Levi-Civita tensor from Misner, Thorne, and Wheeler (1973), √ (2.1) 𝜀 𝑎𝑏𝑐𝑑 = −𝑔 [𝑎𝑏𝑐𝑑] , where 𝑔 is the determinant of spacetime metric and [𝑎𝑏𝑐𝑑] = ±1 with [0123] = +1 is the flat-space antisymmetric symbol.3 2.2 General relativistic force-free electrodynamics We begin by outlining the mathematical description used to numerically study magnetospheric dynamics. This includes a general relativistic formulation of electrodynamics in a curved spacetime, which is then specialized to the force-free case: general relativistic force-free electrodynamics (GRFFE). 2.2.1 Electrodynamics The dynamics of electric and magnetic fields is governed by the Maxwell equations. In covariant form, they are given by ∇𝑏 𝐹 𝑎𝑏 = J 𝑎 (2.2) ∇𝑏 ∗𝐹 𝑎𝑏 = 0 (2.3) where 𝐹 𝑎𝑏 and ∗𝐹 𝑎𝑏 = 𝜀 𝑎𝑏𝑐𝑑 𝐹𝑐𝑑 /2 are the electromagnetic field tensor and its dual, and J 𝑎 is the electric 4-current density. For the standard 3+1 decomposition of the spacetime metric 𝑑𝑠2 = −𝛼2 𝑑𝑡 2 + 𝛾𝑖 𝑗 (𝑑𝑥 𝑖 + 𝛽𝑖 𝑑𝑡)(𝑑𝑥 𝑗 + 𝛽 𝑗 𝑑𝑡), (2.4) where 𝛼 is lapse, 𝛽𝑖 is the shift vector, and 𝛾𝑖 𝑗 is the spatial metric, the normal to spatial hypersurfaces is given by 𝑛𝑎 = (1/𝛼, −𝛽𝑖 /𝛼), 𝑛𝑎 = (−𝛼, 0). (2.5) 3 Note that an opposite sign convention is sometimes adopted in the literature (e.g. Palenzuela, Lehner, and Yoshida, 2010; Palenzuela, 2013) 8 In terms of the normal vector 𝑛𝑎 , electromagnetic field tensor 𝐹 𝑎𝑏 and its dual ∗𝐹 𝑎𝑏 can be decomposed as 𝐹 𝑎𝑏 = 𝑛𝑎 𝐸 𝑏 − 𝑛 𝑏 𝐸 𝑎 − 𝜀 𝑎𝑏𝑐𝑑 𝐵𝑐 𝑛 𝑑 , (2.6) ∗ 𝑎𝑏 (2.7) 𝐹 = −𝑛𝑎 𝐵 𝑏 + 𝑛 𝑏 𝐵𝑎 − 𝜀 𝑎𝑏𝑐𝑑 𝐸 𝑐 𝑛 𝑑 , where 𝑛𝑎 𝐸 𝑎 = 𝑛𝑎 𝐵𝑎 = 0. (2.8) 𝐸 𝑎 = (0, 𝐸 𝑖 ) and 𝐵𝑎 = (0, 𝐵𝑖 ) are electric and magnetic fields in the frame of an Eulerian observer. One can read off 𝐸 𝑎 and 𝐵𝑎 from 𝐹 𝑎𝑏 using the following relations 𝐸 𝑎 = 𝐹 𝑎𝑏 𝑛 𝑏 , 1 𝐵𝑎 = − 𝜀 𝑎𝑏𝑐𝑑 𝑛 𝑏 𝐹𝑐𝑑 = − ∗𝐹 𝑎𝑏 𝑛 𝑏 . 2 (2.9) (2.10) While analytically complete, Maxwell equations cannot be directly evolved numerically, as any violation of the divergence constraints (Eqs. (2.2) and (2.3) with 𝑎 = 0) will break strong hyperbolicity of the system (Schoepe, Hilditch, and Bugner, 2018; Hilditch and Schoepe, 2019). This can be avoided by either using constrained transport approaches (Evans and J. F. Hawley, 1988) or extending the system using effective Lagrange multipliers (Dedner et al., 2002). We here adopt the latter approach. The extended (or augmented) Maxwell equations (Komissarov, 2007; Palenzuela, 2013) are ∇𝑎 (𝐹 𝑎𝑏 + 𝑔 𝑎𝑏 𝜓) = −J 𝑏 + 𝜅 𝜓 𝑛 𝑏 𝜓 (2.11) ∇𝑎 ( ∗𝐹 𝑎𝑏 + 𝑔 𝑎𝑏 𝜙) = 𝜅 𝜙 𝑛 𝑏 𝜙 (2.12) ∇𝑎 J 𝑎 = 0 (2.13) where auxiliary scalar fields 𝜓 and 𝜙 propagate divergence constraint violations of electric field and magnetic field. 𝜅 𝜓 and 𝜅 𝜙 are damping constants, leading to an −1 . exponential damping of the constraints in the characteristic timescales 𝜅 𝜓,𝜙 Performing a standard 3+1 decomposition of the extended Maxwell’s equations (2.11)–(2.13) using the normal vector 𝑛𝑎 and the spatial projection operator ℎ𝑎 𝑏 ≡ 𝛿 𝑎 𝑏 + 𝑛𝑎 𝑛 𝑏 , we get 𝑖𝑗𝑘 (𝜕𝑡 − L 𝛽 )𝐸 𝑖 − 𝜀 (3) 𝐷 𝑗 (𝛼𝐵 𝑘 ) + 𝛼𝛾 𝑖 𝑗 𝐷 𝑗 𝜓 = −𝛼𝐽 𝑖 + 𝛼𝐾 𝐸 𝑖 , (2.14a) 9 𝑖𝑗𝑘 (𝜕𝑡 − L 𝛽 )𝐵𝑖 + 𝜀 (3) 𝐷 𝑗 (𝛼𝐸 𝑘 ) + 𝛼𝛾 𝑖 𝑗 𝐷 𝑗 𝜙 = 𝛼𝐾 𝐵𝑖 , (2.14b) (𝜕𝑡 − L 𝛽 )𝜓 + 𝛼𝐷 𝑖 𝐸 𝑖 = −𝛼𝜅 𝜓 𝜓 + 𝛼𝑞, (2.14c) (𝜕𝑡 − L 𝛽 )𝜙 + 𝛼𝐷 𝑖 𝐵𝑖 = −𝛼𝜅 𝜙 𝜙, (2.14d) (𝜕𝑡 − L 𝛽 )𝑞 + 𝐷 𝑖 (𝛼𝐽 𝑖 ) = 𝛼𝑞𝐾, (2.14e) where 𝐷 𝑖 = ℎ𝑎𝑖 ∇𝑎 is the spatial covariant derivative, 𝐾 is the trace of extrinsic curvature, 𝑞 = −𝑛 𝜇 J 𝜇 is the electric charge density measured by an Eulerian observer, and 𝐽 𝑖 = ℎ𝑖 𝑎 J 𝑎 is the spatial electric current density. Here we also defined the spatial Levi-Civita tensor associated with the spatial metric as 𝑑𝑎𝑏𝑐 𝜀 𝑎𝑏𝑐 . (3) ≡ 𝑛 𝑑 𝜀 (2.15) The Lie derivative along the shift vector applied to a spatial vector 𝐸 𝑖 is L 𝛽 𝐸 𝑖 = 𝛽 𝑗 𝜕 𝑗 𝐸 𝑖 − 𝐸 𝑗 𝜕 𝑗 𝛽𝑖 , and same for 𝐵𝑖 on the left hand side of Eq. (2.14), while it is simply a directional derivative (e.g. L 𝛽 (𝑞) = 𝛽𝑖 𝜕𝑖 𝑞) when applied to a scalar variable. Evolution equations (2.14) can be cast into conservative form 𝜕𝑡 U + 𝜕 𝑗 F 𝑗 = S, (2.16)  𝐸 𝑖   𝐸˜ 𝑖           𝐵𝑖   𝐵˜ 𝑖      √     U = 𝛾  𝜓  ≡  𝜓˜  ,      𝜙   𝜙˜           𝑞   𝑞˜      (2.17)  −𝛽 𝑗 𝐸˜ 𝑖 + 𝛼(𝛾 𝑖 𝑗 𝜓˜ − 𝜀𝑖 𝑗 𝑘 𝐵˜ 𝑘 )    (3)    −𝛽 𝑗 𝐵˜ 𝑖 + 𝛼(𝛾 𝑖 𝑗 𝜙˜ + 𝜀𝑖 𝑗 𝑘 𝐸˜ )   (3) 𝑘     F𝑗 =  −𝛽 𝑗 𝜓˜ + 𝛼 𝐸˜ 𝑗 ,     𝑗 𝑗 ˜ ˜   −𝛽 𝜙 + 𝛼 𝐵       ˜ 𝑗 − 𝛽 𝑗 𝑞˜ 𝐽   (2.18) with evolved variables fluxes 10 and source terms −𝛼√𝛾𝐽 𝑖 − 𝐸˜ 𝑗 𝜕 𝑗 𝛽𝑖 + 𝜓(𝛾 ˜ 𝑖 𝑗 𝜕 𝑗 𝛼 − 𝛼𝛾 𝑗 𝑘 Γ𝑖 )   𝑗𝑘     ˜ 𝑖 𝑗 𝜕 𝑗 𝛼 − 𝛼𝛾 𝑗 𝑘 Γ𝑖 ) − 𝐵˜ 𝑗 𝜕 𝑗 𝛽𝑖 + 𝜙(𝛾   𝑗𝑘     𝑘 ˜ ˜ S=  𝐸 𝜕𝑘 𝛼 + 𝛼 𝑞˜ − 𝛼𝜓(𝐾 + 𝜅 𝜓 ) ,     𝑘 ˜ 𝜕𝑘 𝛼 − 𝛼 𝜙(𝐾 ˜ + 𝜅𝜙)   𝐵       0   (2.19) where Γ𝑖𝑗 𝑘 are the Christoffel symbols associated with the spatial metric. A prescription for the electric current density 𝐽 𝑖 (Ohm’s law) needs to be supplied to close the system. 2.2.2 Force-free limit In the magnetospheres of neutron stars and black holes, we expect copious production of electron-positron pairs (Goldreich and Julian, 1969). The resulting plasma will be highly conductive, effectively screening electric field components parallel to the magnetic field. In addition, the magnetization of the plasma will be very high, allowing us to consider the limit in which the Lorentz force density vanishes and the plasma becomes force-free. The force-free conditions are given as 𝐹 𝑎𝑏 J𝑏 = 0, (2.20) ∗ 𝑎𝑏 𝐹 𝐹𝑎𝑏 = 0, (2.21) 𝐹 𝑎𝑏 𝐹𝑎𝑏 > 0. (2.22) In terms of 𝐸 𝑖 , 𝐵𝑖 , 𝑞 and 𝐽 𝑖 , these conditions are 𝑖𝑗𝑘 𝑞𝐸 𝑖 + 𝜀 (3) 𝐽 𝑗 𝐵 𝑘 = 0, (2.23) 𝐸 𝑎 𝐵𝑎 = 𝐸 𝑖 𝐵𝑖 = 0, (2.24) 𝐵2 − 𝐸 2 > 0, (2.25) where 𝐸 2 = 𝐸 𝑎 𝐸 𝑎 = 𝐸𝑖 𝐸 𝑖 and 𝐵2 = 𝐵𝑎 𝐵𝑎 = 𝐵𝑖 𝐵𝑖 . The first condition (2.23) corresponds to the vanishing Lorentz force density, and the second one (2.24) shows the screening of electric field along magnetic field lines. The third condition (2.25) is called magnetic dominance, and violation of this constraint flags the breakdown of force-free electrodynamics; characteristic speeds associated with Alfvén modes 11 become complex and Maxwell equations are no longer hyperbolic (Pfeiffer and MacFadyen, 2013). Physically, 𝐸 2 ≈ 𝐵2 means that the plasma drift speed approaches the speed of light, beyond which the FFE approximation breaks down. The force-free conditions also give constraints on the electric current density. Eq. (2.23) gives 𝐽 𝑖 in the form 𝑖𝑗𝑘 𝐽𝑖 = 𝑞 𝜀 (3) 𝐸 𝑗 𝐵 𝑘 𝐵2 + (𝐽𝑙 𝐵𝑙 ) 𝑖 𝐵, 𝐵2 (2.26) which leaves the parallel component 𝐽𝑙 𝐵𝑙 undetermined. The first term on the right hand side of (2.26), the drift current, is perpendicular to both electric and magnetic fields and shows that electric charge moves collectively with the drift 𝑖𝑗𝑘 velocity 𝑣 𝑑 = 𝜀 (3) 𝐸 𝑗 𝐵 𝑘 /𝐵2 . Requiring Eq. (2.24) to always be satisfied, we obtain a closed form expression of the parallel current 𝐽𝑙 𝐵𝑙 as (McKinney, 2006a; Paschalidis and Shapiro, 2013)
𝑖𝑗𝑘 𝐽𝑙 𝐵𝑙 = 𝜖 (3) 𝐵𝑖 𝐷 𝑗 𝐵 𝑘 − 𝐸𝑖 𝐷 𝑗 𝐸 𝑘 − 2𝐸 𝑖 𝐵 𝑗 𝐾𝑖 𝑗 , (2.27) 𝐽𝑙 𝐵𝑙 = 𝐵𝑖 (∇ × 𝐵) 𝑖 − 𝐸 𝑗 (∇ × 𝐸) 𝑗 (2.28) which reduces to in the special relativistic limit (Gruzinov, 1999). The parallel current Eq. (2.27) contains the spatial derivatives of 𝐸 and 𝐵, the dynamical variables that we evolve. Including these derivatives in the source terms changes the principal part of the Maxwell PDE system, and the resulting system of equations is not strongly hyperbolic (Pfeiffer and MacFadyen, 2013). A straightforward way to keep the force-free conditions satisfied in numerical simulations is to algebraically impose Eq. (2.24) and (2.25) in the time evolution (Spitkovsky, 2006; Palenzuela, Garrett, et al., 2010; Petri, 2012; Parfrey, Beloborodov, and Hui, 2012; Cao, L. Zhang, and Sun, 2016; A. Y. Chen, Yuan, and Vasilopoulos, 2020; Mahlmann, Aloy, et al., 2021a). This commonly employed approach exactly ensures the force-free conditions, but reduces the numerical accuracy to first-order convergence in time. As we aim to implement a higher-order numerical scheme for GRFFE, we consider an alternative strategy. We adopt the driver term approach first implemented in Alic et al. (2012) and Moesta et al. (2012) and applied in later studies (e.g. Most and Philippov, 2020; Most and Philippov, 2022). In this method, a stiff relaxation term 12 is added to the electric current density 𝐽 𝑖 to continuously damp the violation of the force-free conditions. We adopt the following electric current density prescription (Most and Philippov, 2022) 𝑖𝑗𝑘   𝜀 (3) 𝐸 𝑗 𝐵 𝑘 𝐸 𝑗 𝐵 𝑗 𝑖 R (𝐸 2 − 𝐵2 ) 𝑖 𝑖 𝐽 =𝑞 +𝜂 𝐵 + 𝐸 , (2.29) 𝐵2 𝐵2 𝐵2 where R (𝑥) ≡ max(𝑥, 0) is the rectifier function and 𝜂 is a relaxation parameter. The parallel current consists of the terms in the square bracket in Eq. (2.29), each being proportional to the violation of the force-free conditions (2.24) and (2.25). They are coupled to the evolution of electric field and drive the solution to the force-free limit with the characteristic damping time scale 𝜂−1 . The limiting case 𝜂 → ∞ corresponds to the ideal force-free limit. A caveat to the FFE simulations with a parallel electric current is that the energy loss from an Ohmic dissipation 𝐽𝑖 𝐸 𝑖 is removed out from and no further tracked in simulations; therefore, total electromagnetic energy is not conserved.4 While numerical dissipation will also contribute to the energy loss, the amount of energy dissipation in current sheets (corresponding to the rectifier term in Eq. (2.29)) dominates, albeit likely at a different rate compared to a full kinetic reconnection model (e.g. Cerutti, Philippov, Parfrey, et al., 2015; Philippov, Cerutti, et al., 2015). 2.3 Numerical implementation In this section, we describe the details of our numerical scheme and its implementation. We present our method of spatial discretization in Sec. 2.3.1, time integration in Sec. 2.3.2, and the adaptive discontinuous Galerkin-finite difference hybrid solver in Sec. 2.3.3. Our numerical scheme described here is implemented in the open source numerical relativity code SpECTRE (Deppe, Throwe, et al., 2025). 2.3.1 Domain decomposition and spatial discretization The computational domain typically used in astrophysics or numerical relativity simulations is simple enough to be decomposed into a set of non-overlapping deformed cubes. We divide the domain into these deformed cubes, which are called subdomain elements (hereafter simply elements). Neighboring elements share their boundaries at an element interface between them. Within each element, a spectral expansion can be performed to represent a field of interest. We also need to define a prescription for handling boundary corrections 4 In a MHD model, it is captured as the same amount of increase in the internal (thermal) energy of the plasma. 13 from element interfaces. This family of numerical methods is broadly called spectral element methods (Kopriva, 2009). We choose to adopt the nodal discontinuous Galerkin discretization (Hesthaven and Warburton, 2007), so our approach is formally referred to as a discontinuous Galerkin spectral element method (DG-SEM), which is often simply called a discontinuous Galerkin (DG) method. Each element is mapped to a reference cube spanning {𝜉 1 , 𝜉 2 , 𝜉 3 } ∈ [−1, 1] 3 in the reference coordinate system {𝜉 𝑖 }. A coordinate map 𝑥 𝑖 (𝜉 𝑗 ) relates the reference coordinates 𝜉 𝑗 to physical coordinates 𝑥 𝑖 . A set of collocation points {𝜉𝑖1 , 𝜉 2𝑗 , 𝜉 𝑘3 } are chosen to represent the solution ∑︁ 𝑢(𝜉) = 𝑢𝑖, 𝑗,𝑘 𝜙𝑖, 𝑗,𝑘 (𝜉) (2.30) 𝑖, 𝑗,𝑘 where 𝑢𝑖, 𝑗,𝑘 = 𝑢(𝜉𝑖1 , 𝜉 2𝑗 , 𝜉 𝑘3 ) is the value of the solution at the collocation point (𝜉𝑖1 , 𝜉 2𝑗 , 𝜉 𝑘3 ), and 𝜙𝑖, 𝑗,𝑘 (𝜉) is the nodal basis function     1, for 𝑖 = 𝑙, 𝑗 = 𝑚, 𝑘 = 𝑛     .   0,  otherwise   𝜙𝑖, 𝑗,𝑘 (𝜉𝑙1 , 𝜉𝑚2 , 𝜉𝑛3 ) =  (2.31) We use the tensor product basis 𝜙𝑖, 𝑗,𝑘 (𝜉) = 𝑙𝑖 (𝜉 1 ) 𝑙 𝑗 (𝜉 2 ) 𝑙 𝑘 (𝜉 3 ) (2.32) where 𝑙 𝑑 (𝑥) is the 1D Lagrange polynomial interpolating collocation points along the 𝑑-th axis. We choose to use an isotropic DG mesh with the same polynomial degree 𝑁 for each spatial dimension. The resulting nodal expansion of the solution is 𝑁 ∑︁ 𝑁 ∑︁ 𝑁 ∑︁ 𝑢(𝜉) = 𝑢𝑖, 𝑗,𝑘 𝑙𝑖 (𝜉 1 )𝑙 𝑗 (𝜉 2 )𝑙 𝑘 (𝜉 3 ). (2.33) 𝑖=0 𝑗=0 𝑘=0 The solution (2.33) can be also represented in a modal form 𝑢(𝜉) = 𝑁 𝑁 ∑︁ 𝑁 ∑︁ ∑︁ 𝑐 𝑝,𝑞,𝑟 𝐿 𝑝 (𝜉 1 )𝐿 𝑞 (𝜉 2 )𝐿 𝑟 (𝜉 3 ). (2.34) 𝑝=0 𝑞=0 𝑟=0 where 𝐿 𝑝 (𝑥) is the Legendre polynomial of degree 𝑝. See also Teukolsky (2016) for a detailed derivation of formulating the DG scheme in a curved spacetime. In this article, we denote a scheme using the 𝑁-th degree polynomial basis (i.e. 𝑁 + 1 collocation points) in each spatial dimension as a DG-𝑃 𝑁 scheme. For instance, a DG-𝑃5 scheme uses 63 collocation points in each element and a solution 14 is approximated as a fifth degree polynomial in each spatial direction. When the solution is smooth, a DG-𝑃 𝑁 scheme exhibits O (𝐿 𝑁+1 ) spatial convergence where 𝐿 is the spatial size of an element. We mainly use a DG-𝑃5 scheme, although we present results for different DG orders where necessary. We use the Legendre-Gauss-Lobatto collocation points with the mass lumping approximation (Teukolsky, 2015a). For a reduced aliasing error, an exponential filter is applied to rescale the modal coefficients 𝑐 𝑝,𝑞,𝑟 in Eq. (2.34):     Ö 𝑛 2𝑏 𝑐 𝑝,𝑞,𝑟 → 𝑐 𝑝,𝑞,𝑟 exp −𝑎 (2.35) 𝑁 𝑛={𝑝,𝑞,𝑟} after every DG time (sub)step. We use 𝑎 = 36 and 𝑏 = 50, which effectively zeros only the highest mode (𝑖 = 𝑁) and leaves other modes intact. Filtering out the highest mode reduces expected spatial converge of a DG-𝑃 𝑁 scheme from O (𝐿 𝑁+1 ) to O (𝐿 𝑁 ). We note that this is a common practice adopted in spectral methods for curing aliasing and has marginal effects on capturing discontinuities, since typically a Gibbs phenomenon near a discontinuity excites not only the highest mode but multiple high modes simultaneously. 2.3.2 Time integration Based on the spatial discretization presented in the previous section, evolution equations can be integrated over time using the method of lines. The maximum admissible time step size for a DG-𝑃 𝑁 scheme is (Cockburn and Shu, 2001; Dumbser, Zanotti, Loubère, et al., 2014) Δ𝑡 ≤ 𝑐 𝐿 𝜆max (2𝑁 + 1) 𝐷 (2.36) where 𝐿 is the minimum (Cartesian) edge length of an element, 𝜆 max is the maximum characteristic speed inside the element, 𝑐 is a stability constant specific to a time stepper, which is usually of order unity,5 and 𝐷 is the number of spatial dimensions. However, usage of a nontrivial coordinate map 𝑥(𝜉) and a complex geometry of elements deforms the spatial distribution of grid points, and an actual upper bound can differ from Eq. (2.36). As a practical strategy, we adopt the following expression Δ𝑡 = 𝑓 (Δ𝑥)min 𝑐 𝜆 max 𝐷 (2.37) 5 For example, the classic 4th-order Runge-Kutta method has 𝑐 ≈ 1.39 (Hairer, Nørsett, and Wanner, 1993). 15 for the DG time step size, where (Δ𝑥)min is the minimum grid spacing between DG collocation points in physical coordinates and 𝑓 is the CFL factor. In order to keep the force-free constraint violations as small as possible during evolution, we aim to use a large value of the damping coefficient 𝜂 for the driver term in Eq. (2.29), possibly up to 𝜂Δ𝑡 ≳ 10. This implies that the characteristic time scale of constraint damping 𝜂−1 is smaller than the time step size, which introduces stiffness in evolution equations and makes explicit time integration unstable unless an unreasonably small time step is used. To address the stiffness from rapid constraint damping, we adopt the implicitexplicit (IMEX) time stepping technique. In particular, we make use of the IMEXSSP3(4,3,3) scheme by Pareschi and Russo (2005), which is third order in time. In this IMEX approach, we evolve all quantities explicitly using a standard 3rdorder Runge-Kutta scheme, and treat only the stiff part of the source terms (2.19) implicitly. Specifically, in the evolution of electric fields this requires us to solve the following nonlinear algebraic equation at all substeps, "
# 2 − 𝐵2 𝑗 R 𝐸 𝐸 𝐵 𝑗 𝐵𝑖 + 𝐸𝑖 (2.38) 𝐸 𝑖 = (𝐸 𝑖 ) ∗ − 𝛼𝜂Δ𝑡 ′ 𝐵2 𝐵2 where (𝐸 𝑖 ) ∗ are provided values and Δ𝑡 ′ is an IMEX-scheme-dependent corrector step size. When 𝐸 2 < 𝐵2 , the solution to this equation is analytical whereas in general cases we employ a three-dimensional Newton-Raphson solver with a specific initial guess. In addition to the stiff electric current, we also apply the IMEX time integration to the hyperbolic divergence cleaning parts to ensure stability, 𝜓 = 𝜓 ∗ − 𝜅 𝜓 Δ𝑡 ′𝜓, (2.39) 𝜙 = 𝜙∗ − 𝜅 𝜙 Δ𝑡 ′ 𝜙, (2.40) which are linear equations and have exact analytic inversions. Because of the simplicity of the implicit equations in our evolution system, the cost overhead from using an IMEX scheme is less than 5% of the total runtime. Being able to use much larger time steps more than compensates for this. 2.3.3 The discontinuous Galerkin-finite difference hybrid method This section describes our implementation of the DG-FD hybrid solver for GRFFE equations. We closely follow the original implementation of Deppe, Hébert, et al. 16 (2022), which was designed for GRMHD, with several improvements and adaptations. Overview of the algorithm Consider an element performing a time step on the DG grid. After each substep of the time integrator, the candidate solution is monitored by the troubled cell indicator (TCI) to check if the solution is admissible on the DG grid. If it is admissible, we continue with the updated solution on the DG grid. If the candidate solution is inadmissible, the troubled cell indicator is flagged, we undo the DG substep, project the DG solution onto the sub-element FD grid, then repeat the substep using the FD solver. Evolution on the FD grid proceeds in a similar way; after every time step the solution gets monitored by the troubled cell indicator, which determines whether the solution needs to stay on the FD grid or it is admissible on the DG grid. If the candidate solution looks admissible on the DG grid, the solution is projected back to the DG grid and the evolution proceeds using the DG solver. An optimal number of sub-element finite-difference grid points for a DG-𝑃 𝑁 scheme is 2𝑁 + 1 (Dumbser, Zanotti, Loubère, et al., 2014). We follow such a prescription, and an element with (𝑁 + 1) 𝐷 collocation points on the DG grid is switched to (2𝑁 + 1) 𝐷 FD cells with a uniform grid spacing Δ𝜉 𝑖 = 2/(2𝑁 + 1) in the reference coordinates. At the code initialization phase, all physical quantities are evaluated on the FD grid to avoid potential spurious oscillations arising from a spectral representation of the initial data. Next, each element projects evolved variables onto the DG grid, then either switches to the DG grid or stays on the FD grid depending on the decision made by the troubled cell indicator. The projection algorithm of scalar and tensor quantities between DG and FD grids is described in detail in Deppe, Hébert, et al. (2022). We use a general sixth-order accurate interpolation scheme. Since the scheme is general and does not respect the physical constraints,6 repeated applications (i.e., switching back and forth between DG and FD too frequently) can introduce spurious errors in the solution. To suppress this behavior, we need to design the troubled cell indicator to apply tighter criteria when switching back from FD to DG grid. 6 For example, the interpolation scheme between the DG and FD grids does not strictly preserve the force-free conditions or the divergence (Gauss) constraints. 17 Finite difference solver Evolution on the finite-difference grid is performed using a conservative finitedifference scheme (Shu and Osher, 1988; Shu and Osher, 1989). For an element using a DG-𝑃 𝑁 scheme, we divide the reference coordinate interval [−1, 1] into 2𝑁 + 1 finite-difference cells and project a solution from the DG grid onto cellcentered values {𝑈𝚤ˆ }. A flux-balanced law (2.16) is discretized as 𝑗  𝑘  𝐹ˆ 𝑗 − 𝐹ˆ𝚤ˆ−1/2 𝑑𝑈𝚤ˆ 𝜕𝜉 𝚤ˆ+1/2 + = 𝑆(𝑈𝚤ˆ) 𝑑𝑡 𝜕𝑥 𝑗 Δ𝜉 𝑘 (2.41) where we used hat indices to label FD cells and plain indices to label spatial directions. Computation of a numerical flux 𝐹ˆ𝚤ˆ+1/2 is dimensionally split, and closely follows that of the ECHO scheme (Del Zanna, Zanotti, et al., 2007). At the left and right sides of the FD cell interface 𝑥𝚤ˆ+1/2 , evolved variables are reconstructed using their cell-centered values {𝑈𝚤ˆ }. In our implementation, densitized electric current density 𝐽˜𝑖 is also reconstructed to compute fluxes associated with 𝑞˜ (see also Palenzuela, 2013). 𝑗 Once face-centered values 𝑈𝚤ˆ𝐿,𝑅 are reconstructed, the interface Riemann flux +1/2 ∗ 𝐹𝚤ˆ+1/2 is computed using the Rusanov (local-Lax-Friedrichs) flux formula (Rusanov, 1962). Since the principal part of our equations is linear, this solver will reduce to the exact solution (see Dedner et al., 2002). In order to achieve high-order accuracy, a high-order derivative corrector is added to the interface Riemann flux to obtain the final numerical flux: . 𝐹ˆ𝚤ˆ+1/2 = 𝐹𝚤ˆ∗+1/2 − 𝐺 𝚤ˆ(4) +1/2 (2.42) The original ECHO scheme uses the Riemann fluxes from cell interfaces (e.g. 𝐹𝚤ˆ∗±3/2 ) for the higher-order correction term 𝐺 𝚤ˆ(4) . Since we do not employ a +1/2 constrained-transport algorithm requiring a consistent and fixed stencil, we opt for simpler cell-centered fluxes (e.g., 𝐹𝚤ˆ±1 ) for a more compact stencil and reduced amount of data communications (see Nonomura and Fujii, 2013; Y. Chen, Tóth, and Gombosi, 2016). For the simulations presented in this work, we use the WENO5-Z reconstruction with the nonlinear weight exponent 𝑞 = 2 (Borges et al., 2008). The high-order finite-difference corrector is currently implemented only on Cartesian meshes. We 18 therefore use it for all of our one-dimensional test problems, where we assess numerical convergence of the scheme, and defer to future work its applications in multi-dimensional contexts. Consistent with previous assessments, we find it sufficient to use only a fourth-order accurate derivative correction when combined with WENO5-Z (Most, Papenfort, and Rezzolla, 2019). Troubled cell indicator In order to decide when to switch between DG and FD grids, our numerical scheme requires a robust criterion to identify regions of non-smoothness. Such an approach somewhat shares its idea with popular adaptive-mesh-refinement criteria. These criteria are inherently problem dependent, and an optimal design of the troubled cell indicator is at the heart of the DG-FD hybrid method. Requirements on the indicator include (i) A relatively low computational cost (ii) Early and robust detection of spurious oscillations developing on the DG grid. (iii) Being unflagged as soon as the oscillation no longer exists, so that evolution can be performed by a more efficient DG solver. A solution approximated with an 𝑁-th degree polynomial on the DG grid has nodal and modal representations where 𝐿 𝑝 (𝑥) is the Legendre polynomial of degree 𝑝. Motivated by the idea of the modal shock indicator devised by Persson and Peraire (2006), we adopt the oscillation detection criterion √︄ Í 2 𝑖 𝑢ˆ 𝑖 (2.43) Í 2 > (𝑁 + 1 − 𝑀) −𝛼 , 𝑖 𝑢𝑖 where 𝑢ˆ is the solution with the lowest 𝑀 modes filtered out i.e. 𝑢(𝜉) ˆ = 𝑁 ∑︁ 𝑁 ∑︁ 𝑁 ∑︁ 𝑐 𝑝,𝑞,𝑟 𝐿 𝑝 (𝜉 1 )𝐿 𝑞 (𝜉 2 )𝐿 𝑟 (𝜉 3 ), (2.44) 𝑝=𝑀 𝑞=𝑀 𝑟=𝑀 Í and the summations 𝑖 in Eq. (2.43) with the nodal values 𝑢𝑖 , 𝑢ˆ𝑖 are performed over all DG grid points. The exponent 𝛼 in the criterion (2.43) controls the sensitivity of the indicator. Since we filter out the highest mode on the DG grid, the troubled cell indicator needs to use 𝑀 ≥ 2. We use 𝑀 = 3 for the troubled cell indicator, effectively monitoring power from the second and third highest modes. Empirically 19 we find that 𝑀 ≲ ⌊(𝑁 + 1)/2⌋ provides robust detections of discontinuities without the indicator being excessively triggered. We use 𝛼 = 4.0 following Deppe, Hébert, et al. (2022) and Deppe et al. (2022). To avoid an element switching back and forth between DG and FD grid in an unnecessarily frequent manner, we use 𝛼′ = 𝛼 + 1 when an element is evolving on FD grid. The tighter bound 𝛼′ ensures an extra smoothness of solution when the grid is switched back to DG, preventing it from switching again to FD within only a few time steps. Depending on the specific type of an evolved system, one may consider additional physical admissibility criteria (e.g. positivity of the mass density in the case of hydrodynamics) for the troubled cell indicator. Since the only physical constraints in our evolution system, the force-free conditions, are handled by the stiff parallel electric current, we do not impose any physics-motivated criteria. In our implementation of the DG-FD hybrid scheme for GRFFE, we adopt only one criterion for the troubled cell indicator: application of the modal sensor (2.43) to the magnitude of 𝐵˜ 𝑖 . While it looks somewhat oversimplified that the information of a single scalar quantity is used for monitoring a system with nine evolution variables ˜ 𝜙, ˜ 𝑞}, { 𝐸˜ 𝑖 , 𝐵˜ 𝑖 , 𝜓, ˜ we show in Sec. 2.4 that it is capable of detecting troubled elements in a satisfactory manner. 2.3.4 Outer boundary condition In 3D simulations, the outer boundary of the computational domain is usually placed far out to avoid spurious boundary effects leaking into the internal evolution. Still, in order to suppress potential unphysical noise or reflections at the outer boundary, we implement a no-incoming Poynting flux boundary condition as follows. The ˜ 𝜙, ˜ 𝑞) evolved variables at the outer boundary Uout = ( 𝐸˜ 𝑖 , 𝐵˜ 𝑖 , 𝜓, ˜ out are prescribed as follows. First, we copy the values of { 𝐸˜ 𝑖 , 𝐵˜ 𝑖 , 𝑞} ˜ from the outermost grid points. Then, if the Poynting flux is pointing inward, we set ( 𝐸˜ 𝑖 )out to zero. Divergence ˜ out and ( 𝜙) ˜ out are always set to zero.7 On the DG grid, Uout cleaning scalar fields ( 𝜓) is fed as an external state when computing the boundary correction terms. On the FD grid, ghost zones are filled with Uout during the FD reconstruction step. 7 Normally, the level of errors associated with the divergence cleaning part (𝜓, 𝜙) is much smaller than that of the physical variables (𝐸 𝑖 , 𝐵𝑖 , 𝑞). Spurious reflections, if any, in the divergence cleaning parts are subdominant to Poynting fluxes transmitting through the outer boundaries. 20 Table 2.1: Simulation setup for 1D tests in Sec. 2.4.1. Grid resolution is increased with 𝑛 = 0 (Low), 𝑛 = 1 (Med), and 𝑛 = 2 (High). For the FFE breakdown problem, we use 𝑛 = 3 as a reference solution. Each resolution, if all elements are switched to FD, is equivalent to 352 × 2𝑛 finite-difference grid points along the 𝑥 axis. Fast wave Alfvén wave FFE breakdown 2.4 Domain size ( ×[−0.1, 0.1] 2 ) [−0.5, 1.5] [−1.5, 1.5] [−0.5, 0.5] DG Grid points ( × 62 ) (192 × 2𝑛 ) 𝜂 CFL factor 106 0.3 Δ𝑡 (×2−𝑛 ) 9.22 × 10−4 1.38 × 10−3 4.61 × 10−4 Results In this section, we test and assess our implementation of the DG-FD hybrid method for evolving GRFFE equations with a suite of robust code validation problems. We perform 1D tests in Sec. 2.4.1, curved spacetime tests with black holes in Sec. 2.4.2, and pulsar magnetosphere tests in Sec. 2.4.3. We also discuss accuracy and efficiency aspects of the DG-FD hybrid method in Sec. 2.4.4. 2.4.1 One-dimensional test problems One-dimensional test problems evolve initial data that only has dependence in the 𝑥 direction. We use a computational domain consisting of a single element along the 𝑦 and 𝑧 axes, and impose periodic boundary condition on those directions. Our lowest grid resolution has 32 elements along the 𝑥 axis, resulting in 192 DG grid points. To facilitate comparisons with other results available in the literature, we note that this resolution is equivalent to 352 grid points if all elements are switched to an FD grid. The number of elements along the 𝑥 axis is increased by a factor of two to run medium (64 elements) and high (128 elements) resolutions. Dirichlet boundary conditions are applied at both ends of the 𝑥 axis. We use the CFL factor 0.3 and parallel conductivity 𝜂 = 106 . Simulation setups are summarized in Table 2.1. 21 1.0 |Ez | 0.9 0.8 0.7 Numerical Exact 10−2 |∆Ez | 10−4 Low Med High 10−6 10−8 10−10 0.00 0.25 0.50 0.75 1.00 x Figure 2.1: Fast wave at 𝑡 = 0.5. Top: comparison between the exact solution and a numerical solution with the lowest grid resolution. Bottom: error of 𝐸 𝑧 for three different grid resolutions. Fast wave Originally due to Komissarov (2002), this test problem evolves a pure fast mode propagating in an electrovacuum. The initial profile 𝐵𝑥 = 1.0,   1.0    𝑦 𝐵 = −1.5𝑥 + 0.85    0.7    if 𝑥 < −0.1    if − 0.1 < 𝑥 < 0.1 ,    if 𝑥 > 0.1  (2.45) 𝐵 𝑧 = 0, 𝐸 𝑥 = 0, 𝐸 𝑦 = 0, 𝐸 𝑧 = −𝐵 𝑦 , advects to the +𝑥 direction with the wave speed 𝜇 = 1. The analytic solution is 𝑄(𝑥, 𝑡) = 𝑄(𝑥 − 𝑡, 0) for any physical quantity 𝑄. As shown in Figure 2.1, our scheme shows good convergence in flat regions with increasing grid resolution. We observe that the accuracy and numerical convergence 22 1.3 Numerical Exact Bz 1.2 1.1 1.0 10 −3 Low Med High |∆Bz | 10−4 10−5 10−6 −0.3 −0.2 −0.1 0.0 0.1 0.2 0.3 x Figure 2.2: Stationary Alfvén wave at 𝑡 = 2.0. Same plot description as Fig. 2.1. of the solution is substantially lost around two kinks present in the initial data (corresponding to 𝑥 = 0.5 ± 0.1 in Fig. 2.1) at which spatial derivatives of fields are discontinuous. Alfvén wave The stationary Alfvén wave problem (Komissarov, 2004a) has a transition layer |𝑥| < 0.1 that maintains a strong parallel current, and the accuracy of the test results essentially reflects how well a numerical scheme can maintain the force-free conditions. In the rest frame of the wave, electric and magnetic fields are 𝐵𝑥 = 𝐵 𝑦 = 1.0,    1.0 if 𝑥 < −0.1        𝐵 𝑧 = 1.15 + 0.15 sin(5𝜋𝑥) if |𝑥| < 0.1 ,      1.3  if 𝑥 > 0.1   𝐸 𝑥 = −𝐵 𝑧 , 𝐸 𝑦 = 0, 𝐸 𝑧 = 1.0. (2.46) 23 2.5 B2 − E2 2.0 1.5 1.0 0.5 0.0 t=0 t = 0.02 t = 0.04 |∆Bz | 10−2 10−4 Low Med High 10−6 10−8 −0.2 −0.1 0.0 0.1 0.2 x Figure 2.3: FFE breakdown problem. Top: initial data (𝑡 = 0) and a numerical solution with the lowest grid resolution at 𝑡 = 0.02 and 𝑡 = 0.04. Bottom: Error of 𝐵 𝑧 with respect to the reference solution. The case with a nonzero wave speed −1 < 𝜇 < 1 can be tested by performing an appropriate Lorentz boost to the initial conditions (2.46) (see e.g. Paschalidis and Shapiro, 2013). We show the result at 𝑡 = 2.0 in Figure 2.2. It needs to be noted that time derivatives of fields at 𝑡 = 0, from the initial condition (2.46), vanish only if the parallel current 𝐽𝑖 𝐵𝑖 equals Eq. (2.27). In our approach, the region |𝑥| < 0.1 initially develops a small transient until the stiff relaxation term becomes fully active within several time steps and effectively recovers the same value of 𝐽𝑙 𝐵𝑙 . The amplitude of the initial transient rapidly decreases at higher grid resolutions. Owing to the higherorder accuracy of the discontinuous Galerkin discretization, our result shows good convergence and low amounts of grid dissipation. 24 FFE breakdown The force-free electrodynamics breakdown problem, originally designed by Komissarov (2002), demonstrates that a state initially satisfying the force-free conditions can later develop into a state violating them. The initial state is 𝐵𝑥 = 1,     1 if 𝑥 < −0.1       𝑦 𝑧 𝐵 = 𝐵 = −10𝑥 if − 0.1 < 𝑥 < 0.1 ,      −1  if 𝑥 > 0.1   (2.47) 𝐸 𝑥 = 0, 𝐸 𝑦 = 0.5, 𝐸 𝑧 = −0.5. 𝐵2 − 𝐸 2 decreases over time toward zero in the transition layer |𝑥| < 0.1 and the magnetic dominance condition eventually breaks down. Figure 2.3 shows numerical results. At 𝑡 ≳ 0.02, the rectifier term restoring the 𝐵2 −𝐸 2 > 0 condition is switched on and robustly maintains the magnetic dominance at later times. Since this problem does not have a closed form solution, we perform an additional higher resolution run using 256 elements along the 𝑥 axis and use it as a reference solution to check the convergence. Similar to the fast wave test, we note the loss of accuracy and numerical convergence near the kinks present in the solution. 2.4.2 Three-dimensional tests: Black hole magnetospheres We perform a set of 3D tests in a curved spacetime using black hole magnetosphere problems. The grid structure of the computational domain is portrayed in Figure 2.4. A spherical shell spanning the radius [𝑟 in , 𝑟 out ] is split into six cubed-sphere wedges, which are then further refined into elements. We use an equiangular coordinate map along angular directions and a logarithmic map along the radial direction. Exact Wald solution Wald (1974) found a stationary electrovacuum solution of Maxwell’s equations in the Kerr spacetime. The solution for the 4-potential is given as 𝐴𝑏 = 𝐵0 [(𝜕𝜙 ) 𝑏 + 2𝑎(𝜕𝑡 ) 𝑏 ], 2 (2.48) where 𝐵0 is the field amplitude, 𝜕𝑡 and 𝜕𝜙 are the Killing vector fields in time and azimuthal directions, and 𝑎 = 𝐽/𝑀 is the rotational parameter of the Kerr black 25 Figure 2.4: A half-cut illustration of the spherical grid used for black hole tests in Sec. 2.4.2. Blue lines show boundaries between the cubed-sphere wedges, where black lines show boundaries between each element (in this example, there are 8 elements in each wedge). The DG-𝑃5 mesh consisting of 63 Legendre-GaussLobatto collocation points is shown with gray lines. The total number of elements in this example is 𝑁𝑟 × 𝑁Ω = 2 × 24. hole. See Appendix B for the explicit expressions of the Wald solution (2.48) in the spherical Kerr-Schild coordinates. The Wald solution with 𝑎 = 0 satisfies the force-free conditions outside the horizon. Electric and magnetic fields in Kerr-Schild coordinates are given by 𝐵˜ 𝑥 = 𝐵˜ 𝑦 = 0, 𝐵˜ 𝑧 = 𝐵0 , 2𝑀 𝐵0 𝑦 ˜ 𝑦 2𝑀 𝐵0 𝑥 ˜ 𝑧 , 𝐸 = , 𝐸 = 0. 𝐸˜ 𝑥 = − 𝑟2 𝑟2 (2.49) We evolve the initial condition (2.49) to 𝑡 = 5𝑀 and measure the L2 error norm v t 𝑛  1 ∑︁  𝑥 2 𝑦 𝑖 𝐿 2 (𝑣 ) ≡ (𝑣 𝑘 ) + (𝑣 𝑘 ) 2 + (𝑣 𝑧𝑘 ) 2 , (2.50) 𝑛 𝑘=1 where 𝑣 𝑖 = 𝐵˜ 𝑖numerical − 𝐵˜ 𝑖exact and 𝑛 is the number of grid points. The inner domain boundary is placed at 𝑟 in = 1.99𝑀, at which no specific boundary condition is imposed. A Dirichlet boundary condition is imposed at the outer boundary 𝑟 out = 20𝑀. Conductivity of the magnetosphere is turned off by setting 𝜂 = 0. 26 Table 2.2: Convergence tests of different DG-𝑃 𝑁 schemes on the exact Wald solution (Sec. 2.4.2). For each level of grid resolution, we show the number of elements used in radial and angular directions, time step size Δ𝑡/𝑀, L2 error norm of 𝐵˜ 𝑖 at 𝑡 = 5𝑀, and the measured order of numerical convergence. Resolution DG-𝑃5 DG-𝑃7 DG-𝑃9 Low Medium High Low Medium High Low Medium High Elements (𝑁𝑟 × 𝑁Ω ) 1×6 2 × 24 4 × 96 Δ𝑡/𝑀 Error( 𝐵˜ 𝑖 ) 2.90 × 10−2 1.15 × 10−2 5.76 × 10−3 1.62 × 10−2 6.29 × 10−3 3.14 × 10−3 1.03 × 10−2 3.95 × 10−3 1.97 × 10−3 8.71 × 10−4 2.74 × 10−5 4.11 × 10−8 1.23 × 10−5 1.51 × 10−7 1.29 × 10−9 2.75 × 10−7 2.03 × 10−9 8.04 × 10−12 Convergence order 4.99 4.72 6.35 6.87 7.08 7.98 In Table 2.2, we show convergence studies for different orders of DG schemes 𝑁 = 5, 7, 9. Measured convergence of DG-𝑃5 and DG-𝑃7 schemes is consistent with the order of DG discretization. A somewhat slower convergence of the DG-𝑃9 scheme can be attributed to other limiting factors such as the truncation error from time integration or the sixth-order interpolation from the initial FD grid to DG grid. In all test cases shown in Table 2.2, all elements stayed on the DG grid throughout the evolution. Vacuum Wald problem A time-dependent evolution of electromagnetic fields around a Kerr black hole can be simulated with the initial magnetic fields given by the Wald solution (2.48) where electric fields are set to zero at 𝑡 = 0. The system reaches a steady state that depends on the spin of the black hole and electrical conductivity of the magnetosphere. We first simulate the electrovacuum case. The background spacetime is the Kerr metric with 𝑎 = 0.999𝑀 in spherical Kerr-Schild coordinates. The electrical conductivity of the magnetosphere is switched off by setting 𝜂 = 0. We use 𝑁𝑟 × 𝑁Ω = 16 × 96 elements and use CFL factor 0.25, resulting in the time step size Δ𝑡 = 1.97 × 10−3 𝑀. The inner domain boundary is located at 𝑟 in = 𝑀, and the no-incoming Poynting flux boundary condition (see Sec. 2.3.4) is applied at the outer domain boundary 𝑟 out = 125𝑀. The evolution reaches a stationary state after 𝑡 ≳ 80𝑀. We show the structure 27 Figure 2.5: Vacuum Wald problem (at 𝑡 = 125𝑀) with black hole spin 𝑎 = 0.999𝑀. Top: Toroidal component of the magnetic field and its in-plane field lines on the meridional plane. We show the interior of the outer horizon 𝑟 = 𝑟 + with a black disk and the ergosphere with black solid lines. Bottom: A three-dimensional visualization illustrates the magnetic field lines (silver lines) expelled from the horizon (black sphere). 28 Φ/(πM 2 B0 ) 4 3 2 1 0 0.0 0.2 0.4 0.6 0.8 1.0 a Figure 2.6: Vacuum Wald problem: total magnetic flux through the upper hemisphere of the outer horizon versus the spin of the Kerr black hole. Numerical results at 𝑡 = 125𝑀 (black dots) are shown on top of the analytic prediction (dotted line, Eq. (2.51)). of magnetic fields at 𝑡 = 125𝑀 in Figure 2.5. The Kerr black hole expels magnetic field lines, successfully demonstrating the “Meissner effect” of black hole electrodynamics (Komissarov and McKinney, 2007). In a stationary state, total magnetic flux through the upper hemisphere of the outer horizon has an analytic expression (King, Lasota, and Kundt, 1975)   ∮ 𝑎4 𝑖 2 (2.51) Φ= 𝐵 𝑑Σ𝑖 = 𝜋𝑟 + 𝐵0 1 − 4 , 𝑟+ 𝑟=𝑟 + , 𝑧>0 √ where 𝑎 is the rotational parameter of the Kerr black hole and 𝑟 + = 𝑀 + 𝑀 2 − 𝑎 2 is the outer horizon radius in spherical Kerr-Schild coordinates. We perform additional simulations varying the black hole spin 𝑎 using the same grid setup, all reaching stationary states at 𝑡 ≳ 80𝑀. We plot the obtained magnetic flux at 𝑡 = 125𝑀 in Figure 2.6; our numerical results are in an excellent agreement with the analytic prediction. The troubled cell indicator is flagged at several innermost elements only for the highly spinning cases with 𝑎/𝑀 ≥ 0.90. Magnetospheric Wald problem First performed by Komissarov (2004a), this problem models a highly conductive magnetosphere around a Kerr black hole. The initial condition is the same as the 29 Figure 2.7: Magnetospheric Wald problem at 𝑡 = 125𝑀. In both panels, the interior of the outer horizon and the ergosphere are shown with a black disk and a thick black line, respectively. Top: toroidal component of the electric current density and magnetic field. In-plane magnetic field lines are shown with thin black lines in the right half. Bottom: distribution of troubled elements evolved on the FD grid. 30 vacuum Wald problem but now the electrical conductivity of the magnetosphere is switched on. Compared to the electrovacuum case, the presence of highly conductive plasma dramatically changes the behavior of the magnetosphere, since the parallel components of electric fields 𝐸𝑖 𝐵𝑖 can be neutralized by the parallel electric current. There is no analytic solution to the evolution of this initial value problem, where numerical simulations (e.g. Komissarov, 2004a; Paschalidis and Shapiro, 2013; Etienne, Wan, et al., 2017; Parfrey, Philippov, and Cerutti, 2019; Mahlmann, Aloy, et al., 2021a) show that the system reaches a quasisteady state that resembles the analytically derived solutions of a stationary force-free magnetosphere (Nathanail and Contopoulos, 2014). We perform a test with the black hole spin 𝑎 = 0.999𝑀 using 𝑁𝑟 × 𝑁Ω = 32 × 384 elements with the CFL factor 0.25 (Δ𝑡 = 9.86×10−4 𝑀). At this grid resolution, if all elements are on the FD grid, there are 176 FD grid points along the 𝜃 direction with the minimum radial grid spacing Δ𝑟 = 0.014𝑀 at the inner boundary 𝑟 = 𝑀. Parallel conductivity is set to 𝜂 = 105 𝑀 −1 . Small numerical errors and resulting constraint violations naturally introduce electric charge density into the computational domain via the parallel current Eq. (2.29), filling up the magnetosphere. The system reaches a stationary state at 𝑡 ≳ 80𝑀. We show the result at 𝑡 = 125𝑀 in Figure. 2.7. Inside the ergosphere, magnetic field lines are dragged by the rotation of the black hole and a thin current sheet is formed in the equatorial plane. The overall configuration and topology of the magnetic fields agree well with previous results reported in the literature. The troubled cell indicator is always flagged at the elements encompassing the equatorial current sheet, while several more elements sparsely distributed near the ergosphere are also switched to the FD grid (bottom panel of Fig. 2.7). In the high electrical conductivity limit, magnetic field lines entering the ergosphere end up crossing the outer horizon (Komissarov, 2007; Parfrey, Philippov, and Cerutti, 2019), apart from a small portion reconnecting at the equatorial current sheet. Because of a large grid resistivity in our setup (the ergosphere is radially ∼50 FD grid points across on the equatorial plane), we see that only about half of the magnetic field lines penetrate the horizon. Some temporal variations of the current sheet and magnetic field lines near the ergosphere are observed, but the details of magnetic reconnection and plasmoid formations at the equatorial current sheet (Parfrey, Philippov, and Cerutti, 2019) are not fully resolved at the current grid resolution. 31 2.4.3 Three-dimensional tests: Pulsar magnetospheres A conducting sphere threaded with a dipolar magnetic field and rotating in free space serves as a toy model of pulsars. In the flat spacetime, we set the initial dipolar magnetic field as (𝑥 2 + 𝑦 2 ) 𝐴𝜙 = 𝜇 2 , (2.52) (𝑟 + 𝛿2 ) 3/2 where 𝜇 is the magnetic dipole moment, 𝑟 2 = 𝑥 2 + 𝑦 2 + 𝑧2 , and 𝛿 is a small number to regularize the field at 𝑟 = 0. All other variables, including electric fields, are set to zero everywhere in the initial data. Rotation of the star is turned on at 𝑡 = 0 with a fixed angular velocity Ω𝑧ˆ. Inside the star (𝑟 ≤ 𝑅), we enforce the perfect conductor condition 𝑖𝑗𝑘 𝐸 𝑖 + 𝜀 (3) 𝑣 𝑗 𝐵 𝑘 = 0, (2.53) 𝑖𝑧 𝑗 with the (rigid) rotation velocity field 𝑣 𝑖 = 𝜀 (3) Ω𝑥 𝑗 . In practice this is implemented by overwriting electric fields 𝐸 𝑖 with those consistent with (2.53) at every substep of time integration. By this means, the magnetic field is effectively anchored and corotates within 𝑟 ≤ 𝑅, whereas fields at 𝑟 > 𝑅 are freely evolved. For consistent behavior of other evolved variables, we also fix 𝜓˜ = 0 and 𝑞˜ = 0 inside the star. The magnetic part of the evolution equations is freely evolved everywhere. We use 𝛿 = 0.1𝑅 for this test. Denoting the grid refinement level by an integer 𝑙, the computational domain consists of an inner cube (23𝑙 elements) and six cubed-sphere wedges (𝑁𝑟 × 𝑁Ω = 2𝑙−3 × 22𝑙 elements for each) surrounding it, wrapped with an outer spherical shell (𝑁𝑟 × 𝑁Ω = 2𝑙−3 × (6×22(𝑙−1) )). The outer shell uses a logarithmic map along the radial direction and is fixed to stay on the DG grid. Figure 2.8 shows the grid structure for 𝑙 = 4. The wedges and the outer shell use an equiangular map for the angular directions, leading to non-uniform sizes of the elements in the inner cube: elements closer to the √ origin have smaller sizes. Vertices of the inner cube are located at 𝑟 cube = 10 3𝑅, and the cubed-sphere wedges fill the region up to 𝑟 in = 20𝑅. The outer shell extends to the outer domain boundary 𝑟 out = 60𝑅, at which the no-incoming Poynting flux boundary condition is imposed. At the 𝑙 = 4 grid resolution, the total number of grid points is 𝑛1/3 ≈ 120 on the DG grid, and the radius of the rotator 𝑅 is a single grid element wide at the center. We test with the angular velocity Ω = (5𝑅) −1 and use a parallel conductivity 𝜂 = 105 𝑅 −1 . Our simulation grid is rotated along the 𝑧 axis with the same angular speed as the rotator. 32 Figure 2.8: A zoom-in view of the computational domain used for the pulsar magnetosphere tests in Sec. 2.4.3. A sphere domain is divided into an inner cube at the center, a layer of cubed-sphere wedges, and an outer spherical shell (not fully shown in this figure). Aligned rotator An aligned rotator (𝜃 = 0) is a simple model of a rotating magnetized neutron star with magnetic moment aligned with the axis of rotation. This problem is relatively simple because it is axisymmetric, and has been treated in a large volume of studies (e.g., Goldreich and Julian, 1969; Contopoulos, Kazanas, and Fendt, 1999; Spitkovsky, 2006; Komissarov, 2006; McKinney, 2006b; Parfrey, Beloborodov, and Hui, 2012; Cao, L. Zhang, and Sun, 2016) We use the 𝑙 = 5 resolution, which has 22 FD grid points across the rotator radius 1/3 and 𝑛grid ≈ 240 total grid points across the DG grid, along with the CFL factor 0.25 (Δ𝑡 = 6.03 × 10−3 𝑅). Following an initial numerical transient, the magnetosphere 33 Figure 2.9: Aligned rotator after two rotation periods. Magnetic field lines are shown with white solid lines on the left half and black solid lines on the right. Electric charge density is shown with a colormap on the left, and the distribution of troubled elements is shown with gray shades on the right. The light cylinder radius (𝑟 LC = Ω−1 ) is shown with magenta dashed lines. of the rotator gradually expands and the system reaches a quasi-steady state after one rotation period. In Figure 2.9, we show the distribution of electric charge density and the structure of magnetic fields after two rotation periods (𝑡 = 20𝜋𝑅). Our scheme successfully reproduces all characteristic features of the aligned rotator magnetosphere. An equatorial current sheet is formed outside the light cylinder radius 𝑟 LC = Ω−1 , and magnetic field lines far from the equatorial plane open up to form a monopolelike configuration. Because of the grid resistivity, magnetic field lines 𝑟 ≳ 2𝑟 LC spuriously reconnect through the equatorial current sheet. The troubled cell indicator faithfully tracks the current sheet and the regions with rapid variations of the magnetic field, which are likely to develop oscillations on the DG grid, switching elements to the more robust FD grid. The widening of the distribution of the troubled elements is observed in the outer region 𝑟 > 10. While elements near the center of the domain has a cubic shape, the elements in this outer region are deformed (curved) cubes that build up an outer spherical shell (see Fig. 2.8), and the Jacobian matrix that maps logical and physical coordinates is no longer constant within an element. Ideally, this should have marginal effects on the behavior of the troubled cell indicator, where empirically we find that the indicator becomes slightly more 34 sensitive on the elements with curved shapes. Oblique rotator Having a misalignment angle between the magnetic moment and the rotation axis, an oblique rotator serves as a more realistic model of astrophysical pulsars (Spitkovsky, 2006; Kalapotharakos and Contopoulos, 2009; Petri, 2012; Petri, 2016; Carrasco, Palenzuela, and Reula, 2018). Since the configuration is no longer axisymmetric, a full 3D simulation is required to study this problem. We use the same simulation setup as the aligned rotator test, but tilt the initial magnetic field by an inclination angle 𝜃 = 𝜋/4. The system reaches a steady state after about one rotation period. Figure 2.10 shows simulation snapshots on the equatorial plane (left panel) and meridional plane (right panel) after two periods of rotation. Generic features of the solution are similar to the aligned rotator. Beyond the light cylinder radius, a current sheet is formed and magnetic field lines are opened up. Now that the magnetic axis is misaligned with the rotation axis, the current sheet has a periodically modulated curved 3D geometry, appearing as a spiral pattern on the equatorial plane. It is clearly visible that the troubled cell indicator robustly captures and tracks magnetic reconnection points and the spiral current sheet so that the solution can be evolved on the FD grid in those regions. The remainder of the domain keeps evolving on the DG grid, which is computationally more efficient. One can compute the spin-down luminosity of the rotator ∮ 𝐿= 𝑆𝑖 𝑑Σ𝑖 , (2.54) where 𝐸 2 + 𝐵2 𝑎 𝑛 + 𝜀 𝑎𝑏𝑐 (2.55) (3) 𝐸 𝑏 𝐵 𝑐 2 is the Poynting vector. We perform simulations with a lower grid resolution 𝑙 = 4 for different inclination angles and compute the spin-down luminosity Eq. (2.54) after two rotation periods at 𝑟 = 6𝑅. Figure 2.11 shows the measured values. The inclination dependence of the spin-down luminosity 𝐿 is well fitted with the relation (Spitkovsky, 2006) 𝐿 = 𝐿 0 (𝑘 1 + 𝑘 2 sin2 𝜃) (2.56) 𝑆𝑎 = yielding 𝑘 1 = 1.04 and 𝑘 2 = 1.24, where 𝐿 0 is the luminosity of the aligned configuration (𝜃 = 0). 35 Figure 2.10: Oblique rotator: Simulation snapshot on the equatorial (top) and meridional (bottom) plane after two periods of rotation. Plotted physical quantities and their visualizations are the same as Fig. 2.9. 36 2.50 Fit 2.25 L/L0 2.00 1.75 1.50 1.25 1.00 0.75 0 15 30 45 60 75 90 Inclination Angle (◦ ) Figure 2.11: Oblique rotator: Inclination angle dependence of the spin-down luminosity. Numerical results (black dots) are fitted with the formula (2.56), shown with a gray dashed line. 2.4.4 Performance comparison between DG and FD grids One of the main goals of this work is to assess the performance and cost-saving potential of using a DG-FD hybrid method for global FFE simulations of compact binary magnetospheres. We do so in two steps. First, we establish an accuracy benchmark to identify corresponding DG and FD resolution requirements for the same level of accuracy. Second, using this optimal choice, we estimate the cost-savings/speed-up factor of the DG-FD hybrid methods over traditional FD approaches for the problems presented in this work. Accuracy comparison Depending on which scheme is taken as the baseline, the DG-FD hybrid method can be interpreted either as a sophisticated shock-capturing technique for the DG method, or an FD method that compresses a group of cells into a high-order spectral representation on smooth regions (Deppe et al., 2022). The ‘exchange ratio’ between these two grids, namely (𝑁 + 1) 𝐷 DG grid points and (2𝑁 + 1) 𝐷 FD grid points, has been determined by equalizing the maximum admissible time step sizes (Dumbser, Zanotti, Loubère, et al., 2014). 37 10−2 Error(B i ) 10−3 10−4 10−5 FD2 FD4 DG-P5 DG-P5 (unfiltered) 10−6 0 5 10 t/T 15 20 Figure 2.12: Error norm of magnetic field over time from a test evolving a sinusoidal fast wave in a periodic box (described in Sec. 2.4.4). For the FD runs, we perform the test without (FD2) and with (FD4) the high-order flux correction. An important follow-up question is comparing the accuracy between the DG and FD grids when the number of grid points is subject to the above ratio. For example, does a compression of 113 FD grid points into a 𝑃5 DG mesh with 63 grid points in a smooth region lead to an increase or decrease in accuracy? Clearly, the answer is highly dependent on the details of DG (e.g. order of the polynomial, how filtering is applied) and FD solvers (e.g. reconstruction scheme, high-order corrections), which needs to be assessed on a case-by-case basis. However, it is desirable that the DG and FD solvers have similar levels of accuracy in smooth regions. For instance, coupling a low-order DG scheme with a very high-order FD reconstruction is not ideal since the evolution on the FD grid is computationally too expensive considering the overall achievable accuracy with such a choice. This may possibly make adopting a low-order FD scheme and using an increased number of elements overall more efficient. On the other hand, hybridizing a high-order DG scheme with a low-order FD scheme introduces a relatively large numerical diffusion on the FD grid, artificially smearing out important features, 38 especially on smooth regions close to a discontinuity. In this case, the quality of the solution from the DG-FD hybridization, despite its algorithmic complication, might be no better than simply applying an aggressive DG limiter. A desired sweet spot is setting the DG and FD discretization to have the same order of convergence. As a fiducial case, we consider the same setup used for the 1D test problems: a DG-𝑃5 with the highest mode filtered out and a FD solver using the WENO5-Z reconstruction with 𝑞 = 2 along with the fourth-order derivative corrector. Both discretizations are fifth-order convergent for smooth solutions. We perform a simple numerical experiment as follows. The 1D fast wave problem in Sec. 2.4.1 is modified to a smooth initial profile 𝐸 𝑧 = −𝐵 𝑦 = sin(2𝜋𝑥/𝜆) with the wavelength 𝜆 = 2. The computational domain [0, 𝜆] 3 is split into four elements in each spatial direction. The initial condition is evolved up to 20 wave crossing times using periodic boundary condition. Figure 2.12 shows the time evolution of the error norm of the magnetic field. While both the DG and FD discretizations used in this test have the same fifthorder convergence, the FD grid has a twice smaller grid spacing and shows better accuracy for 𝑡 ≤ 20𝜆. We note the boundedness of the error norm on the DG grid, demonstrating a lower numerical dissipation and its resulting suitability for problems involving long-range wave propagation. By contrast, the error norm on the FD grid increases monotonically with time, approaching the same level of error as the DG grid at 𝑡 ≥ 20𝜆. We cautiously interpret this result in the following way. In long-term magnetospheric simulations, replacing the group of (2𝑁 + 1) 𝐷 FD cells with a DG-𝑃 𝑁 spectral mesh in smooth regions likely does not harm global accuracy. In simulations with realistic astrophysical scenarios, the solution is not smooth everywhere but will have localized large gradients such as current sheets separated by smooth regions. The global numerical error will then be dominated by the regions with large gradients, since a shock-capturing FD scheme (such as WENO5-Z) will fall back to a lower order. As an additional example, in Fig. 2.12 we also show results from an unfiltered DG-𝑃5 scheme and an FD scheme without the high-order flux correction. The accuracy of the solution on the FD grid is significantly lower without the high-order corrections, the error being even higher than the DG grid using twice fewer grid points per spatial dimension. The unfiltered DG-𝑃5 has a sixth-order convergence 39 9.6x Relative Speedup 10 8 6 5.2x 4.4x 3.5x 4 3.3x 2.6x 2 0 50 40 30 20 10 Portion of FD elements (%) 0 Figure 2.13: Single-core wall-clock speedup of the 1D Alfvén wave problem for different fractions of elements using the FD grid. An ideal scaling between all-DG and all-FD is shown with a dashed gray line. The measured scaling (yellow dashed line) fit shows that a DG element runs 4.4× faster on average than an FD element. 10%/20% fraction of FD elements give 3.3× (5.2×) / 2.6× (3.5×) overall (ideal) speedup as shown by the solid (dashed) gray line. and shows smaller error than the FD grid at 𝑡 ≳ 10𝜆. As soon as the DG grid has a higher order of discretization than the order of FD discretization, DG shows a better accuracy in spite of having fewer grid points. In summary, in particular for the hybridization of a DG-𝑃5 and a WENO5-Z FD scheme, we conclude that switching from the FD to the DG grid results in a marginal loss of local accuracy in smooth regions, which is unlikely to affect the global error in actual simulations. Efficiency Having confirmed that the DG and FD grids show a comparable level of accuracy in the setup we use, we now assess potential computational cost savings when using the hybrid scheme. Since the number of grid points on the DG grid is fewer than the FD grid by a factor of (11/6) 3 = 6.2 in 3D, a similar amount of computational speedup is naturally anticipated. In order to quantify the actual speedup in our implementation, 40 we run the stationary Alfvén wave test (Sec. 2.4.1) using (𝑁𝑥 , 𝑁 𝑦 , 𝑁 𝑧 ) = (8, 32, 8) elements. We manually force elements to stay on the DG grid for 𝑦 > 0 and on the FD grid for 𝑦 < 0.8 The fraction of elements running on the FD grid is changed by shifting the upper and lower bounds of the 𝑦 coordinate. This allows us to vary the fraction of FD to DG grid points in a controlled way. We additionally disable parallelization and carry out all tests in this section on a single CPU core to disentangle parallel scaling from algorithmic performance. Our fiducial benchmark is then given by the overall wall-clock time of the evolution algorithm. Figure 2.13 shows the relative speedup compared to the case when all elements are using the FD grid. Since the DG solver does not involve computationally expensive reconstruction steps and has less data communication, it can perform ∼50% more grid point updates per second compared to the FD solver, which results in a combined 9.6× overall speedup. In absolute terms, the measured zone-cycles per CPU second are 108K when all elements are on the DG grid, and 69K when all elements are on the FD grid. Assuming perfect scaling, the overall speedup relative to the all-FD case can be estimated with the simple formula 1 𝑥 + (1 − 𝑥)/ 𝑓 (2.57) where 𝑥 is the portion of elements using the FD grid and 𝑓 is the speedup factor of the DG grid with respect to the FD grid. In Fig. 2.13, we show the ideal speedup scaling with 𝑓 = 9.6 (gray dashed curve). However, our measurements show that as soon as there is any portion of FD elements, the effective speedup factor drops down to 𝑓 = 4.4, implying the presence of an algorithmic bottleneck. When half the elements are using the FD grid, measured zone-cycles per second are 65K, even a bit slower than the all-FD case. This somewhat unexpected drop in performance is likely to be related to an extra interpolation step required at the interface between DG and FD elements to convert the ghost zone data sent between the elements. As a representative number, we quote the achievement of 3.3× (5.2×) / 2.6× (3.5×) overall (ideal) speedup when 10% / 20% of elements are using the FD grid. A separate, detailed profiling of the code suggests at most 10% overhead from the controlling part of the DG-FD hybrid algorithm (applying the TCI to the solution and 8 Therefore, in this controlled experiment, the overhead related to (i) execution of the TCI and (ii) rolling back the time step on troubled elements are excluded. Each element still sends out ghost-zone data to neighboring elements at every time step. 41 undoing a time step if an element is troubled), which was excluded in the speedup test described above. Comparing simulations of a smooth wave solution using the DG grid on all elements with the adaptive DG-FD scheme turned on and off showed less than 4% of difference in total runtime. 2.5 Conclusions We have developed a new numerical scheme for general-relativistic FFE based on a DG-FD hybrid method. The numerical scheme combines a high-order spatial discretization with IMEX time stepping to handle stiff source terms associated with maintaining the FFE constraints. We have further implemented a troubled cell indicator capable of flagging spurious features in the DG evolution, allowing the associated elements to transition to a more dissipative conservative FD scheme. In this way, the scheme achieves high-order convergence for smooth problems while robustly tracking and capturing large gradients present in solutions such as current sheets. Our implementation is based on the open-source SpECTRE code and successfully passes and reproduces a suite of standard test problems in one- and threedimensions. In particular, we achieve up to eighth-order numerical convergence in smooth vacuum problems. A quantitative measure of the numerical resistivity in our scheme, in particular using the approaches by (Rembiasz et al., 2017; Mahlmann, Aloy, et al., 2021b), will be explored in future works. In order to assess potential cost savings of this approach over more traditional FDonly schemes, we have performed a quantitative assessment of its accuracy and efficiency. We find that our approach has a potential to speed up FD simulations by the factor of 2–3 with little to no loss of accuracy. We further demonstrate an additional optimization potential of (in some cases) up to a factor 2, when compared to the ideal speed up of the code. Similar or even larger performance gains have been reported when adopting GPU-based parallelization strategies (e.g. Liska et al., 2022; Del Zanna, Landi, et al., 2024; Grete, Glines, and B. W. O’Shea, 2021). Additional improvements may come from a more optimal set of troubled cell indicator criteria or a dynamic power monitor (e.g. see Szilágyi, 2014), which can potentially facilitate a more economical grid switching between DG and FD. The DG-FD hybrid scheme presented here is particularly well suited to study wave propagation as well as accuracy-limited problems, such as steady-state twists or magnetospheric explosion dynamics that evolve on long timescales (e.g. Parfrey, Beloborodov, and Hui, 2013; Yuan, Spitkovsky, et al., 2019; Yuan, Y. Levin, et al., 42 2021). Such studies will be the subject of future work. Acknowledgements We are grateful to Kyle Nelli and Nils Vu for helpful discussions and technical support during various stages of this project. Y.K. acknowledges Cristòbal Armaza for encouraging conversations. Computations were performed on the Wheeler cluster and Resnick HPC Center at Caltech. 43 Chapter 3 GENERAL RELATIVISTIC MAGNETIZED BONDI-HOYLE-LYTTLETON ACCRETION Kim, Yoonsoo and Elias R. Most (2025). “General relativistic magnetized BondiHoyle-Lyttleton accretion with a spin-field misalignment: Jet nutation, polarity reversals, and Magnus drag”. In: Physical Review D 111 (8), p. 083025. doi: 10.1103/PhysRevD.111.083025. url: https://link.aps.org/doi/10. 1103/PhysRevD.111.083025. The dynamics of a black hole traveling through a plasma – a general relativistic extension of the classic Bondi-Hoyle-Lyttleton accretion problem – can be related to a variety of astrophysical contexts, including the aftermath of binary black hole (BBH) mergers in gaseous environments. We perform three-dimensional general relativistic magnetohydrodynamics simulations of Bondi-Hoyle-Lyttleton accretion onto a spinning black hole when magnetic field of the incoming wind is inclined to the spin axis of the black hole. Irrespective of inclination but dependent on the wind speed, we find that the accretion flow onto the black hole can become magnetically arrested, launching an intermittent jet whose formation is assisted by a turbulent dynamo-like process in the inner disk. The upstream ram pressure of the wind bends the jet, and confines the angular extent into which the magnetic flux tubes ejected from quasi-periodic eruptions are released. Recoil from magnetic flux eruptions drives strong oscillations in the inner accretion disk, resulting in jet nutation at the outer radii and occasionally ripping off the inner part of the accretion disk. When the incoming magnetic field is perpendicular to the spin axis of the black hole, we find that the magnetic polarity of the jets can undergo a stochastic reversal. In addition to dynamical friction, the black hole experiences a perpendicular drag force analogous to the Magnus effect. Qualitative effects of the incoming magnetic field orientation, the strength of the magnetization, and the incoming wind speed are investigated as well. 3.1 3.1.1 Introduction BBH merger remnant in AGN disk The merger of a binary black hole (B. P. Abbott et al., 2016; B. P. Abbott et al., 2019; R. Abbott et al., 2021a; R. Abbott et al., 2023) within the accretion disk 44 of an active galactic nucleus (AGN) (N. C. Stone, Metzger, and Haiman, 2017; Tagawa, Haiman, and Kocsis, 2020; Gröbner et al., 2020; Ishibashi and Gröbner, 2020; McKernan et al., 2022; Ford and McKernan, 2022; Kaaz, Schrøder, et al., 2023) is thought to be one of major channels of the observed BBH mergers, but is also an interesting astrophysical scenario within the context of multi-messenger astronomy. If asymmetry is present in a black hole binary, the resulting anisotropic emission of gravitational waves from the merger can impart a recoil onto the postmerger remnant black hole (Gonzalez, Sperhake, et al., 2007; Campanelli et al., 2007a). Since the kicked remnant will be moving through a gas-rich environment, a luminous accretion flow onto or a relativistic jets from the BH may give rise to an observable post-merger electromagnetic signal (Graham et al., 2020; K. Chen and Z.-G. Dai, 2024). The gaseous environment of the AGN disk can affect the long-term evolution of an embedded BBH, often putting constraints on its orbital configuration. Newtonian studies suggest that orbital and spin axes of a BBH embedded in a gaseous environment align over time (Bogdanovic, Reynolds, and Miller, 2007; Coleman Miller and Krolik, 2013), and the orbit is driven to be aligned with the AGN disk plane as well (Dittmann, Dempsey, and H. Li, 2024). These findings, put together, indicate that both the orbital angular momentum and spin of the BBH are likely aligned with the AGN disk. Unless anisotropic gravitational radiation induces a significant torque on the system, the remnant would retain its prior spin direction. Recent large-scale cosmological simulations revealed that the magnetic field of AGN disks are predominantly toroidal (parallel to the disk plane) (Hopkins, Grudic, et al., 2024; Hopkins, Squire, et al., 2024); see also Gaburov, Johansen, and Y. Levin (2012) for an earlier work. The presence of mixed poloidal-toroidal configurations is also in line with simulations of magnetized circumbinary disks around BBHs as well (Most and H.-Y. Wang, 2024b; Most and H.-Y. Wang, 2024a). Overall, this motivates a theoretical investigation on a recoiled black hole flying through a plasma embedded with a magnetic field misaligned with the spin of the black hole. 3.1.2 Bondi-Hoyle-Lyttleton accretion Bondi-Hoyle-Lyttleton (BHL) accretion (Bondi, 1952; Hoyle and Lyttleton, 1939) is a classic problem in astrophysics involving a gravitational accretor traveling through a uniform fluid. Despite being highly simplified, it can be applied to a wide range of astrophysical systems including common envelope phases of binary 45 star evolution (MacLeod and Ramirez-Ruiz, 2015; MacLeod, Antoni, et al., 2017; Murguia-Berthier et al., 2017; López-Cámara, Moreno Méndez, and De Colle, 2020), wind-fed X-ray binaries (El Mellah, Sundqvist, and Keppens, 2018), star clusters (Kaaz, Antoni, and Ramirez-Ruiz, 2019) or protoplanetary disks (Moeckel and Throop, 2009). Owing to its astrophysical significance, a large volume of analytical and numerical studies exist in the literature on Newtonian (see Edgar, 2004; Foglizzo, Galletti, and Ruffert, 2005 for a review) and relativistic regimes (Petrich, Shapiro, and Teukolsky, 1988; Petrich, Shapiro, Stark, et al., 1989; Font and Ibanez, 1998b; Font, Ibanez, and P. Papadopoulos, 1999; Font and Ibanez, 1998a; Dönmez, Zanotti, and Rezzolla, 2011; Dönmez, 2012; Zanotti, Roedig, et al., 2011; Penner, 2011; Penner, 2013; Lora-Clavijo and Guzman, 2013; Koyuncu and Dönmez, 2014; Lora-Clavijo, Cruz-Osorio, and Méndez, 2015; Blakely and Nikiforakis, 2015; Cruz-Osorio, Lora-Clavijo, and Guzman, 2012; Cruz-Osorio and Lora-Clavijo, 2016; Tejeda and Aguayo-Ortiz, 2019; Cruz-Osorio and Rezzolla, 2020; Gracia-Linares and Guzmán, 2015; Kaaz, Murguia-Berthier, et al., 2023; Gracia-Linares and Guzmán, 2023). Basic physical scales associated with BHL accretion are the accretion radius1 2𝐺 𝑀 , 𝑣 2∞ (3.1) 𝑅𝑎 2𝐺 𝑀 = 3 , 𝑣∞ 𝑣∞ (3.2) 𝑅𝑎 = the accretion timescale 𝜏𝑎 = and the Bondi-Hoyle-Lyttleton mass accretion rate 2 2 4𝜋𝐺 𝑀 𝜌∞ , 𝑀¤ BHL = 𝜋𝑅𝑎2 𝜌∞ 𝑣 ∞ = 𝑣 3∞ (3.3) where 𝐺 is the gravitational constant, 𝑀 is the mass of the accreting object, 𝑣 ∞ is the asymptotic relative velocity, and 𝜌∞ is the asymptotic mass density of the fluid. A major challenge in computational approaches to this problem is its inherent multiscale nature, namely simultaneously resolving the size of the accreting object 𝑟 0 and 1 An alternate definition of the accretion radius exists in the literature 𝑅𝑎 = 2𝐺 𝑀 𝑐2𝑠,∞ + 𝑣 2∞ , where 𝑐 𝑠,∞ is the asymptotic sound speed of the incoming fluid. We adopt the definition (3.1) throughout this chapter. 46 the accretion radius 𝑅𝑎 on a single numerical grid. For example, a black hole with mass 𝑀 has 𝑟 0 ≈ 𝑟 𝑔 where 𝐺𝑀 𝑟𝑔 = 2 (3.4) 𝑐 is the gravitational radius of the black hole. The ratio between the two length scales is 𝑅𝑎  𝑣 ∞  −2 ∼ . (3.5) 𝑟𝑔 𝑐 Also, time integration needs to be performed at least several times of 𝜏𝑎 to reach a steady state, which is longer than the dynamical timescale associated with the black hole by a factor of  𝑣  −3 𝜏𝑎 ∞ ∼ . (3.6) (𝑟 𝑔 /𝑐) 𝑐 A large separation in both length and time scales, which is especially severe for a compact accretor such as a black hole, rapidly increases the computational cost for realistic values of 𝑣 ∞ . As a result, many studies are often forced to assume an unrealistically fast velocity of the BH relative to the fluid. Due to its high computational cost, most numerical studies on general relativistic BHL accretion have considered hydrodynamic flows on either 2D planar or 3D axisymmetric geometry. However, inclusion of magnetic fields can dramatically alter the flow morphology, and a restrictive nature of the assumed spatial symmetry might fail to fully capture multi-dimensional effects. Following the first study of 2D magnetohydrodynamic (Penner, 2011) and 3D hydrodynamic (Gracia-Linares and Guzmán, 2015) flows, the first simulations of general relativistic magnetohydrodynamics (GRMHD) BondiHoyle-Lyttleton accretion in full 3D have been carried out only recently by Kaaz, Murguia-Berthier, et al. (2023) and Gracia-Linares and Guzmán (2023). Each of these studies respectively explored jet launching from the BH (Kaaz, MurguiaBerthier, et al., 2023) and the effect of the BH spin-wind orientation on the shock morphology (Gracia-Linares and Guzmán, 2023), which can only be properly captured in a 3D MHD simulation. In this work, we perform GRMHD simulations of a spinning black hole traveling through a fluid embedded with a magnetic field inclined to the spin axis of the BH. The physical scenario is approximated by a relativistic Bondi-Hoyle-Lyttleton accretion problem with a magnetized wind. We examine large-scale morphology and temporal evolution of the accretion flow, both of which are closely related to intermittent jet launching and magnetic flux eruptions from the BH. The impacts 47 of magnetic field orientation, magnetization, and the wind speed are assessed by systematically varying simulation parameters. Another purpose of our numerical experiment is to measure the outflow luminosity (power) and determine the efficiency with which 𝑀¤ BHL 𝑐2 can be converted into an energy outflow. The drag force exerted on the accreting BH is also measured and its astrophysical implications are discussed. This chapter is organized as follows. In Sec. 3.2, we describe our numerical setup and methods. We present our results in two steps, focusing on a specific parameter set first in Sec. 3.3, before generalizing it in Sec. 3.4. We present discussions on the results in Sec. 3.5, then summarize our main findings along with limitations and future perspectives in Sec. 3.6. 3.2 Methods The background spacetime is set to the Kerr metric in (Cartesian) Kerr-Schild coordinates (see also Appendix A). The spin of the black hole is aligned with the 𝑧ˆ axis of the computational domain. The exact form of the spacetime metric is 2𝐺 𝑀𝑟 3 /𝑐2 𝑑𝑠2 = −𝑐2 𝑑𝑡 2 + 𝑑𝑥 2 + 𝑑𝑦 2 + 𝑑𝑧 2 + 4 𝑟 + 𝑎2 𝑧2  2 𝑟 (𝑥𝑑𝑥 + 𝑦𝑑𝑦) + 𝑎(𝑦𝑑𝑥 − 𝑥𝑑𝑦) 𝑧𝑑𝑧 , × 𝑐 𝑑𝑡 + + 𝑟 𝑟 2 + 𝑎2 (3.7) where 𝑀 is the mass and 𝑎 is the spin parameter of the black hole (with the unit of length). The coordinate variable 𝑟 is defined as 𝑥 2 + 𝑦2 𝑧2 + = 1. 𝑟 2 + 𝑎2 𝑟 2 (3.8) We solve the equations of ideal GRMHD, which are given in terms of the rest-mass density current 𝐽 𝜇 = 𝜌𝑢 𝜇 , (3.9) and the stress-energy tensor, 𝑇 𝜇𝜈  = 𝜌+𝑒+𝑝+𝑏 2   𝑏 2 𝜇𝜈 𝑢 𝑢 + 𝑃+ 𝑔 − 𝑏 𝜇 𝑏𝜈 , 2  𝜇 𝜈 (3.10) with its electromagnetic component 𝑇EM = 𝑏 2 𝑢 𝜇 𝑢 𝜈 + 𝜇𝜈 𝑏 2 𝜇𝜈 𝑔 − 𝑏 𝜇 𝑏𝜈 , 2 (3.11) 48 where 𝜌 is the rest mass density, 𝑒 is the internal energy density, 𝑝 is the pressure, 𝑢 𝜇 is the four-velocity, and 𝑏 𝜇 the comoving magnetic field of the fluid, with 𝑏 2 = 𝑏 𝜇 𝑏 𝜇 . The electromagnetic field is evolved using the dual field strength tensor, ∗ 𝜇𝜈 𝐹 = 𝑏 𝜇𝑢𝜈 − 𝑢 𝜇 𝑏𝜈 . (3.12) ∇𝜇 𝐽 𝜇 = 0 , (3.13) ∇ 𝜇 ∗𝐹 𝜇𝜈 = 0 , (3.14) ∇ 𝜇𝑇 𝜇𝜈 = 0 . (3.15) The evolution equations are We find it advantageous to define a normal magnetic field as2 𝐵¯ 𝑖 = ∗𝐹 0𝑖 . (3.16) We further model the gas dynamics using an ideal fluid equation of state 𝑝 = 𝑒 (Γ − 1) , (3.17) where Γ = 5/3 is the adiabatic index. 3.2.1 Numerical setup We numerically solve the ideal GRMHD system using AthenaK (J. M. Stone, Mullen, et al., 2024), a rewrite of Athena++ code (J. M. Stone, Tomida, et al., 2020) with a performance portability library Kokkos (Trott et al., 2022). The computational domain [−40960𝑟 𝑔 , 40960𝑟 𝑔 ] 3 is discretized into a uniform Cartesian grid with the base grid resolution 2563 . 13 levels of static mesh refinement are applied around the coordinate origin, with the innermost mesh [−5𝑟 𝑔 , 5𝑟 𝑔 ] 3 providing a resolution of ∼26 grid points per 𝑟 𝑔 . Time integration is performed using a second order Runge-Kutta stepper, piecewise parabolic reconstruction (Colella and 2 This definition of the magnetic field is not covariant, as it is different by a factor of 𝛼 (the lapse function in 3+1 decomposition) from the Eulerian magnetic field 𝐵𝑖 commonly adopted in numerical relativity or relativistic electrodynamics literature (e.g., Baumgarte and Shapiro, 2010; Paschalidis and Shapiro, 2013; Komissarov, 2004a). However, we adopt the definition (3.16) here since it simplifies the handling of the solenoidal (divergence-free) constraint on the magnetic field as, 𝜕𝑖 𝐵¯ 𝑖 = 0 , in the Kerr-Schild coordinates. 49 Woodward, 1984), an HLLE Riemann solver (Harten, Lax, and Leer, 1983; Einfeldt et al., 1991), and a constrained transport algorithm (Gardiner and J. M. Stone, 2008). We use the mass density and internal energy floor values 𝜌floor = 10−14 𝜌∞ , 𝑒 floor = 𝜌floor 𝑐2 /3, and cap the maximum Lorentz factor of the fluid to 𝑊max = 20. The drift frame flooring technique (Ressler, Tchekhovskoy, et al., 2017) is applied to limit the comoving magnetization to 𝜎max = 50. AthenaK can be compiled and run on graphics processing unit (GPU) devices at scale, providing O (107 ) cell updates per second per each GPU card for large scale GRMHD simulations. The simulations presented here overall have been performed on 132 GPU nodes (792 NVIDIA Volta cards) at OLCF Summit cluster, costing 900 node hours per 104𝑟 𝑔 /𝑐 integration time on average. The total computational cost used for all simulations is about 34,000 node hours. 3.2.2 Initial data Over the whole computational domain, matter profile is initialized with a uniform rest mass density 𝜌∞ and spatial velocity ! ′ 𝑣 ∞ , 0, 0 , (3.18) 𝑢𝑖 = − √︁ 1 − 𝑣 2∞ /𝑐2 where 𝑣 ∞ is the asymptotic incoming speed of the fluid, 𝑐 is the speed of light, and ′ 𝑢𝑖 = 𝑢𝑖 + 𝛽𝑖 𝑢 0 is the normal-frame spatial velocity which is a primitive variable used in the code.3 Given the sound speed 𝑐 𝑠,∞ , the fluid internal energy density 𝑒∞ = 𝑐2𝑠,∞ 𝜌∞ Γ(Γ − 1 − 𝑐2𝑠,∞ /𝑐2 ) , (3.19) and pressure 𝑝 ∞ = 𝑒 ∞ (Γ − 1), (3.20) can be initialized accordingly. The inclination between the magnetic field of the incoming wind and its velocity can be arbitrary in general. In this study, in order to narrow down the parameter space and focus on the misalignment between the BH spin and the magnetic field, we assume that the incoming magnetic field is perpendicular to the wind velocity. We initialize the magnetic field as 𝐵¯ 𝑖 = 𝐵0 (0, sin 𝜃 𝐵 , cos 𝜃 𝐵 ), (3.21) 3 See section 4 of J. M. Stone, Tomida, et al. (2020) for the definition of GRMHD primitive variables used in Athena++ or the section 3 of J. M. Stone, Mullen, et al. (2024) for AthenaK. 50 where 𝐵0 is the field strength and 𝜃 𝐵 is the inclination angle between the BH spin and the magnetization of the incoming fluid. In the asymptotic limit (𝑥 𝑖 → ∞), the magnetic field strength 𝐵0 is related to the magnetization of the fluid 𝜎 as 𝐵02 /(1 − 𝑣 2∞ /𝑐2 ) (𝑏 2 )∞ 𝜎∞ = = . (𝜌𝑐2 + 𝑒 + 𝑝)∞ 𝜌∞ 𝑐2 + Γ𝑒 ∞ (3.22) The plasma 𝛽-parameter and the magnetization 𝜎 are related via 𝛽∞ = 𝑝 gas 𝑝∞ 2 (𝑐 𝑠,∞ /𝑐) 2 = 2 = . 𝑝𝑏 (𝑏 )∞ /2 Γ 𝜎∞ (3.23) Input parameters of our simulations are the asymptotic fluid mass density 𝜌∞ , accretion radius 𝑅𝑎 , asymptotic sound speed 𝑐 𝑠,∞ , the plasma parameter 𝛽∞ , and the magnetic field inclination angle 𝜃 𝐵 . The incoming speed of the fluid 𝑣 ∞ is computed from 𝑣 2∞ = 2𝐺 𝑀/𝑅𝑎 and used to initialize the spatial velocity (3.18). From the sound speed 𝑐 𝑠,∞ , both internal energy density and pressure can be initialized using (3.19) and (3.20). The plasma parameter 𝛽∞ determines the relativistic magnetization 𝜎∞ via (3.23), in turn giving 𝐵0 from (3.22), which we use to initialize the magnetic fields as (3.21). Primitive variables are initialized everywhere in the computational domain with the values prescribed as above. Simulation parameters are listed in Table 3.1. We particularly focus on the cases that the magnetic field of the incoming wind is perpendicular or inclined with respect to the BH spin (see Kaaz, Murguia-Berthier, et al., 2023 for the aligned magnetic field configuration). For all simulations we adopt a BH spin 𝑎/𝑟 𝑔 = 0.9 and the asymptotic sound speed 𝑐 𝑠,∞ = 0.05𝑐. Our representative, fiducial model assumes 𝛽∞ = 10 with a horizontal orientation of the magnetic field 𝜃 𝐵 = 90◦ , along with the fluid incoming speed 𝑣 ∞ = 0.1𝑐 corresponding to 𝑅𝑎 = 200𝑟 𝑔 ; this model is labeled as 𝛽10 -𝜃 90 -𝑅200 . To explore the influence of magnetic field orientation relative to the BH spin, we run the identical setup but varying 𝜃 𝐵 = 22.5◦ , 45◦ , 67.5◦ , each of which is labeled with 𝜃 23 , 𝜃 45 , and 𝜃 68 . We perform an experiment on the effect of a weaker magnetization 𝛽∞ = 100 while keeping other parameters fixed from the fiducial model. Lastly, we run two more models with a smaller (𝑅𝑎 = 50𝑟 𝑔 ) and a larger (𝑅𝑎 = 400𝑟 𝑔 ) accretion radius to test the influence of the fluid incoming speed, also keeping the other parameters fixed to the same values as the fiducial model. 51 Table 3.1: List of the models and parameters considered in this work. Each column denotes the accretion radius 𝑅𝑎 , asymptotic fluid incoming speed 𝑣 ∞ , asymptotic Mach number M, accretion time scale 𝜏𝑎 , the plasma beta parameter of the incoming fluid 𝛽∞ , and the magnetic field inclination angle 𝜃 𝐵 . The spin of the BH is set to 𝑎 = 0.9𝑟 𝑔 for all simulations. Label 𝛽10 -𝜃 90 -𝑅200 𝛽10 -𝜃 68 -𝑅200 𝛽10 -𝜃 45 -𝑅200 𝛽10 -𝜃 23 -𝑅200 𝛽100 -𝜃 90 -𝑅200 𝛽10 -𝜃 90 -𝑅50 𝛽10 -𝜃 90 -𝑅400 3.2.3 𝑅𝑎 [𝑟 𝑔 ] 200 200 200 200 200 50 400 𝑣 ∞ /𝑐 M 0.1 0.1 0.1 0.1 0.1 0.2 0.07 2.0 2.0 2.0 2.0 2.0 4.0 1.4 𝜏𝑎 [𝑟 𝑔 /𝑐] 2000 2000 2000 2000 2000 250 5660 𝛽∞ 𝜃𝐵 Comments 10 10 10 10 100 10 10 90◦ 67.5◦ 45◦ 22.5◦ 90◦ 90◦ 90◦ Fiducial setup Varying 𝜃 𝐵 High 𝛽 Faster 𝑣 ∞ Slower 𝑣 ∞ Boundary conditions At the outer boundary of the computational domain facing the +𝑥ˆ (upstream) direction, we impose a Dirichlet boundary condition injecting a constant wind profile same as the initial data. At all other sides of the domain boundary, primitive variables at the outermost grid points are copied into the ghost zones to impose a free streaming boundary condition. 3.2.4 Analysis In addition to the magnetohydrodynamics variables evolved on the grid, we compute following integral quantities from simulation data in order to monitor time evolution of the system: • Mass accretion rate 𝑀¤ = ∮ √ (−𝜌𝑢𝑟 ) −𝑔 𝑑𝜃𝑑𝜙, (3.24) where 𝜌 is the rest mass density and 𝑢𝑟 is the radial component of the fourvelocity. Note the minus sign in the integrand, which makes a positive value indicate mass inflow. • Total energy (in-)flux 𝐸¤ = ∮ √ 𝑇 𝑟𝑡 −𝑔 𝑑𝜃𝑑𝜙 . (3.25) 52 • Angular momentum (out-)flux 𝐽¤ = ∮ √ 𝑇 𝑟𝜙 −𝑔 𝑑𝜃𝑑𝜙. • Total magnetic fluxes threading the horizon ∮ 1 √ ΦBH = | 𝐵¯ 𝑟 | −𝑔 𝑑𝜃𝑑𝜙. 2 horizon (3.26) (3.27) For our discussions presented here and in the following sections, we will often refer to the dimensionless total magnetic fluxes threading the BH horizon ΦBH . 𝜙BH ≡ √︃ ¤ 𝑔2 𝑐 𝑀𝑟 (3.28) The dimensionless magnetic flux can be used as an indicator for the accretion state, showing that for 𝜙BH ≳ 20 jets can be launched (Tchekhovskoy, Narayan, and McKinney, 2011). For larger values of 𝜙BH , the accretion flow will become fully magnetically arrested (e.g., Tchekhovskoy, Narayan, and McKinney, 2011; McKinney, Tchekhovskoy, and Blandford, 2012; White, J. M. Stone, and Quataert, 2019; Chatterjee and Narayan, 2022). • Drag force exerted on the BH 𝑖 𝑖 𝐹 𝑖 = −𝐹mom + 𝐹grav , (3.29) 𝑖 where 𝐹mom is the total momentum (out-)flux through a spherical surface ∮ √ 𝑖 (3.30) 𝐹mom = 𝑇 𝑟𝑖 −𝑔 𝑑𝜃𝑑𝜙, 𝑖 and 𝐹grav is the gravitational drag computed with a modified Newtonian formula (Cruz-Osorio and Rezzolla, 2020; Kaaz, Murguia-Berthier, et al., 2023) ∫ 𝑥𝑖 𝑖 𝐹grav = 𝜌 3 𝑑𝑉 . (3.31) 𝑟 A minus sign on the right hand side of Eq. (3.29) accounts for that the momentum loss in a closed volume results in a reaction force to the opposite direction. For 𝑥 example, in our simulation setup in which BH travels to +𝑥 direction, 𝐹mom >0 corresponds to the deceleration of the BH with respect to the ambient medium, 𝑥 where 𝐹grav > 0 corresponds to the acceleration. 53 The flow in the vicinity of the BH is strongly magnetized in our simulations, frequently triggering density and energy floors on troubled grid cells. Since the artificial injection of mass and energy density from numerical flooring can contaminate some of the diagnostics introduced above, we perform surface integrals around the BH at a slightly larger radius than the outer event horizon (e.g., Kaaz, Murguia-Berthier, et al., 2023). In our analysis, the horizon magnetic fluxes ΦBH is integrated over the 𝑖 ¤ 𝐸, ¤ 𝐽, ¤ and 𝐹mom outer event horizon whereas 𝑀, are extracted at 𝑟 = 3𝑟 𝑔 . The gravi𝑖 tational drag 𝐹grav is integrated over the whole volume of the computational domain excluding 𝑟 < 3𝑟 𝑔 in the same regard. We find this approach has no significant effect our diagnostics, see Appendix 3.A for a detailed analysis. The turbulent nature of the accretion flow leads to high-frequency fluctuations of the time series data. For improved readability, we have averaged all time series data over a sliding time window of 100𝑟 𝑔 /𝑐, unless otherwise stated. Computations are performed in a scale-free manner by setting 𝑐 = 𝐺 𝑀 = 𝜌∞ = 1. From simulation results, physical values can be recovered as 𝑥 𝑖 = 𝑟 𝑔 𝑥ˆ𝑖 (3.32) 𝑡 = (𝑟 𝑔 /𝑐) 𝑡ˆ (3.33) 𝜌 = 𝜌∞ 𝜌ˆ (3.34) 𝑀¤ = (𝑟 𝑔2 𝜌∞ 𝑐) 𝑀¤̂ (3.35) 𝐸¤ = (𝑟 𝑔2 𝜌∞ 𝑐3 ) 𝐸¤̂ (3.36) 𝐽¤ = (𝑟 𝑔3 𝜌∞ 𝑐2 ) 𝐽¤̂ (3.37) 1/2 𝐵𝑖 = (𝜌∞ 𝑐) 𝐵ˆ 𝑖 (3.38) 1/2 ΦBH = (𝑟 𝑔2 𝜌∞ 𝑐) Φ̂BH (3.39) 𝐹 𝑖 = (𝑟 𝑔2 𝜌∞ 𝑐2 ) 𝐹ˆ 𝑖 (3.40) where hat variables are the results in the (scale-free) code unit. 3.3 Fiducial model: Jet launching from horizontally magnetized wind In this section, we analyze the results from the baseline model 𝛽10 -𝜃 90 -𝑅200 in detail to highlight several key phenomena observed in the simulation. This model features a wind speed 𝑣 ∞ = 0.1𝑐 and a magnetic field perpendicular (horizontal) to the black hole spin axis, with its strength set by the plasma parameter 𝛽 = 10. 54 Figure 3.1: Initial jet launching process in the 𝛽10 -𝜃 90 -𝑅200 simulation. Shown here are in-plane magnetic field lines (cyan solid lines), magnetization 𝜎, vertical magnetic field 𝐵 𝑧 , and its normalized value 𝐵 𝑧 /|𝐵|. Each column displays physical quantities on the vertical (𝑦𝑧, the face-on direction with respect to the incoming wind) and the equatorial (𝑥𝑦) plane at each simulation time. 3.3.1 Overview Initial jet launching process Shortly following the beginning of the simulation, a shocked region around the BH quickly expands and forms a bow-shaped shock front, which is a common characteristic of supersonic BHL accretion flows (e.g., Foglizzo, Galletti, and Ruffert, 2005; Edgar, 2004; Shima et al., 1985; Font and Ibanez, 1998b). This initial accretion phase is largely hydrodynamical. Over time, the spin of the BH drags the flow into rotation around the horizon, starting to form an accretion disk, and magnetic flux is accumulated around the BH. In Figure 3.1, we show simulation snapshots at an early time (𝑡 ≲ 2.5𝜏𝑎 ) focusing 55 on the evolution of magnetic fields near the BH. While the magnetic field is initially perpendicular to the BH spin, turbulent fluid motion developing in the accretion flow generate vertical (poloidal) magnetic flux, akin to an MHD dynamo. This can be seen from the third and fourth rows of Fig. 3.1, showing the vertical magnetic field (𝐵 𝑧 ) on the equatorial plane. As the turbulent dynamo continues to supply vertical magnetic fluxes into the accretion flow with various eddy sizes, we find that a coarse large-scale magnetic flux separation emerges at 𝑡 ≈ 1.8𝜏𝑎 (the fifth column in Fig. 3.1, appearing as a two-sided spiral shape). This serves as a reservoir of unilateral vertical magnetic field lines attached to the BH, which enables jet launching via the Blandford-Znajek mechanism (Blandford and Znajek, 1977). From the time series plots in Fig. 3.3, we can see that the magnetization around the BH quickly reaches half-MAD (𝜙 𝐵 ≃ 25) levels within 𝑡 ≈ 3𝜏𝑎 , as indicated by the dimensionless magnetic flux 𝜙BH , where a magnetically arrested disk (MAD) state is characterized by 𝜙BH ≃ 50 (Tchekhovskoy, Narayan, and McKinney, 2011). This built-up of magnetic flux coincides with the formation of a magnetized polar funnel region near the black hole as described above. Large-scale flow morphology Fig. 3.2 shows hydrodynamic and magnetic properties of the accretion flow on the meridional (𝑥𝑧) and equatorial (𝑥𝑦) plane at 𝑡 = 20400𝑟 𝑔 /𝑐 ≈ 10𝜏𝑎 . Relativistic jets powered by the BH appear as low-density, highly magnetized (𝜎 ≫ 1) funnel regions traversing the shock cone. The wind from the upstream collides with the jets and deflect them downstream with its ram pressure. The amount of jet bending correlates directly with the wind speed. A pair of these bent, relativistic jets are observed from tailed radio sources such as bent-tail radio galaxies (e.g. Hardcastle and Sakelliou, 2004; O’Dea and Owen, 1986; Miley, Wellington, and van der Laan, 1975; Giacintucci and Venturi, 2009). An accretion disk forms around the BH with the same direction of rotation as the BH spin, and spans out to 𝑟 ∼ 30𝑟 𝑔 on the equatorial plane although its spatial extent is varying over time. At the outermost radius of the disk, its circulatory flow is mixed with the regular downstream flow entering the shock cone, forming a stagnation point, as can be seen from the distribution of velocity streamlines near (𝑥, 𝑦) = (−20, −30)𝑟 𝑔 of the lower left panel of Fig. 3.2. The matter inflow entering the shock cone on the co-rotating side (+𝑦) is smoothly connected to the circulatory accretion flow, where that on the counter-rotating side (−𝑦) collides with 56 Figure 3.2: Transient jets and magnetic flux eruptions in GRMHD Bondi-HoyleLyttleton accretion. Shown here are mass density, 𝜌, (left) and relativistic magnetization, 𝜎, (right) on the meridional plane (top) and the equatorial plane (bottom) from the fiducial model 𝛽10 -𝜃 90 -𝑅200 at 𝑡 = 20400 𝑟 𝑔 /𝑐. The in-plane velocity and magnetic fields are shown with white and cyan streamlines on the left and right panels, respectively. the accretion flow to be pushed radially outward. This leads to the bulk motion of the downstream flow being deflected into − 𝑦ˆ direction. In contrast to accretion simulations initialized or supplied with a finite angular momentum,4 the incoming flow has zero net angular momentum in our setup. Spin-induced frame dragging of the BH is the only source of angular momentum imparted onto the flow, naturally limiting the radial size of the accretion disk. The BH is in a near MAD state (Igumenshchev, Narayan, and Abramowicz, 2003; Narayan, Igumenshchev, and Abramowicz, 2003). MAD states feature the built-up of strong magnetic flux near the horizon, leading to the establishment of a mag4 e.g., Fishbone-Moncrief torus (Fishbone and Moncrief, 1976), or BHL accretion scenarios with a density/velocity gradient (Lora-Clavijo, Cruz-Osorio, and Méndez, 2015; Cruz-Osorio and Lora-Clavijo, 2016) 57 netically arrested flow structure near the innermost stable circular orbit. Once the horizon magnetic flux rises above a threshold, reconnection triggers an ejection of magnetic flux bundles from the BH magnetosphere, accompanied by decay of the horizon magnetic flux (Ripperda, Bacchini, and Philippov, 2020; Ripperda, Liska, et al., 2022; Chatterjee and Narayan, 2022). Shearing instabilities, such as the Rayleigh-Taylor instability (Kulkarni and Romanova, 2008), ultimately trigger a magnetic flux eruption with a simultaneous mass accretion inwards, via interchanging magnetically buoyant low-density bubbles with less magnetized, dense parcel of fluid (Igumenshchev, 2008). Through this process a MAD state is re-established over the viscous timescale of the accretion flow. As a consequence, the BH does not sustain steady outflows but exhibits fluctuations in the jet power and intermittent flux eruptions. The quasi-periodic MAD cycle results in two notable features in the flow morphology compared to unmagnetized axisymmetric BHL accretion flows. (1) Each eruption event launches a pressure wave that expands outward from the black hole. This wavefront with a relatively higher density pushes the boundary of the bow shock further upstream, expanding the shock cone, before it shrinks back due to the ram pressure of the incoming fluid. As a consequence, the bow shock exhibits a breathing motion in lockstep with each of the magnetic flux eruptions. This results in a deformed morphology of the shockfront, rather than a smooth parabolic shape commonly observed from unmagnetized cases (e.g., Shima et al., 1985; Penner, 2013; Lora-Clavijo and Guzman, 2013; Gracia-Linares and Guzmán, 2015). This eruption-driven expansion of the bow shock and multiple pressure waves approaching the bow shock can be seen from the left panels of Fig. 3.2 or in Fig. 3.5. (2) A series of magnetic flux tubes with high magnetization are formed near the BH and drift downstream on the equatorial plane as they are released, a few of which can be identified from the region 𝑥 < 0, 𝑦 < 0 in the lower panels of Fig. 3.2. This situation is reminiscent of unboosted MAD BH flows (e.g., Chatterjee and Narayan, 2022; Ripperda, Liska, et al., 2022; Porth et al., 2021). When a flux eruption event occurs, magnetic pressure pushes the matter outward via an interchange instability and forms highly magnetized, hot, lowdensity voids near the horizon (e.g., see the simulation snapshots included in Fig. 3.3 and Fig. 3.4). The ejected flux tubes are spiralling outwards 58 from the BH (Porth et al., 2021), getting sheared and ultimately reach the stagnation point, where they are fragmented by ram pressure and released into the downstream flow as a mushroom-shaped feature. In magnetic flux eruptions of axially symmetric accretion flows, flux tubes would have been launched without a preferred direction. Here, due to a preferential direction of the ambient flow, the flux tubes can be only released toward a particular range of angle relative to the direction of the upstream wind. The ejected flux tubes are filled with hot non-thermal plasma produced by the near-horizon magnetic reconnection that initially created the tubes (Ripperda, Bacchini, and Philippov, 2020; Ripperda, Liska, et al., 2022). This plasma has the potential to power TeV and X-ray flares (Porth et al., 2021; Hakobyan, Ripperda, and Philippov, 2023), in a similar manner to what has been proposed as an explanation for galactic flares near Sgr A* (Dexter et al., 2020; Antonopoulou, Loules, and Nathanail, 2025). Time evolution Having described the overall properties of the accretion flow, we now focus on its time variability. In Fig. 3.3, we provide the time series data of the dimensionless ¤ energy outflow efficiency 𝜂 = horizon magnetic flux 𝜙BH , mass accretion rate 𝑀, ¤ 2 )/ 𝑀, ¤ (outwards directed) angular momentum flux 𝐽, ¤ and the total drag ( 𝑀¤ − 𝐸/𝑐 force 𝐹 𝑖 . Based on these, we examine the time evolution of our fiducial accretion flow setup in this section. After the initial development of relativistic polar outflows from the BH (𝑡 ∼ 3𝜏𝑎 ), the accretion flow undergoes an oscillatory MAD cycle showing a transient behavior of 𝜙BH associated with episodes of magnetic flux eruptions. In active phases, when the jet is present, the mass accretion rate is suppressed, whereas it is enhanced in quiet (accreting) phases when the jet is absent. The overall evolution observed in this model is broadly consistent with previous studies on MADs (e.g. White, J. M. Stone, and Quataert, 2019; Porth et al., 2021; Chatterjee and Narayan, 2022; Tchekhovskoy, Narayan, and McKinney, 2011). The continued eruption cycle is maintained until 𝑡 ∼ 10𝜏𝑎 , which we hereafter refer to as the first eruption epoch. Following the final flux eruption of the first eruption epoch at 𝑡 ∼ 10𝜏𝑎 , the magnetic flux of the BH drops and the system enters a fully quiescent period, 10𝜏𝑎 ≤ 𝑡 ≤ 12𝜏𝑎 with the jet being fully quenched. The accretion flow temporarily enters a standard 59 0 5 (a) 10 (b) (c) t [τa ] 15 20 25 30 30000 40000 50000 60000 30 φBH 20 10 Ṁ / ṀBHL 0 0.3 0.2 0.1 0.0 3 η 2 1 0.0 F i / (ṀBHL v∞ ) ˙ ṀBHL rg c) J/( 0 0.2 1 0 −1 x y z −2 0 10000 20000 t [rg /c] Figure 3.3: Time evolution of physical quantities for the representative (𝛽10 -𝜃 90 𝑅200 ) model. Each panel of the line plots, from top to bottom, presents the dimen¤ energy outflow efficiency sionless horizon magnetic flux 𝜙BH , mass accretion rate 𝑀, 𝑖 ¤ 𝜂, angular momentum flux 𝐽, and the total drag force 𝐹 . Quiescent periods (shaded) with a duration ∼ 2000𝑟 𝑔 /𝑐 are separated by an epoch of continued flux eruptions lasting ∼ 24000𝑟 𝑔 /𝑐. Color plots on top of this figure show the mass density, 𝜌, on the 𝑥𝑧 (first row) and 𝑥𝑦 (second row) plane in active states 𝑡 (𝑎) = 1.8 × 104𝑟 𝑔 /𝑐, 𝑡 (𝑐) = 2.5 × 104𝑟 𝑔 /𝑐 and a SANE-like quiescent state 𝑡 (𝑏) = 2.15 × 104𝑟 𝑔 /𝑐, which are indicated with dashed and dotted vertical lines in the line plots. 60 and normal evolution (SANE, Narayan, Sadowski, et al., 2012) regime in which magnetized polar funnel are replaced by a smooth inflow of matter toward the BH and the accretion exhibits a relavitely laminar flow on the equatorial plane (with only mild turbulent eddies) without a strong vertical stratification (Chatterjee and Narayan, 2022). As a result, the mass accretion rate is notably increased and the BH experiences a temporary spin up (𝐽¤ < 0) from the falling matter. In the middle of the SANE-like quiescent period, the horizon magnetic flux and the energy outflow efficiency rises again, mass accretion decreases, and the angular momentum flux transitions from inflow (−) to outflow (+). The evolution of the system in this time window is similar to the process of the first jet launching phase, indicating the revival of it. This revival process takes roughly a viscous time scale of the disk. The system re-enters the MAD state and the jet is launched again at 𝑡 ∼ 12𝜏𝑎 , which marks the beginning of the second eruption epoch lasting 12𝜏𝑎 ≤ 𝑡 ≤ 23𝜏𝑎 . In the upper half of Fig. 3.3, we include simulation snapshots displaying the mass density distribution of the accretion flow at three different times 𝑡 (𝑎) = 1.8×104𝑟 𝑔 /𝑐, 𝑡 (𝑏) = 2.15 × 104𝑟 𝑔 /𝑐, and 𝑡 (𝑐) = 2.5 × 104𝑟 𝑔 /𝑐, each of which corresponds to (𝑎) a MAD-like active state by the end of the first eruption epoch, (𝑏) a quiet SANE-like state with no jet present, and (𝑐) an active state in the second eruption epoch after the revival. This process overall repeats periodically. 3.3.2 Magnetic flux eruptions In the preceding sections, we have described the global dynamics leading to the establishment of a MAD accretion state and subsequent magnetic flux eruption events. In the following, we provide a detailed description of the flux eruption event at 𝑡 = 16500𝑟 𝑔 /𝑐 as a representative example of the process, and elucidate its effect on the jet morphology. Near black-hole dynamics Along with the three sections of plots comprising Fig. 3.4, we now illustrate a comprehensive picture of a single magnetic flux eruption event. • Evolution of the horizon magnetic flux (Fig. 3.4, top panel): The eruption cycle begins with a magnetically relaxed state after the previous eruption has settled down. The horizon magnetic flux 𝜙BH starts to increase from 𝑡 = 15500𝑟 𝑔 /𝑐, 61 t [τa ] 7.8 8.0 8.2 8.4 8.6 φBH 60 40 20 0 15600 15800 16000 16200 16400 16600 16800 17000 17200 t [rg /c] Figure 3.4: A magnetic flux eruption event at 𝑡 = 16500𝑟 𝑔 /𝑐. Top panel: change of dimensionless magnetic flux 𝜙BH on the black hole (BH) over the eruption. Three vertical dashed lines mark 𝑡 = 16400, 16500, 16600 𝑟 𝑔 /𝑐 respectively. Data points are displayed without smoothing. Middle panels: mass density, 𝜌, (left) and radial 1/2 magnetic field 𝐵𝑟 in units ([𝜌∞ 𝑐]), (right) on a spherical surface 𝑟 = 5𝑟 𝑔 are shown with the Mollweide projection aligned with the BH spin axis. The radial component of the magnetic field is shown in color and the angular components are shown with black streamlines. The center of the plot corresponds to the +𝑥ˆ direction. From top to bottom, each row corresponds to 𝑡 = 16400, 16500, 16600 𝑟 𝑔 /𝑐. Bottom panels: mass density on the equatorial plane (left), on the meridional plane (center), and the relativistic magnetization 𝜎 on the meridional plane (right) at 𝑡 = 16500𝑟 𝑔 /𝑐. In-plane velocity and magnetic field are shown with white and black streamlines. 62 reaching 𝜙BH ≈ 20 at 16300𝑟 𝑔 /𝑐. Then it rapidly rises to 𝜙BH > 50 around 𝑡 = 16500𝑟 𝑔 /𝑐, and relaxes down to 𝜙BH ≈ 10 at 𝑡 = 16700𝑟 𝑔 /𝑐 after the eruption. • Nutation of the accreting plane (middle panels in the Mollweide projection in Fig. 3.4): These panels, showing mass density and magnetic field on a spherical surface 𝑟 = 5𝑟 𝑔 at 𝑡 = 16400𝑟 𝑔 /𝑐, 16500𝑟 𝑔 /𝑐 and 16600𝑟 𝑔 /𝑐, visualize angular distribution of the accretion flow and the geometry of magnetic field near the BH during the eruption event. In the pre-eruption stage, the accretion disk develops precession with an increasingly large amplitude over time. Left column of the panels shows that the accretion disk is subject to a significant tilt and distortion during the eruption (which otherwise would appear as a smooth strip along the equator). The ejection of the magnetic flux tube during the eruption is off the equator and exerts a strong recoil (and corresponding torque) onto the system, which leads to a nutation of the accretion disk. • Tearing of accretion disk (bottom panels of Fig. 3.4): The asymmetric ejection of the magnetic flux tube (low density/high magnetization region) and the resulting recoil strongly tilting and tearing the inner accretion flow can be clearly seen from the figure. The dynamical time scale of the fluid at which the accretion disk is torn can be estimated via the local Keplerian orbital period as  3  1/2 𝑟 2𝜋 = 370𝑟 𝑔 /𝑐 = 2𝜋 𝑇≈ Ω𝐾 𝐺𝑀 (3.41) for 𝑟 = 15𝑟 𝑔 , which is in good agreement with the disk precession period ≈ 400𝑟 𝑔 /𝑐, which we estimate empirically. While the inner part of the accretion disk has a shorter dynamical timescale (𝑟 < 15𝑟 𝑔 ) than the driving frequency (𝑇 ≈ 400𝑟 𝑔 /𝑐) and is able to reorganize itself, the outer part of the disk with a longer dynamical timescale (𝑟 > 15𝑟 𝑔 ) cannot dynamically react (synchronize) to the driving and is decoupled from the inner part. In a different context, highly tilted accretion disks have been observed to suffer more tearing, precession, and fragmentation, when the inner part of the disk is subject to torque (Liska et al., 2021). Jet morphology High-resolution simulations of tilted accretion disks have shown that the motions of the disk and the jet are coupled (Liska et al., 2018; Liska et al., 2021), and we 63 (a) 𝑡 = 16500𝑟 𝑔 /𝑐 (b) 𝑡 = 21500𝑟 𝑔 /𝑐 Figure 3.5: A three-dimensional rendering of the simulation depicting an active state (left) and a quenched state (right). Shown here are low-density, highly magnetized outflow (filtered by 𝜌 < 0.05𝜌∞ , red-white colors), accretion flow inside the bow shock (filtered by 𝜌 > 2𝜌∞ and half-cut to vertical, blue-yellow colors), and magnetic field lines emanating from the accretion disk. Each colormap shows the magnitude of the spatial components of the four-velocity and the normalized mass density 𝜌/𝜌∞ . A vertical tearing of the accretion disk (see Sec. 3.3.2) is visible in the left panel. observe a similar phenomenon in our simulations. While the structure of the jet remains more or less aligned with the BH spin very close the horizon (𝑟 ≲ 5𝑟 𝑔 ), the rapid nutation of the accretion disk and a consequent wobbling of the the polar funnels surrounded by the accretion flow results in fluctuations in the direction of the jet at 𝑟 ≳ 10𝑟 𝑔 (see also Liska et al., 2018). From a three-dimensional visualization in Fig. 3.5, one can observe the twisted morphology of the jet associated with the nutation of the near-BH accretion flow. This non-smooth jet morphology may further enhance kink-like instabilities naturally present in these systems (Appl, Lery, and Baty, 2000; L.-X. Li, 2000; Narayan, J. Li, and Tchekhovskoy, 2009; Bromberg and Tchekhovskoy, 2016). At larger distances, the interaction with the ambient wind gradually smooths out these features. We can also see how the jet is quenched at a later time. In Fig. 3.6, we show the distribution of the radially outgoing electromagnetic flux (𝑇EM ) 𝑡 𝑟 at 𝑟 = 100𝑟 𝑔 . The opening angle of the jet shows variations between 15◦ 64 Figure 3.6: Distribution of the radial Poynting flux, (𝑇EM ) 𝑡 𝑟 , on the upper hemisphere of a spherical surface 𝑟 = 100𝑟 𝑔 at 𝑡 = 19600 𝑟 𝑔 /𝑐 (top), 𝑡 = 19800 𝑟 𝑔 /𝑐 (center), and 𝑡 = 20000 𝑟 𝑔 /𝑐 (bottom). 65 and 25◦ , where the position of the peak Poynting flux (center of the jet) precesses with an amplitude ≲ 15◦ . While the jet still exhibits an oscillatory behavior at this radius to some extent, its angular variations remain much smaller than the inner accretion disk attached to the BH. We expect that these variations will be further attenuated at a larger radius. 3.3.3 Drag and spin-down Here we look into the transport of linear and angular momentum between the BH and the accretion flow, which are responsible for the deceleration and spin-down of the BH. A time scale which we will frequently refer to for the discussions in this ¤ which can be rescaled using the section is the mass doubling timescale 𝜏𝑀 = 𝑀/ 𝑀, BHL mass accretion rate 𝑀¤ BHL as  −1   3 𝑣∞ 𝑀¤ 4 𝜏𝑀 = 2.8 × 10 1000 km s−1 𝑀¤ BHL (3.42)  −1   −1  𝜌∞ 𝑀 yr . × 100𝑀⊙ 10−10 g cm−3 ¤ 𝑀¤ BHL ∼ 0.1 in our simulation (see The normalized mass accretion rate is 𝑀/ Fig. 3.3). Drag force An accretor traveling through a gaseous medium is subject to a dynamical friction (Chandrasekhar, 1943). The reference scale of this drag is 𝑀¤ BHL 𝑣 ∞ , which is the drag force in the ballistic (dust) limit, and a multiplicative correction factor needs to be included in generic cases.5 For convenience, we define the fudge factor 𝑖 𝑓BHL ≡ 𝐹 𝑖 /( 𝑀¤ BHL 𝑣 ∞ ) (3.43) where 𝐹 𝑖 is the measured drag force in each directions. Since our simulation is performed in a fixed spacetime and does not consistently capture the slow-down of the BH in dynamical general relativity, the drag force is estimated by means of an approximate formula Eq. (3.29). The bottom panel of Fig. 3.3 shows the time variation of the total drag force normalized with the BHL drag scale 𝑀¤ BHL 𝑣 ∞ , effectively displaying the factor 𝑖 𝑓BHL for each spatial directions. The tangential drag (dynamical friction) reaches a 5 Newtonian studies suggest that the correction is not expected to exceed a factor of ten in hydrodynamic BHL accretion (Edgar, 2004). 66 𝑥 steady value of 𝑓BHL ∼ 2.5 around 𝑡 = 5 × 104𝑟 𝑔 /𝑐. The linear momentum accretion 𝑥 rate 𝐹mom is nearly zero when time averaged, yet it is very oscillatory. It is the 𝑥 gravitational drag 𝐹grav that constitutes a dominant net portion of the total drag. See 𝑥 𝑥 . A similar trend is Appendix 3.A for the raw time series data of 𝐹mom and 𝐹grav observed for 𝐹 𝑦 and 𝐹 𝑧 as well, suggesting that the accretion flow plunging into 𝑦 the BH is mostly symmetric when averaged over time. However, we note that 𝐹mom shows a small positive average ∼ 0.1 𝑀¤ BHL 𝑣 ∞ . The drag force results in a slow-down of the BH relative to the surrounding medium. The deceleration time scale can be estimated by dividing the initial BH linear momentum by the drag force 𝜏dec = 𝑀𝑣 ∞ = ( 𝑓BHL ) −1 𝜏𝑀 ¤ 𝑓BHL ( 𝑀BHL 𝑣 ∞ ) (3.44) ¤ 𝑀¤ BHL ∼ where the mass doubling timescale 𝜏𝑀 is given as Eq. (3.42). Plugging in 𝑀/ 0.1 and 𝑓BHL ∼ 2.5 from our fiducial simulation, we get 𝜏dec ∼ 105 yr for the representative values of 𝑣 ∞ , 𝜌∞ , 𝑀 chosen in (3.42). On top of the drag force 𝐹 𝑥 tangential to the direction of BH motion, there exist nonzero transverse drag (𝐹 𝑦 ) and vertical drag (𝐹 𝑧 ) which are perpendicular and parallel to the BH spin. These drags can potentially induce a deflection or oscillatory features in the trajectory of the accreting BH. The transverse drag force 𝐹 𝑦 shows an average magnitude ≈ 0.5 𝑀¤ BHL 𝑣 ∞ , which is about 20% of the tangential drag 𝐹 𝑥 . The vertical drag 𝐹 𝑧 has a similar magnitude as the transverse part 𝐹 𝑦 . The transverse drag 𝐹 𝑦 maintains a positive finite value where the vertical drag periodically flips its sign between eruption epochs. Each of these components are intimately related with the gravitational Magnus effect (Sec. 3.5.2) and the magnetic reversal of jets (Sec. 3.3.4) observed in our simulations. Spin-down As previously noted, in our setup, the spin effect from the BH is the only physical origin of circulation introduced in the accretion flow. From Fig. 3.3, we see that the angular momentum transport rate from the BH into accretion flows is 𝐽¤ ≈ 0.1 𝑀¤ BHL𝑟 𝑔 𝑐 during eruption epochs where it is reversed during quiescent periods. Angular momentum in the disk is mainly transported in flux eruption episodes (Chatterjee and Narayan, 2022), where the overall spin-down of the BH is largely affected by the jet as well (Lowell et al., 2024). Since our problem setup (BHL accretion) is different from a commonly referenced MAD configuration which is 67 being reached starting from an accreting torus, it is noted that an additional feedback mechanism might contribute to the angular momentum transport. A systematic investigation on the spin-down of a BH in the MAD state showed that the characteristic decaying time of the BH dimensionless spin 𝐽𝑐/𝐺 𝑀 2 = 𝑎/𝑟 𝑔 is about 10% of the mass doubling time scale (Lowell et al., 2024). To explore the same aspect but for the BHL accretion, we estimate the spin-down rate of the BH during an active phase in our simulation as follows. The angular momentum loss timescale of the BH can be estimated as  −1     𝐽 𝑀¤ 𝑎 𝐽¤ 𝜏𝐽 = = 𝜏𝑀 . (3.45) 𝐽¤ 𝑀¤ BHL𝑟 𝑔 𝑐 𝑀¤ BHL 𝑟 𝑔 ¤ 𝑀¤ BHL𝑟 𝑔 𝑐) ≈ 0.1 and 𝑀/ ¤ 𝑀¤ BHL ≈ 0.05, Our simulation results (Fig. 3.3) shows 𝐽/( indicating 𝜏𝐽 ∼ 𝜏𝑀 /2. Then the spin-down timescale of the dimensionless spin 𝑎/𝑟 𝑔 can be estimated as 𝑎 𝐽 𝑒 −𝑡/𝜏𝐽 (3.46) ∝ 2 ∝ 2𝑡/𝜏 ∼ 𝑒 −4𝑡/𝜏𝑀 , 𝑟𝑔 𝑀 𝑒 𝑀 yielding a slightly longer spin-down timescale 𝜏𝑀 /4 than ≈ 𝜏𝑀 /10 from Lowell et al. (2024). This rather lower rate of the angular momentum extraction from the BH can be attributed to the fact that in our case the accretion flow stays in a mildly MAD state (𝜙BH ≈ 20). 3.3.4 Magnetic reversal Within the simulation time of the model (𝑡 ≤ 6 × 104 𝑟 𝑔 /𝑐), the BH and the accretion flow undergo two quiescent periods at 𝑡 = 10𝜏𝑎 and 𝑡 = 24𝜏𝑎 . While all other quantities show a recurring pattern of rise and fall in every eruption epoch, the vertical drag force 𝐹 𝑧 (see the bottom line plot of Fig. 3.3) shows a distinct behavior in that it flips its sign during a quiescent period and maintains that opposite sign for the next eruption epoch, before coming back to the original sign after another quiescent period. We find that the polarity of the horizon magnetic fluxes and jets is reversed during these quiescent periods, namely that the MAD state experiences a magnetic reversal between eruption epochs; see Fig. 3.7. This reversal behavior is observed from all models with a purely horizontal magnetic field (𝜃 𝐵 = 90◦ ) of the incoming fluid. Unfortunately, our simulation is too short to conclusively decide whether the polarity selection always alternates or is stochastic. We postpone a detailed discussion on the origin of this phenomena to future studies. 68 Figure 3.7: Magnetic field reversal of the jets between three eruption epochs in the fiducial model (𝛽10 -𝜃 90 -𝑅200 ). The relativistic magnetization 𝜎 is shown in color and the in-plane components of the magnetic field are shown with black streamlines. 3.4 Dependence on accretion parameters In this section, we present a systematic survey of the accretion parameters. We change the inclination angle 𝜃 𝐵 between the incoming magnetic field and the BH spin (Sec. 3.4.1), then explore a weakly magnetized case with 𝛽∞ = 100 (Sec. 3.4.2), followed by different incoming speeds of the fluid 𝑣 ∞ (Sec. 3.4.3). Rather than delving into the same level of detail as the previous section, here we provide a broader overview on qualitative impacts of the physical parameter chosen to be varied. 3.4.1 Mixed magnetic fields Keeping other parameters fixed as the fiducial setup, the inclination angle of the initial magnetic field was varied to 𝜃 𝐵 = 22.5◦ , 45◦ , 67.5◦ . The purpose of this experiment is to investigate how much the inclination of the incoming magnetic field relative to the BH spin affects jet launching and the time evolution of accretion flow. In Fig. 3.8, we present the time series data from the three simulations varying 𝜃 𝐵 . The result from the fiducial model 𝛽10 -𝜃 90 -𝑅200 is overlayed with a transparent 69 θB = 22.5◦ 0 5 θB = 45◦ t [τa ] 10 15 20 25 0 5 θB = 67.5◦ t [τa ] 10 15 20 25 0 5 2 3 4 5 0 1 t [τa ] 10 15 20 25 2 3 4 5 30 φBH 20 10 0 Ṁ / ṀBHL 0.4 0.2 0.0 3 η 2 1 0.2 0.0 F i / (ṀBHL v∞ ) ˙ ṀBHL rg c) J/( 0 0 x y z −2 0 1 2 3 t [104 rg /c] 4 5 0 1 t [104 rg /c] t [104 rg /c] Figure 3.8: Time evolution of physical quantities for three models with 𝜃 𝐵 = 23.5◦ , 45◦ and 67.5◦ , all with 𝛽 = 10 and 𝑅𝑎 = 200. See Fig. 3.3 for a description of the quantities shown. The result from the 𝜃 𝐵 = 90◦ (fiducial) model is overlayed with a transparent line. line in each panels to highlight deviations. While the overall correlations between each physical quantities is similar to that of the fiducial setup, we compile several observations below. (i) The time it takes from the beginning of the simulation to the first successful jet launching and transition to the MAD state is longer for smaller inclination angle, namely when the incoming magnetic field has more vertical component. Interestingly, the maximally misaligned case (𝜃 90 ) launches the jet the earliest.rm (ii) Within the simulation time 𝑡 ≤ 5 × 104𝑟 𝑔 /𝑐, both 𝜃 𝐵 = 45◦ and 𝜃 𝐵 = 67.5◦ models did not show any quiescent period; the first eruption epoch continued throughout the final time. In contrast, the 𝜃 𝐵 = 22.5◦ model turned into a quiescent state after a single eruption epoch of a duration 10𝜏𝑎 , and did not revive its jet activity until the end of the simulation, showing a quiescent 70 MAD epoch duration [τa ] 25 20 15 10 5 0 0 45 90 ◦ θB ( ) Figure 3.9: Duration of a magnetically arrested accretion epoch in units of the accretion timescale 𝜏𝑎 = 2000𝑟 𝑔 /𝑐 for the four models with 𝑅𝑎 = 200𝑟 𝑔 and 𝛽∞ = 10. period with the duration at least longer than 5𝜏𝑎 . The BH fails to establish a MAD state during this quiescent period, where its episodic launching of weak outflows (𝜂 ≲ 10%) into random directions appears to be very similar to what was observed from low angular momentum accretion flows (Ressler, Quataert, et al., 2021; Kwan, L. Dai, and Tchekhovskoy, 2023; Lalakos et al., 2024; Galishnikova et al., 2025). ¤ (iii) However, once the BH enters the eruption (MAD) state, values of 𝜙BH , 𝑀, 𝜂 and 𝐽¤ are almost same as the 𝜃 𝐵 = 90◦ case for all models. This implies that quasi-stationary properties of the MAD state in this setup do not depend on the large scale magnetic field geometry, which only determines the time period between eruption epochs.6 In Fig. 3.9, we plot the 𝜃 𝐵 dependence of the duration of eruption epochs we could observe from our simulations, while we could only constrain lower bounds for 𝜃 𝐵 = 45◦ and 𝜃 𝐵 = 67.5◦ cases. Our results indicate a non-monotonic behavior of the eruption epoch duration 6 Kaaz, Murguia-Berthier, et al. (2023) adopts a similar parameter regime as in this work, apart ¤ 𝑀¤ BHL ≈ 0.05 and from M = 2.45 and 𝜃 𝐵 = 0◦ . In an active (jet) state, the mass accretion rate 𝑀/ the energy outflow efficiency 1 ≲ 𝜂 ≲ 3 observed in our simulation agree with those from Kaaz, Murguia-Berthier, et al., 2023, while the average value of 𝜙BH appears to be somewhat lower in our study. 71 β = 100 Ra = 50rg t [τa ] 0 10 Ra = 400rg t [τa ] 20 30 40 0 25 50 75 t [τa ] 100 125 150 0 2 4 6 8 10 30 φBH 20 10 Ṁ / ṀBHL 0 1.00 0.75 0.50 0.25 0.00 3 η 2 1 ˙ ṀBHL rg c) J/( 0 0.4 0.2 0.0 F i / (ṀBHL v∞ ) −0.2 2 0 −2 −4 x y z −6 0 1 2 3 4 5 6 7 8 t [104 rg /c] 0 1 2 t [104 rg /c] 3 4 0 1 2 3 4 5 6 t [104 rg /c] Figure 3.10: Time series data for the 𝛽100 -𝜃 90 -𝑅200 (left), 𝛽10 -𝜃 90 -𝑅50 (center), and 𝛽10 -𝜃 90 -𝑅400 (right) models. See Fig. 3.3 for a description of the quantities shown. We overlay the result from the fiducial (𝛽10 -𝜃 90 -𝑅200 ) model with a transparent line only for the 𝛽100 -𝜃 90 -𝑅200 model (first column). with the magnetic field orientation 𝜃 𝐵 , but an exact functional relationship between them is highly uncertain due to an insufficient number of data points and limited numerical resolution surveys. 3.4.2 Lower magnetization The first column of Fig. 3.10 shows the time series of accretion quantities for the model 𝛽100 -𝜃 90 -𝑅200 , which has ten times weaker magnetization of the incoming fluid compared to the fiducial model. Several peculiar features observed in this model compelled us to perform a longer time integration up to 𝑡 = 8.7 × 104𝑟 𝑔 /𝑐. We outline those below. The evolution of the horizon magnetic flux 𝜙BH roughly follows a similar cycle. 72 However, it undergoes more gradual increase and fall of 𝜙BH between eruption and quiescent periods, exhibiting relatively short duration of eruptions and a longer quiescent period. The BH cannot sustain the MAD state long enough as the 𝛽 = 10 case and takes longer to revive jets once it goes quiescent, which is consistent with the upstream wind providing a smaller amount of magnetic flux per time. These ¤ 𝑀¤ BHL ≈ 0.5 in a quiescent period and decreased trends, along with an increased 𝑀/ average values of 𝜙BH , 𝜂 during an active period, is similar to the results of Kaaz, Murguia-Berthier, et al. (2023) despite a different incoming magnetic field geometry. Another observation that can be made from the result, owing to a prolonged duration of a quiescent period, is a reduced (or a reduced accumulation of) dynamical drag 𝐹 𝑥 over an eruption period. A strong, low-density bipolar jets launched from the BH pushes the gas outward, dropping the average mass density in the region trailing behind the BH (X. Li et al., 2020). The third eruption epoch (28𝜏𝑎 ≲ 𝑡 ≲ 36𝜏𝑎 ), though not as conspicuous as the first two, ends up staying only in the mildly MAD regime 𝜙BH ≤ 10. We observe the fourth round of rising magnetic fluxes at 𝑡 = 40𝜏𝑎 during which horizon magnetic fluxes also stayed below 𝜙BH = 10. In brief, the system exhibits a periodic eruptionquiescence cycle similar to what is observed in our fiducial model, but with a decreasing level of activity over time. We caution that the decay we see may be correlated with numerical resolution and scale separation between the BH and the outer simulation domain boundary. 3.4.3 Faster/slower incoming speed Time evolution of the models with a faster (𝑅𝑎 = 50𝑟 𝑔 , 𝑣 ∞ = 0.2𝑐) and a slower (𝑅𝑎 = 400𝑟 𝑔 , 𝑣 ∞ = 0.07𝑐) speed of the incoming fluid are shown on the second and third columns in Fig. 3.10. We show the mass density distribution on the meridional plane for the two models in Fig. 3.11, highlighting differences in the shape and radius of the bow shock, as well as the bending angle of outflows from the BH. We first examine the case with a faster incoming speed (𝑅𝑎 = 50𝑟 𝑔 ). The outflow launched in the polar directions is choked by a strong ram pressure. The BH exhibits sporadic flux eruptions and launches outflows intermittently, but fails to maintain a continuous jet. While the horizon magnetic flux is kept below 𝜙BH ≤ 10, the mass accretion rate shows a large oscillation between 0.3–0.9 𝑀¤ BHL , highly anti-correlated with 𝜙BH . While the dynamical time of the accretion flow around the BH is the same independent of the wind speed, the replenishing timescale of the flux changes 73 (a) 𝑅 𝑎 = 50𝑟 𝑔 (at 𝑡 = 2 × 104 𝑟 𝑔 /𝑐) (b) 𝑅 𝑎 = 400𝑟 𝑔 (at 𝑡 = 4 × 104 𝑟 𝑔 /𝑐) Figure 3.11: Distribution of the mass density 𝜌 in the meridional plane for the simulations 𝛽10 -𝜃 90 -𝑅50 (top) and 𝛽10 -𝜃 90 -𝑅400 (bottom). 𝑅𝑎 is the accretion radius, 𝑟 𝑔 is the gravitational radius of the black hole, and 𝜌∞ is asymptotic mass density of the wind. 74 with 𝑅𝑎 . For faster wind speeds, the cross section of the bow shock shrinks, with less mass being accreted on the BH, though more relative to the changed reference rate, 𝑀¤ BHL . The BH then preferentially accretes in a SANE regime, even if the magnetic properties of the wind at large scales remain the same. This likely implies that questions about flux accumulation on the black hole horizon in BHL accretion cannot be separated from the effective speed of the black hole. We also observe 𝑥 that the tangential drag reaches 𝑓BHL ≈ 6 by the end of the simulation; recall that 𝑥 𝑓BHL ≈ 2.5 in our fiducial model with 𝑅𝑎 = 200𝑟 𝑔 . Next, we examine the model with a slower incoming speed of fluid (𝑅𝑎 = 400𝑟 𝑔 ). The accretion flow reaches the MAD state relatively early at 𝑡 ∼ 1.5𝜏𝑎 . The shorter time for the flow to become magnetically arrested is consistent with the scaling argument presented in Kaaz, Murguia-Berthier, et al. (2023), 𝜏MAD /𝜏𝑎 ∝ 𝑅𝑎−3/4 , (3.47) while we note the exponent can be slightly different for an inclined magnetization of the inflow. The accretion flow enters a quiescent period at 𝑡 ∼ 9𝜏𝑎 and revives jets at 𝑡 ∼ 10𝜏𝑎 , also showing a magnetic reversal behavior. The overall time evolution of this model is qualitatively very similar to the fiducial model discussed in Sec. 3.3. 𝑥 The dynamical friction measured by the end of the simulation was 𝑓BHL ∼ 1.0. However, we expect a higher value in practice as the measured value of the drag had not fully reached a steady state during our integration time. We also observe that 𝑧 the magnitude of the vertical drag is reduced to | 𝑓BHL | ∼ 0.1 and the correlation of its sign and the magnetic polarity of the jet has become weaker compared to the fiducial model. The vertical drag occasionally turns to a positive value during 2.5𝜏𝑎 ≤ 𝑡 ≤ 4.5𝜏𝑎 where the direction of magnetic field threading the BH and the jet has been steadily kept to +𝑧ˆ during the period. Considering that the astrophysically realistic speed of the BH is still much beyond the lower limit of 𝑣 ∞ explored in this study, it follows that the vertical drag 𝐹 𝑧 may be effectively uncorrelated or only weakly correlated with the magnetic polarity of jets in realistic situations. 3.5 3.5.1 Discussion Energy outflow ¤ 2 − 𝐸¤ for all models. Once Fig. 3.12 compares the net energy outflow power 𝑃 = 𝑀𝑐 the accretion enters the MAD state, the energy outflow power does not show much dependence on the inclination angle 𝜃 𝐵 of the incoming magnetic field. A slight decrease in the power is observed when the magnetization of the incoming medium 75 P = Ṁ c2 − Ė [103 rg2 ρ∞ c3 ] θB = 22.5◦ θB = 45◦ θB = 67.5◦ 1 0.5 0 β = 100 1 0.5 0 Ra = 50rg Ra = 400rg 1 0.5 0 0 1 2 3 4 5 t [104 rg /c] ¤ 2 − 𝐸¤ from all simulations. The gray Figure 3.12: The energy outflow power 𝑃 = 𝑀𝑐 ¤ 𝑀¤ BHL , line shows the result from the fiducial model. For the scaled efficiency 𝜂 𝑀/ see Fig. 3.13. 76 η Ṁ /ṀBHL θB = 22.5◦ θB = 45◦ θB = 67.5◦ 0.10 0.05 0.00 β = 100 0.10 0.05 0.00 Ra = 50rg Ra = 400rg 0.10 0.05 0.00 0 1 2 3 4 5 t [104 rg /c] ¤ 𝑀¤ BHL from all simulations. The Figure 3.13: The energy conversion efficiency 𝜂 𝑀/ gray line shows the result from the fiducial model 𝛽10 -𝜃 90 -𝑅200 . 77 is lower, which can be attributed to a reduced supply of magnetic energy to the BH per unit time. Likewise, an increased energy outflow for a slower speed of incoming fluid can be understood as that of a slowly moving BH with a large accretion radius, leading to a higher mass accretion rate ( 𝑀¤ BHL ∝ 𝑣 −3 ∞ ), and consequently a higher rate of magnetic flux injection onto BH, resulting in a more powerful energy outflow (and vice versa). Our results indicate that among the accretion parameters varied in this study, the energy outflow is most significantly influenced by the fluid speed 𝑣 ∞ and to a lesser extent by the magnetization of the incoming matter (note that the BH speed is 𝑣 ∞ varied by a factor of two at most, where magnetization is set to be ten times weaker in the 𝛽 = 100 model). The magnetic field inclination angle 𝜃 𝐵 has a negligible impact on the outflow power during eruption epochs. However, it largely effects the time evolution and the intermittency of the jet activity, as previously discussed in Sec. 3.4.1. The finding from this comparative analysis, namely that the BH speed 𝑣 ∞ is a primary factor in determining the outflow luminosity, also aligns with the fact that basic physical scales of the BHL accretion e.g., Eq. (3.1)–(3.3) possess a strong dependence in 𝑣 ∞ with the highest power exponent. A reference scale of the energy outflow power in physical units can be written as   𝑀¤ 𝑃=𝜂 𝑀¤ BHL 𝑐2 𝑀¤ BHL   2   ¤ ¤ 𝑀 𝜌∞ 43 𝜂 𝑀/ 𝑀BHL (3.48) = 1.0 × 10 0.05 100𝑀⊙ 10−10 g cm−3  −3  𝑣∞ × erg s−1 . 1000 km s−1 ¤ 𝑀¤ BHL corresponds to an effective energy conversion efficiency with The factor 𝜂 𝑀/ which the rest mass energy inflow 𝑀¤ BHL 𝑐2 is converted into a net energy outflow. Our baseline setup (𝛽10 -𝜃 90 -𝑅200 ) shows the conversion efficiency ≈ 0.05 during an ¤ 𝑀¤ BHL from all models. eruption epoch. See Fig. 3.13 for the values of 𝜂 𝑀/ 3.5.2 Magnus effect As can be seen from Fig. 3.2, the downstream flow trailing the BH is deflected to the − 𝑦ˆ direction owing to the spin of the BH and a resulting circulatory flow surrounding it. This can result in two effects on the transverse drag force 𝐹 𝑦 : (1) the conservation of linear momentum requires the BH to experience a reaction force to 78 F y [ṀBHL v∞ ] θB = 22.5◦ θB = 45◦ θB = 67.5◦ 1.0 0.5 0.0 −0.5 β = 100 1.0 0.5 0.0 −0.5 Ra = 50rg Ra = 400rg 1.0 0.5 0.0 −0.5 0 1 2 3 4 5 t [104 rg /c] Figure 3.14: The drag force to the transverse (𝑦) direction (Magnus force) in all simulations. The gray line shows the result from the fiducial model. A negative value corresponds to the anti-Magnus effect. 79 + 𝑦ˆ direction, where (2) since a more amount of matter is deposited to 𝑦 < 0 region of the downstream, a net gravitational pull from the flow is enhanced toward − 𝑦ˆ direction. These two effects are competing with each other, and the direction of the net drag will be highly dependent on the nature of the accretion flow. A nonzero transverse drag 𝐹 𝑦 is the manifestation of the (general-relativistic analog of the) Magnus effect, the phenomenon in the classical fluid dynamics that a spinning body moving through a fluid experiencing a drag force normal to both the direction of its motion and spin. Although its physical origin is different, a similar effect is expected to be present in general relativity when a spinning compact object is immersed in a mass-energy current not aligned with its spin axis (e.g., Font, Ibanez, and P. Papadopoulos, 1999; Okawa and Cardoso, 2014; Costa, Franco, and Cardoso, 2018). However, the precise direction of this gravitational Magnus effect has been under debate when considering non-hydrodynamical types of matter (Okawa and Cardoso, 2014; Cashen, Aker, and Kesden, 2017; Costa, Franco, and Cardoso, 2018; Dyson et al., 2024; Z. Wang et al., 2024). Costa, Franco, and Cardoso (2018) argues that the gravitational Magnus force consists of two distinctive components (which they name as ‘Magnus’ and ‘Weyl’ respectively therein) and especially the Weyl component can be highly dependent on the specific scenario and boundary conditions of the physical system under consideration. A recent fully relativistic analysis (Dyson et al., 2024) has shown that the drag is always anti-Magnus for a collisionless particle-like matter field, where a wave-like scalar field shows a mixed behavior. A numerical relativity simulation on the BHL accretion of scalar dark matter (Z. Wang et al., 2024) has reported an anti-Magnus effect. Font, Ibanez, and P. Papadopoulos (1999), to the best of our knowledge, is the only numerical study commenting on this phenomenon in the hydrodynamic regime, and suggested that the enhanced pressure of the accretion flow on the counter rotating side of the Kerr BH gives rise to the (pro-) Magnus effect, albeit without a quantitative argument. We report in here that our physical scenario—3D GRMHD BHL accretion onto a spinning BH—yielded the positive sign of the Magnus force, analogous to the one in classical fluid dynamics. Fig. 3.14 collects and compares the Magnus force 𝐹 𝑦 from all models. For all the cases, the Magnus drag maintained a positive value for the most of the simulation time. While the gravitational drag force Eq. (3.31) we compute is not a fully general relativistic formula (see e.g., Costa, Franco, 80 and Cardoso, 2018; Dyson et al., 2024; Z. Wang et al., 2024), it is very unlikely to change the direction of the Magnus effect we observe from simulations. We mention that previous numerical studies on the gravitational Magnus effect in a scalar field (Okawa and Cardoso, 2014; Z. Wang et al., 2024) have been carried out in a boosted metric, unlike our setup in which the black hole is fixed and the inflow is imposed in terms of fluid velocity, potentially causing a quantitative difference in the drag force. It is also noteworthy that the magnitude of the Magnus force is not small, often rising to a level comparable to the BHL drag force scale 𝑀¤ BHL 𝑣 ∞ . Across all models, we quote a conservative overall estimate that the Magnus force has been observed to be about 10% of the dynamical friction. In the circumstances that the initial linear momentum of the BH has been substantially lost by the dynamical friction, its traveling trajectory could have been largely deflected from its original direction of motion. Our results have implications for a number of astrophysical contexts. In an extreme mass-ratio inspiral of binary black hole, if a secondary BH has spin and is surrounded by gaseous medium, the gravitational Magnus force on the secondary can alter its trajectory or excite an eccentricity to the orbit. The resulting features in gravitational waves can be potentially detectable with next generation gravitational wave detectors e.g., LISA (Afshordi et al., 2023). In the common envelope phase of a binary star, the drag force inside the gaseous envelope is responsible for the orbital decay and expansion of the envelope (Livio and Soker, 1988; Taam and Sandquist, 2000; Ivanova et al., 2013). Our findings imply that the BH orbiting within the envelope could experience the Magnus force when spiralling around the core of the companion star. Depending on the relative orientation of the BH spin to its direction of motion, the Magnus force can increase orbital eccentricity or induce precession of the orbital plane. This may have a considerable impact on the evolution of the BH orbit over long periods and the final configuration of the binary after the common envelope phase. We note that in this context a wind profile varying in the transverse direction needs to be taken into consideration as well, since the nonzero gradient in mass density or velocity leads to a misaligned, rotated shock cone geometry (Cruz-Osorio and Lora-Clavijo, 2016; Cruz-Osorio and Rezzolla, 2020; Lora-Clavijo, Cruz-Osorio, and Méndez, 2015) which in turn can significantly alter the direction of the total drag force. 81 3.6 Conclusion We have conducted three-dimensional general-relativistic magnetohydrodynamic simulations of Bondi-Hoyle-Lyttleton accretion onto a spinning black hole when the magnetic field of the surrounding plasma is inclined with respect to the spin of the BH. Our primary motivation was to investigate the dynamics of a BBH merger remnant kicked into the disk of an active galactic nucleus, but owing to the ubiquitous applicability of the Bondi-Hoyle-Lyttleton accretion problem, our results are also applicable to broader astrophysical contexts. We summarize our main findings below. • The accumulation of magnetic flux onto the BH establishes a magnetically arrested state of the accretion flow and launches relativistic outflows to polar directions, which are bent toward the downstream at a larger radius. The accretion disk surrounding the BH extends up to a few tens of 𝑟 𝑔 from the horizon, and is encompassed by a large-scale downstream flow in the shock cone. • Quasi-periodic magnetic flux eruptions from the BH launch pressure waves expanding the bow-shaped shock cone, and release strongly magnetized blobs (flux tubes)—which can potentially power flare-type electromagnetic transients (e.g., Hakobyan, Ripperda, and Philippov, 2023; Zhdankin, Ripperda, and Philippov, 2023) —along the equatorial plane into a narrow range of angles relative to the wind direction. • Anisotropic recoil from the magnetic flux eruptions drive a strong nutation on the accretion disk, often ripping its inner region off from the outer part. Nutation of the accretion disk is imprinted on the jet as its twisted morphology, potentially aiding the development of a kink instability (Bromberg and Tchekhovskoy, 2016). • For a purely horizontal magnetized wind (𝜃 𝐵 = 90◦ ), the system periodically undergoes a quiescent period with the duration ≳ 𝜏𝑎 , during which jet is quenched and the accretion flow relaxes to the SANE state. Magnetic polarity inversion is observed during this period. • With an increasing inclination of the magnetic field relative to the BH spin, the jet is launched earlier. However, the energy outflow power and efficiency did not show significant differences once the system establishes a MAD state. The orientation of the incoming magnetic field appears to hardly affect steady-state 82 properties of the jet, but determines the temporal behavior of its active and quiescent periods. • The model with a lower magnetization 𝛽 = 100 shows a more gradual evolution of 𝜙BH over active and quiescent epochs, as well as a slightly decreased energy outflow, which can be explained by a reduced supply of magnetic fluxes from the accretion flow. • When subjected to a faster wind speed, a decreased dynamical cross section and an increased ram pressure on the BH results in the suppression of jet launching. On the other hand, the model with a slower wind speed reached the MAD state earliest relative to the accretion timescale 𝜏𝑎 , which is consistent with a qualitative argument made in Kaaz, Murguia-Berthier, et al. (2023). Energy outflow shows the strongest dependency in the wind speed among all parameters considered in this work. This strong dependence of the overall flow dynamics on the wind speed suggests that realistic values of 𝑣 ∞ will be a crucial element for improved models of GRMHD BHL accretion. • The gravitational Magnus effect is observed across all models, with the magnitude of a few tens of percents of the reference drag scale 𝑀¤ BHL 𝑣 ∞ . The direction of the Magnus force is the same as its classical aerodynamic counterpart. Whereas the accretion radius 𝑅𝑎 adopted in this work has one of the largest values in literature for the general relativistic BHL accretion, it is still considerably far from a realistic condition. In the scenario of a kicked BBH post-merger remnant, the recoil velocity is ≲ 200 km s−1 for non-spinning binaries (Gonzalez, Sperhake, et al., 2007) where the spin effects can at most enhance the recoil up to ≈ 4000 km s−1 for ‘superkick’ configurations (Campanelli et al., 2007b; Gonzalez, Hannam, et al., 2007). Newtonian studies suggest that when a BBH is embedded in a gaseous environment, accretion makes their orbital and spin axes aligned (Bogdanovic, Reynolds, and Miller, 2007; Coleman Miller and Krolik, 2013), suppressing the superkick configuration. The maximum recoil for a spin-orbit aligned binary estimated from numerical relativity simulations is ≈ 500 km s−1 (Herrmann et al., 2007; Koppitz et al., 2007; Healy, Lousto, and Yosef Zlochower, 2014). Another physical system that can harbor a fast wind accreting onto a BH is an X-ray binary with a high-speed stellar wind; for example, Cygnus X-1 binary system features 𝑣 ∞ ≳ 1000 km s−1 (Davis and Hartmann, 1983; Gies et al., 2008; Grinberg et al., 2015). However, 83 the orbital motion of the BH and a specific geometry of the wind present in these systems might require a deviation from the conventional BHL accretion setup. Our present study opens up a number of different avenues for future work, with several questions still remaining to be answered. First, the energy outflow power can be affected by parameters other than those we have considered here: the magnitude of the BH spin, the inclination of the magnetic field relative to the wind velocity, the spin-wind orientation (Gracia-Linares and Guzmán, 2023), or hydrodynamic parameters such as the adiabatic index and the Mach number which are known to strongly influence the stability of the shock cone (Foglizzo, Galletti, and Ruffert, 2005). It would be also intriguing to investigate the inflow with a nonzero net angular momentum (Cruz-Osorio and Lora-Clavijo, 2016; Lora-Clavijo, Cruz-Osorio, and Méndez, 2015), which provides a more appropriate scenario for the common envelope phase. The inclusion of radiative effects (Zanotti, Roedig, et al., 2011) will also greatly change the dynamics for super-Eddington accretion flows. Future investigations would help constructing a more detailed physical picture of a black hole moving through a magnetized medium, with a better bridging of the scale gaps between currently available numerical models and realistic astrophysical scenarios. Acknowledgements We are grateful to James Stone, Jacob Fields, and Hengrui Zhu for technical support, and to Saavik Ford, James Fuller, Matthew Graham, Yuri Levin, Barry McKernan, Nicholas Rui, and Alexander Tchekhovskoy for insightful discussions. Simulations were performed on DOE OLCF Summit cluster under allocation AST198, and on DOE NERSC Perlmutter cluster under grant m4575. This research used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725, and resources of the National Energy Research Scientific Computing Center, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231. 84 t [τa ] Ṁ /ṀBHL 5 10 15 20000 30000 20 25 30 40000 50000 60000 r = rH r = 2 rg r = 3 rg r = 4 rg r = 5 rg 0.2 0.1 0.0 10000 t [rg /c] Figure 3.15: Mass accretion rate extracted at different radii for the 𝛽10 -𝜃 90 -𝑅200 model. All data points are displayed without smoothing. t [τa ] x Fmom [ṀBHL v∞ ] 5 10 15 20 25 1.0 0.5 0.0 −0.5 −1.0 −0.5 x Fgrav [ṀBHL v∞ ] 30 r = rH r = 2rg r = 3rg r = 4rg r = 5rg −1.0 −1.5 −2.0 −2.5 −3.0 10000 20000 30000 40000 50000 60000 t [rg /c] Figure 3.16: The 𝑥ˆ component of the momentum drag (top panel) and gravitational drag (lower panel) computed with different radii from the 𝛽10 -𝜃 90 -𝑅200 model. All data points are displayed without smoothing. 3.A Extraction radius of density-related integral quantities As discussed in Sec. 3.2.4, the drift flooring algorithm artificially injects the mass density floor to maintain the comoving magnetization 𝜎 below an upper limit 𝜎max . This often induces spurious increases in the fluid-related integral quantities when computed very close to the BH, where the magnetization is very high. In Figure 3.15, we show the mass accretion rate 𝑀¤ integrated at different radii 𝑟𝑖 = {𝑟 𝐻 , 2𝑟 𝑔 , 3𝑟 𝑔 , 4𝑟 𝑔 , 5𝑟 𝑔 } from our representative model (𝛽10 -𝜃 90 -𝑅200 ) where √︁ 𝑟 𝐻 = 𝑟 𝑔 (1 + 1 − 𝑎 2 /𝑀 2 ) is the outer horizon radius. It can be clearly seen that the artificial effects from numerical flooring becomes almost absent in 𝑟𝑖 ≥ 3𝑟 𝑔 . Fig. 3.16 shows, for the same set of radii, the 𝑥ˆ component of the momentum drag 𝑥 𝑥 𝐹mom computed at the spherical surface 𝑟 = 𝑟𝑖 and the gravitational drag 𝐹grav 85 integrated over the whole computational domain except the spherical volume 𝑟 < 𝑟𝑖 . The momentum drag settles down to near zero for 𝑟𝑖 ≥ 3𝑟 𝑔 . The magnitude of the gravitational drag monotonically decreases for a larger 𝑟𝑖 since the region 𝑟 < 𝑟𝑖 is simply excluded from the volume integral. The differences between them are not significant, indicating that the gravitational drag is mostly contributed from 𝑟 ≥ 5𝑟 𝑔 . 86 Chapter 4 MAGNETOSPHERIC TRANSIENTS FROM BLACK HOLE–NEUTRON STAR MERGER Kim, Yoonsoo et al. (2024). “Black Hole Pulsars and Monster Shocks as Outcomes of Black Hole–Neutron Star Mergers”. In: Astrophysical Journal Letters 982.2, p. L54. doi: 10.3847/2041-8213/adbff9. url: https://iopscience.iop. org/article/10.3847/2041-8213/adbff9. The merger of a black hole (BH) and a neutron star (NS) in most cases is expected to leave no material around the remnant BH; therefore, such events are often considered as sources of gravitational waves without electromagnetic counterparts. However, a bright counterpart can emerge if the NS is strongly magnetized, as its external magnetosphere can experience radiative shocks and magnetic reconnection during/after the merger. We use magnetohydrodynamic simulations in the dynamical spacetime of a merging BH–NS binary to investigate its magnetospheric dynamics. We find that compressive waves excited in the magnetosphere develop into monster shocks as they propagate outward. After swallowing the NS, the BH acquires a magnetosphere that quickly evolves into a split monopole configuration and then undergoes an exponential decay (balding), enabled by magnetic reconnection and also assisted by the ring-down of the remnant BH. This spinning BH drags the split monopole into rotation, forming a transient pulsar-like state. It emits a striped wind if the swallowed magnetic dipole moment is inclined to the spin axis. We predict two types of transients from this scenario: (1) a fast radio burst emitted by the shocks as they expand to large radii and (2) an X/𝛾-ray burst emitted by the 𝑒 ± outflow heated by magnetic dissipation. 4.1 Introduction Merging black hole (BH)–neutron star (NS) binaries are promising sources of gravitational waves (GWs) (see, e.g., R. Abbott et al., 2021b; R. Abbott et al., 2020; Abac et al., 2024, for recent detections). Depending on the mass ratio of the system and spin of the black hole, near-equal mass systems can feature tidal disruption of the neutron star during merger, leading to dynamical mass ejection and the formation 87 of a massive disk (Foucart, 2012; Foucart, Hinderer, and Nissanke, 2018). These can power electromagnetic (EM) counterparts such as kilonova afterglows (Lattimer and Schramm, 1974; L.-X. Li and Paczynski, 1998; Tanaka et al., 2014; Kawaguchi, Kyutoku, et al., 2016; Fernández et al., 2017; Metzger, 2020; Gottlieb et al., 2023b; Kawaguchi, Domoto, et al., 2024) and gamma-ray bursts (GRBs) (Janka et al., 1999; Etienne, Liu, et al., 2012; Etienne, Paschalidis, and Shapiro, 2012; Paschalidis, Ruiz, and Shapiro, 2015; Shapiro, 2017; Ruiz, Shapiro, and Tsokaros, 2018; Hayashi, Fujibayashi, et al., 2022; Gottlieb et al., 2023a; Martineau et al., 2024). However, the high mass ratio typical of such systems (R. Abbott et al., 2021b; Abac et al., 2024) would likely result in a non-disruptive merger, leaving little or no matter surrounding the remnant BH (Foucart, 2012; Foucart, Hinderer, and Nissanke, 2018). Most BH–NS mergers are expected to fall in this latter category and be EM-quiet (Fragione, 2021; Biscoveanu, Landry, and Vitale, 2022), supported by the absence of EM counterparts to previous detections (e.g. Anand et al., 2021). On the other hand, neutron stars can be equipped with strong exterior magnetic fields, leading to potential EM counterparts from magnetospheric interactions with their binary companion. Previously studied scenarios can be broadly split into two groups. Transients before merger (precursors) can be produced through magnetospheric interactions (McWilliams and J. Levin, 2011; Lai, 2012; Piro, 2012; Paschalidis, Etienne, and Shapiro, 2013; Carrasco, Viganò, et al., 2019; Carrasco, Shibata, and Reula, 2021) including flares (Most and Philippov, 2023a; Beloborodov, 2021), or through gravitationally driven resonances in the neutron star such as crustal shattering (Tsang et al., 2012; Penner et al., 2012; Most, Beloborodov, and Ripperda, 2024). Potential transients at merger (concurrent EM counterpart) have been attributed to either a net electric charge of the black hole (J. Levin, D’Orazio, and Garcia-Saenz, 2018; B. Zhang, 2019; Z.-G. Dai, 2019; Pan and Yang, 2019; Zhong, Z.-G. Dai, and Deng, 2019), or magnetic flux shedding during the merger process (D’Orazio and J. Levin, 2013; Mingarelli, J. Levin, and Lazio, 2015; D’Orazio, J. Levin, et al., 2016; East et al., 2021). Predicting magnetospheric dynamics of the merger is intrinsically complicated by various competing processes, some of which can be inferred from previous numerical studies of a NS gravitationally collapsing into a BH. In this related scenario, part of the magnetic field is immediately shed during the collapse (Baumgarte and Shapiro, 2003; Lehner et al., 2012; Palenzuela, 2013; Most, Nathanail, and Rezzolla, 2018). In the absence of resistive dissipation, the resulting BH can in principle 88 acquire a net electric charge (Nathanail, Most, and Rezzolla, 2017). However, pairproduction in realistic environments will lead to an active magnetosphere supporting magnetic flux decay (balding) of the BH (Lyutikov and McKinney, 2011; Bransgrove, Ripperda, and Philippov, 2021; Selvi et al., 2024). On a technical level, most of the studies in numerical relativity have made use of the force-free electrodynamics or vacuum approaches to study magnetospheric dynamics. Compared to magnetohydrodynamic (MHD) approaches explicitly tracking matter dynamics, these crucially miss out the formation of monster radiative shocks from fast magnetosonic waves (Beloborodov, 2023) as was recently demonstrated by Most, Beloborodov, and Ripperda (2024), which could be responsible for some of the high-energy emission in this process. Here, we present GRMHD simulations in full numerical relativity of a merging BH–NS binary, in which the NS is swallowed whole. While BH–NS merger simulations in GRMHD have become common recently (e.g., Chawla et al., 2010; Etienne, Liu, et al., 2012; Etienne, Paschalidis, and Shapiro, 2012; Kiuchi et al., 2015; Ruiz, Shapiro, and Tsokaros, 2018; Ruiz, Paschalidis, et al., 2020; Most, Papenfort, Tootle, et al., 2021; Hayashi, Fujibayashi, et al., 2022; Hayashi, Kiuchi, et al., 2023; Izquierdo et al., 2024), tracking the magnetospheric evolution requires special flooring techniques (Tchekhovskoy and Spitkovsky, 2013; Parfrey and Tchekhovskoy, 2017). We employ such a sophisticated MHD strategy to track the evolution of magnetosphere throughout inspiral and merger. Our simulations identify novel types of shock-powered and reconnection-driven transients from a BH–NS merger. Specifically, we show that monster shocks are formed during the final phase of the inspiral, which can primarily source X-ray and radio bursts. In the post-merger phase, we find that the magnetosphere of the remnant BH re-arranges into a short-lived black hole pulsar state (Selvi et al., 2024), capable of powering X-ray transients that may last for several milliseconds. We describe the simulation setup and the configuration of the binary in Sec. 4.2. This is followed by detailed discussions of two new transients from non-disrupting BH– NS mergers. First, we present the formation of monster shocks in Sec. 4.3. Next, we provide a detailed analysis of the black hole pulsar state that our simulations reveal in Sec. 4.4. We discuss the properties of the expected EM emissions in Sec. 4.5. Finally, we conclude by summarizing our findings in Sec. 4.6. Unless otherwise stated, we adopt Gaussian units with 𝑐 = 𝐺 = 1 throughout this chapter. 89 4.2 Methods We track the time evolution of a BH–NS binary as well as the common binary magnetosphere using ideal GRMHD for dynamical spacetimes (Duez et al., 2005). To this end, we need to specify both initial conditions and evolution parameters. We use the Kadath/FUKA (Papenfort et al., 2021; Grandclement, 2010) initial data framework to construct BH–NS initial data in extended conformal thin sandwich (XCTS) form (Grandclement, 2006; Taniguchi et al., 2007; Taniguchi et al., 2008; Foucart, Kidder, et al., 2008; Tacik et al., 2016). In order to ensure that the NS is fully swallowed at merger,1 we adopt a non-spinning NS with mass 𝑀NS = 1.4𝑀⊙ using the APR4 equation of state (Akmal, Pandharipande, and Ravenhall, 1998), and a BH with mass 𝑀BH = 8.0𝑀⊙ and dimensionless spin 𝑎 = 0.3 aligned with the orbital axis (𝑧ˆ). The initial orbital separation of the binary is 60 km, resulting in ∼ 1.5 orbits before the merger. The neutron star is initially magnetized with a dipolar field with a strength |𝐵∗ | = 1.9 × 1016 G at the magnetic poles on the surface. The precise value of the magnetic field is unimportant for the magnetospheric dynamics we study, since we fix the properties of the magnetosphere in terms of dimensionless quantities such as magnetization 𝜎 = 𝑏 2 /𝜌, and plasma 𝛽 = 2𝑃/𝑏 2 , where 𝑏 2 is the magnetic energy density, 𝜌 the rest-mass density and 𝑃 the pressure. This allows us to rescale the resulting magnetospheric dynamics to arbitrary magnetic field strength. However, for purely numerical reasons we have found that using a stronger field strength eases the transition to a near force-free magnetosphere near the stellar surface in the inspiral computationally. Nevertheless, the chosen strength of the magnetic field hardly impacts the bulk dynamics of the NS (the plasma beta parameter is around 𝛽 ∼ 103 inside the NS during the inspiral). We simulate three models with an initial inclination between the magnetic dipole moment and the orbital axis 𝜃 𝐵 = 0◦ , 30◦ , and 60◦ . The initial NS magnetic field is inclined toward the companion BH at 𝑡 = 0. Dynamical evolutions are performed with the Einstein Toolkit framework (Loffler et al., 2012), using the Frankfurt/IllinoisGRMHD (FIL) (Most, Papenfort, and Rezzolla, 2019; Etienne, Paschalidis, Haas, et al., 2015) code for solving the ideal GRMHD equations in a dynamical spacetime. The spacetime is evolved using FIL’s numerical relativity solver, which implements the Z4c equations (Bernuzzi and Hilditch, 2010; Hilditch, Bernuzzi, et al., 2013) in moving puncture gauge 1 See Foucart (2012) and Foucart, Hinderer, and Nissanke (2018) for the allowed parameter space of a non-disrupting BH–NS merger. 90 (Alcubierre et al., 2003) using a fourth-order finite-difference discretization (Y. Zlochower et al., 2005). The ideal GRMHD equations are solved using the ECHO scheme (Del Zanna, Zanotti, et al., 2007) with upwind constraint transport (Londrillo and Del Zanna, 2004). Similar to our previous work (Most, Beloborodov, and Ripperda, 2024), the fourth-order derivative corrector in the ECHO scheme showed less robust behavior at strongly magnetized shockfronts, and we have disabled it in our runs. A key feature of our simulations is the ability to track the common magnetospheric dynamics in full MHD as opposed to vacuum or force-free electrodynamics. While several studies have evolved magnetic fields in the exterior region in the context of BH–NS mergers (Paschalidis, Ruiz, and Shapiro, 2015; Ruiz, Shapiro, and Tsokaros, 2018; Ruiz, Paschalidis, et al., 2020), reproducing correct (near-) force-free magnetospheric dynamics within the MHD formulation requires the use of robust primitive inversion schemes (Kastaun, Kalinani, and Ciolfi, 2021) and special flooring techniques (Tchekhovskoy and Spitkovsky, 2013; Parfrey and Tchekhovskoy, 2017), unlike floors commonly used in numerical relativity simulations (e.g. Poudel et al., 2020). A detailed prescription of the floors we use here is provided in Most, Beloborodov, and Ripperda (2024). It is precisely this flooring scheme that allows us to correctly capture and uncover the transients we present in this study. Similar to Most, Beloborodov, and Ripperda (2024), we have supplemented the high-density cold equation of state used in the initial data with a thermal equation of state, 𝑃 th = 𝜌𝜖, which primarily governs the magnetospheric dynamics. Here 𝑃 th is the thermal pressure, and 𝜖 the specific internal energy. We use a three-dimensional Cartesian grid with eight levels of nested moving mesh refinement (Schnetter, S. H. Hawley, and Hawke, 2004). The coarsest grid extends to [−3025 km, 3025 km] 3 and the finest resolution is 168 m. The finest grid level consists of two patches covering 2–3× the size of the NS as well as of the BH, being centered to and tracking each of them. A detailed description of the initial evolution of non-disruptive BH–NS mergers can be found elsewhere (see e.g. Kyutoku, Shibata, and Taniguchi, 2021, for a recent review). Since the magnetospheric transients in our simulations are mainly driven during and after the merger, we briefly depict the merger process in Figure 4.1, which highlights the high degree of spatial asymmetry present in the process, and consequently, the need for full numerical relativity not only for the spacetime evolution but particularly to correctly determine the geometry of magnetic field in the post-merger phase. 91 30 20 y [km] 10 0 −10 −20 −30 −30 −20 −10 0 10 x [km] 20 30 −30 −20 −10 0 10 x [km] 20 30 −30 −20 −10 0 10 20 30 x [km] Figure 4.1: Merger of the BH–NS binary in our simulations, where the neutron star is swallowed whole. The entire process shown in this figure happens in less than one millisecond. 4.3 Monster shock The initially dipolar magnetosphere of the NS is sheared in the vicinity of the BH before and during merger. This perturbation will launch waves into the magnetosphere, which will either be transverse (Alfvén) wave propagating along the magnetic field, or be a compressional (fast magnetosonic) wave. In a dipole background field, the compressional waves are expected from any non-toroidal perturbation in the magnetosphere, as happens during the merger. Propagation of fast magnetosonic waves to larger radii 𝑟 is not affected by the background field as long as their wave amplitude 𝐸 = 𝛿𝐵 ∝ 𝑟 −1 is much smaller than the background dipole field 𝐵bg ∝ 𝑟 −3 . With increasing 𝑟, this condition becomes broken, 𝐵2 − 𝐸 2 approaches zero, and the plasma drift speed in the wave approaches the speed of light.2 Recent analytical (Beloborodov, 2023) and numerical (A. Y. Chen, Yuan, X. Li, et al., 2022; Vanthieghem and Levinson, 2025) works have demonstrated that this leads to the formation of monster shocks.3 In particular, in the equatorial plane of the magnetic dipole, the shock appears when 𝛿𝐵 ≈ 𝐵bg /2, which implies 𝐵2 − 𝐸 2 touching zero at the trough of the compressional wave. Near this point, the plasma develops a characteristic negative velocity 𝑣 𝑟 < 0, which leads to shock formation in front of the crest of the wave. In practice, searching for zones with 𝑣 𝑟 < 0 provides a simple way to identify regions of shock formation, 2 Orbiting systems possess an orbital light cylinder, 𝑟 LC ∼ 1/Ωorb , set by the orbital frequency Ωorb . Steepening or distortion of the waves induced by a decreasing 𝐵 bg will only happen efficiently on closed field lines inside the light cylinder, in turn requiring a minimum amplitude of the perturbation (see, e.g., Most, Kim, et al., 2024 for a discussion in the context of non-linear steepening of Alfvén waves). 3 See also Lyubarsky (2003) for an earlier work. 92 Figure 4.2: Poloidal structure (cut in the 𝑦𝑧 plane) of the perturbed magnetosphere of the BH–NS binary 0.9 ms before merger for the aligned (𝜃 𝐵 = 0◦ ) model. Fast magnetosonic waves have toroidal electric fields 𝐸 𝜙 (left), and Alfvén waves have toroidal magnetic perturbations 𝛿𝐵 𝜙 (right). Streamlines show fluid velocity in the left panel and magnetic field lines in the right panel. The BH and NS are shown with a black and blue circle, respectively. in addition to detection of velocity jumps and localized heating spikes. A similar analysis was performed in Most, Beloborodov, and Ripperda (2024) to demonstrate shock formation in the magnetosphere of a collapsing magnetar. In our simulations, the inspiral of the magnetized NS drives a continuous excitation of magnetosonic waves in the magnetosphere, peaking around the plunge of the NS into the BH. The final plunge of the NS injects a strong rarefaction mode into the surrounding magnetosphere as the NS bulk velocity is maximally radially inward at the moment. In Fig. 4.2, we show the excited magnetosphere about half an orbit before the plunge for aligned (𝜃 𝐵 = 0◦ ) magnetic axis. We find that the wave emitted during the plunge leads to the development of a large 𝑣 𝑟 < 0 region characteristic of the monster shock, which we show in the top row of Fig. 4.3. This phenomenology of a leading shock with surrounding weaker shocks resembles the results for the collapsing magnetar (Most, Beloborodov, and Ripperda, 2024), and approximately agrees with the analytical prediction (Beloborodov, 2023, see Fig. 7 therein). The profile of 𝛾𝑣 𝑟 across the shock region is affected by deviations of 𝐵bg from a pure dipole due to the orbital motion of the NS. As an additional validation, we have also confirmed that the regions with 𝑣 𝑟 < 0 develop 𝐸 2 ≈ 𝐵2 plateaus, corroborating 93 Figure 4.3: Monster shocks launched from BH–NS mergers. Shown here are the Lorentz factor (left panels) and the radial spatial velocity (right panels) on the meridional (𝑥𝑧) plane. Dashed orange circles are light spheres with the radius 𝑟 = 𝑐(𝑡 − 𝑡merger ). (Top) Simulation snapshot from the aligned model (𝜃 𝐵 = 0◦ ). A monster shock can be found near 𝑥 ≃ 250 km, with its characteristic feature of a plasma moving radially inward (𝑣 𝑟 < 0) preceding the shockfront. (Bottom) Inclined model 𝜃 𝐵 = 60◦ . A similar feature can be seen near 𝑥 ≃ −400 km. that the observed feature is the monster shock. We have also identified monster shocks for the inclined models. One such model with 𝜃 𝐵 = 60◦ is shown in in the bottom row of Fig. 4.3. We caution that due to the misalignment between magnetic equator and orbital plane, the strongest part of the shock will appear off the shown 𝑦𝑧 plane, and the trough of the wave preceding the monster shock might not strongly exhibit 𝑣 𝑟 < 0. Yet, we can clearly identify a similar leading shock structure as in the aligned case. 4.4 Transient black hole pulsar In this section, we present a detailed analysis of the evolution of the post-merger magnetosphere, defined as the region within the light sphere (𝑟/𝑐 ≤ 𝑡−𝑡merger ), with a 94 special emphasis on the near-horizon dynamics. The merger remnant settles down to a Kerr BH with the mass 𝑀 = 9.2𝑀⊙ and the dimensionless spin 𝑎 = 0.57. Relevant length and time scales are 𝑟 𝑔 ≡ 𝐺 𝑀/𝑐2 = 13.6 km and 𝑟 𝑔 /𝑐 = 46 𝜇s, or equivalently 1 millisecond amounts to ∼ 22𝑟 𝑔 /𝑐. The angular velocity of the outer event horizon √ is Ω𝐻 = 𝑎𝑐/2𝑟 + = 3.43 × 103 s−1 , where 𝑟 + = 𝑟 𝑔 (1 + 1 − 𝑎 2 ) ≈ 25 km is the outer horizon radius. While we will quote values measured from our simulation data in the following discussions, we caution that it is nontrivial to map our results (especially time scales) in a coordinate-independent manner to those obtained from other studies that used a fixed Kerr background, since our merger simulations are performed using dynamically evolved coordinates. 4.4.1 Relaxation into a rotating split-monopole The remnant BH is immersed in a dipole-like magnetic field shortly after the merger. Since black holes cannot support closed magnetic field lines (MacDonald and Thorne, 1982), the dipole gets stretched out, with the magnetic field lines opening up near the magnetic equator. In consequence, the BH magnetosphere transitions into a split-monopole topology (Komissarov, 2004b), and begins to dissipate the magnetic field energy at the current sheet. The inclination of the split-monopole configuration depends on the initial inclination of the NS magnetic field. For all simulations, the topology of magnetic field lines transitions into a split-monopole over a timescale of 1 ms,4 which is consistent with multiple light crossing times across the horizon (2𝑟 + /𝑐 ≈ 0.2 ms). The distribution of magnetic flux on the remnant BH is initially highly localized to the spot through which the NS plunged into (Fig. 4.1). Over the transition period to a split-monopole (𝑡 − 𝑡merger ≲ 1ms), the magnetic flux density on the BH horizon is redistributed, relaxing into a relatively uniform distribution by 𝑡 −𝑡merger ≈ 2 ms. The upper panels of Fig. 4.4 show the post-merger magnetosphere at 𝑡 − 𝑡merger = 3.3 ms for the aligned model (𝜃 𝐵 = 0◦ ), displaying an axisymmetric split-monopole magnetosphere centered on the BH. Magnetic energy of the magnetosphere is partially dissipated via reconnection in the equatorial current sheet, heating the plasma, as can be seen from the plot of 𝑃th /𝜌 in Fig. 4.4. The frame dragging of the remnant BH induces co-rotation of magnetic field lines and forms a rotating split-monopole. The angular velocity of the magnetic field lines in an axisymmetric force-free split-monopole magnetosphere around a Kerr 4 A similar reordering and collimation of the post-merger magnetic field may also have been observed in previous works (East et al., 2021). 95 Figure 4.4: Post-merger magnetosphere of the remnant black hole having settled down to a rotating split monopole. The physical quantities are shown in the 𝑥𝑧 plane. Left: fluid Lorentz factor 𝛾. Right: ratio of the thermal pressure 𝑝 th to the rest energy density 𝜌𝑐2 . Black solid lines show the in-plane magnetic field lines. An equatorial current sheet is formed at which the magnetic field lines in upper and lower hemispheres reconnect, dissipating magnetic energy and causing a flux decay (balding) of the BH. The inner magnetosphere is driven to co-rotate with the BH due to frame dragging. As a result, the post-merger magnetosphere of an inclined model (e.g. 𝜃 𝐵 = 30◦ , bottom panels) exhibits features similar to those of tilted pulsars (see also Fig. 4.10). The spin axis of the BH is along 𝑧ˆ in all models. 96 Figure 4.5: Angular velocity of magnetic field lines threading the apparent BH horizon for 𝜃 𝐵 = 0◦ simulation. Shown are the distribution of Ω𝐹 for each latitude and the stationary axisymmetric force-free solution Ω𝐹 ≃ Ω𝐻 /2 (red dashed line). BH is given as Ω𝐹 = 𝑎/8𝑀 to leading order in the spin (Komissarov, 2004a; Armas et al., 2020). For arbitrary high spins, Ω𝐹 can be calculated either with a perturbative analytic expansion (e.g. Armas et al., 2020) or using an iterative numerical method (e.g. Contopoulos, Kazanas, and D. B. Papadopoulos, 2013; Nathanail and Contopoulos, 2014). The ratio Ω𝐹 /Ω𝐻 , which is 1/2 in the limit 𝑎 → 0, remains ≲ 1% different from 1/2 for the spin 𝑎 ≤ 0.7 (e.g. see Figure 1 of Contopoulos, Kazanas, and D. B. Papadopoulos, 2013). Therefore, in the case of our merger remnant BH with 𝑎 = 0.57, we can safely assume Ω𝐹 /Ω𝐻 ≃ 0.5. For the aligned (𝜃 𝐵 = 0◦ ) model, we measure the rotation angular velocity of magnetic field lines as −𝑦 (𝑢 𝑥 /𝑢 0 ) + 𝑥 (𝑢 𝑦 /𝑢 0 ) , (4.1) 𝜛2 √ where 𝑢 𝜇 is the four-velocity of the plasma and 𝜛 = 𝑟 2 − 𝑧2 is the (coordinate) cylindrical radius. This description is appropriate for the ideal MHD limit we consider. Fig. 4.5 shows the measured Ω𝐹 /Ω𝐻 over a spherical surface 𝑟 = 2.4 𝑟 𝑔 encompassing the remnant BH.5 We observe that the rotation angular velocity of magnetic field lines converges to Ω𝐻 /2 and the asymmetry present in its distribution is decayed out over time. In addition to the magnetic field morphology shown in Ω𝐹 = 5 When computing the horizon angular velocity Ω apparent horizon of the BH. 𝐻 , we used the mean radius of the instantaneous 97 Figure 4.6: A spacetime diagram of the approximate electric current 𝑗 𝜙 on a meridional arc 𝜙 = 0 at 𝑟 = 2.4𝑟 𝑔 in three simulations with different initial inclinations of the NS magnetic dipole moment (𝜃 𝐵 = 0◦ , 30◦ , 60◦ ), displaying the latitude of the post-merger magnetospheric BH current sheet. For the inclined models (𝜃 𝐵 = 30◦ , 60◦ ), the periodic oscillation in the latitude represents the rotation of the inclined current sheet. The orbital current sheet of the NS in the inspiral phase is also visible for 𝑡 − 𝑡merger ≲ −2 ms, where 𝑡 merger indicates the merger time. Fig. 4.4, this provides solid evidence that the post-merger magnetosphere relaxes into a rotating split-monopole. As naturally expected, for inclined magnetic field, the remnant BH settles down to a rotating, inclined split-monopole magnetosphere. The resulting global dynamics of the magnetosphere closely resembles that of a tilted pulsar, akin to a recently proposed black hole pulsar state (Selvi et al., 2024). We present detailed discussions on this transient BH pulsar in later sections. 4.4.2 Rotation and alignment of current sheets The spinning remnant BH induces a rotation of the magnetic field lines and the current sheets via frame dragging with respect to its spin axis. In the following, we would like to track the motion of current sheets. These are easily identified with the (toroidal) electric current, 𝑗 𝜙 , which we here approximate via its Newtonian expression, 𝑗 𝜙 ≈ 𝜖 𝜙𝑖 𝑗 𝜕𝑖 𝐵 𝑗 . (4.2) We then analyze the time evolution of the current on a fixed spherical surface of radius 𝑟 = 2.4𝑟 𝑔 . In Fig. 4.6, we show the distribution of | 𝑗 𝜙 | on the meridional arc 98 0 50 t − tmerger [rg /c] 100 150 200 250 θB = 30◦ θB = 60◦ 80◦ χ 70◦ 60◦ 50◦ 0 2 4 6 8 10 t − tmerger [ms] Figure 4.7: Time evolution of the current sheet inclination angle 𝜒 for the 𝜃 𝐵 = 30◦ , 60◦ models. 𝜙 = 0 (𝑥 > 0, 𝑦 = 0) for all simulations. The equatorial current sheet in the aligned case (𝜃 𝐵 = 0◦ ) does not exhibit notable modulations in its latitude, where we have confirmed in Sec. 4.4.1 that the magnetosphere is in fact rotating with Ω𝐹 = 0.5Ω𝐻 . For the inclined cases, rotation of the current sheet is clearly seen in Fig. 4.6. The time interval between neighboring peaks is about 3.5 ms, revealing that the current sheet is rotating with about half of the horizon angular velocity. Over each meridional lines (𝜙 = const.) on the spherical surface 𝑟 = 2.4𝑟 𝑔 , we collect the latitude 𝛼0 (𝜙) with the maximum value of | 𝑗 𝜙 |, which is effectively the latitude of the current sheet at that azimuthal angle. Then we define the current sheet inclination angle 𝜒 as6 𝜒= max[𝛼0 (𝜙)] − min[𝛼0 (𝜙)] . 2 (4.3) In Fig. 4.7, we show the measured 𝜒(𝑡) from 𝜃 𝐵 = 30◦ , 60◦ simulations. A nonlinear deformation of the NS during the merger is found to greatly enhance the inclination angle of the magnetic field around the merger remnant, resulting in 𝜒 ≈ 60◦ for 6 See the section 3 of Selvi et al. (2024) for an alternative method to measure 𝜒(𝑡) in terms of the magnetic moment. 99 𝜃 𝐵 = 30◦ . We observe a gradual decay in 𝜒(𝑡) for both models, indicating the alignment of the current sheet with respect to the BH spin axis over time, which is consistent with the result of Selvi et al. (2024). However, here we can only provide a crude estimate on the alignment timescale 𝜏𝜒 ≈ 1000–2000 𝑟 𝑔 /𝑐, being limited by a short simulation time and a mild spin of the BH. We also caution that the observed alignment timescale could be affected by a high numerical dissipation (see Sec. 4.4.3). 4.4.3 Balding and ring-down of the remnant BH In a split-monopole magnetosphere of a stationary BH, the total magnetic flux threading the horizon ∮ 1 |𝐵𝑟 |𝑑Ω, (4.4) ΦB = 2 exponentially decays as a result of magnetic reconnection in the current sheet (Lyutikov and McKinney, 2011; Bransgrove, Ripperda, and Philippov, 2021; Selvi et al., 2024). In the top panel of Fig. 4.8, we show the decay of the horizon magnetic flux ΦB for all simulations. Overall, the decay times shown in Fig. 4.8 are an order of magnitude shorter than those from the MHD simulations of Bransgrove, Ripperda, and Philippov (2021) and Selvi et al. (2024). This is likely due to artificially high numerical resistivity (i.e., low spatial grid resolution) compared to those studies. At higher resolution than we use, our MHD solution will equally not be able to recover the correct collisionless reconnection rate (Sironi and Spitkovsky, 2014; Bransgrove, Ripperda, and Philippov, 2021). We therefore treat our results mainly qualitatively, in that the BH pulsar forms and balds, and defer quantitative conclusions to an analytical model discussed in Sec. 4.4.4. The magnetic flux decay timescale 𝜏Φ is almost identical for both of the inclined models, while the aligned model exhibits about 30% faster decay until 𝑡 − 𝑡merger ≲ 6 ms. However, in a split monopole magnetosphere of a stationary Kerr BH, the timescale 𝜏Φ may not notably depend on the current sheet inclination angle 𝜒 (Selvi et al., 2024). We investigate the origin of the accelerated magnetic flux decay by examining additional physical quantities, as follows. Electromagnetic and gravitational perturbations around a BH can be analyzed by means of the Newman-Penrose (NP) scalars (Teukolsky, 1972; Teukolsky, 1973) 𝜓4 = −𝐶𝑎𝑏𝑐𝑑 𝑛𝑎 𝑚¯ 𝑏 𝑛𝑐 𝑚¯ 𝑑 , (4.5) 𝜙2 = 𝐹𝑎𝑏 𝑚¯ 𝑎 𝑛 𝑏 , (4.6) 100 10−1 −50 0 t − tmerger [rg /c] 50 100 150 200 10−2 ΦB 10−3 10−4 0◦ 30◦ 60◦ e−ct/31rg e−ct/23rg 10−5 10−6 10−5 10−6 (1,1) ]| 10−7 |Im [φ2 10−8 10−9 0◦ 30◦ 60◦ e−ct/31rg e−ct/13rg 10−10 10−11 |Im [ψ4 (2,2) ]| 10−4 e−ct/13rg 10−5 10−6 10−7 −2.5 0.0 2.5 5.0 7.5 10.0 t − tmerger [ms] Figure 4.8: Top: total magnetic flux extracted on a spherical surface 𝑟 = 2.4𝑟 𝑔 near the apparent horizon. The result from 𝜃 𝐵 = 30◦ (cyan solid line) and 𝜃 𝐵 = 60◦ (orange solid line) are lying almost on top of each other. Middle: imaginary part of (𝑙, 𝑚) = (1, 1) mode of the Maxwell Newman-Penrose (NP) scalar 𝜙2 extracted at 𝑟 = 4.3𝑟 𝑔 . Bottom: imaginary part of (𝑙, 𝑚) = (2, 2) mode of the NP scalar 𝜓4 extracted at 𝑟 = 4.3𝑟 𝑔 . We only show the result from 𝜃 𝐵 = 0◦ since the result is almost identical for all simulations. Exponential decays with timescales 31𝑟 𝑔 /𝑐, 23𝑟 𝑔 /𝑐 and 13𝑟 𝑔 /𝑐 are indicated by the black dashed, blue dotted, and red dashed lines. 101 Figure 4.9: Imaginary part of the Maxwell Newman-Penrose scalar 𝜙2(𝑙=1,𝑚=1) , corresponding approximately to outgoing fast magnetosonic waves, on the vertical (𝑥𝑧) plane normalized with the magnitude of magnetic field for the aligned case 𝜃 𝐵 = 0◦ at 𝑡 − 𝑡merger = 1.05 ms. where 𝐶𝑎𝑏𝑐𝑑 is the Weyl tensor, 𝐹𝑎𝑏 is the electromagnetic field tensor, and (𝑙 𝑎 , 𝑛𝑎 , 𝑚 𝑎 , 𝑚¯ 𝑎 ) are orthonormal null tetrads √ 𝑙 𝑎 = (𝑡 𝑎 + 𝑟 𝑎 )/ 2, √ 𝑛𝑎 = (𝑡 𝑎 − 𝑟 𝑎 )/ 2, √ 𝑚 𝑎 = (𝜃 𝑎 + 𝑖𝜙𝑎 )/ 2. (4.7) (4.8) (4.9) We use the dominant (𝑙, 𝑚) = (2, 2) quadrupole mode of 𝜓4 to monitor the BH ringdown, and the (𝑙, 𝑚) = (1, 1) dipole mode of 𝜙2 to monitor the electromagnetic modulation in the magnetosphere.7 We show the imaginary part of 𝜙2(𝑙=1,𝑚=1) and 𝜓4(𝑙=2,𝑚=2) (hereafter denoted simply as 𝜙2 and 𝜓4 for brevity) in the middle and lower panels of Fig. 4.8. In the following discussions, we denote the exponential decay time scale of the NP scalar 𝜙2 (𝜓4 ) as 𝜏𝜙NP (𝜏𝜓NP ). We compute the quasi-normal mode (QNM) frequencies of the remnant BH for (𝑠, 𝑙, 𝑚) = (−2, 2, 2) and (𝑠, 𝑙, 𝑚) = (−1, 1, 1) fundamental modes using the qnm package (Stein, 2019). The imaginary parts of the two QNM frequencies are both around 12 𝑟 𝑔 /𝑐. From the real parts, we obtain the oscillation period 0.94 ms for 𝜙2 and 0.59 ms for 𝜓4 . The damped sinusoidal oscillation of 𝜓4 , shown in the bottom panel of Fig. 4.8, agrees well with both real and imaginary parts of the computed QNM frequency. 7 The NP scalar 𝜓 4 corresponds to the outgoing gravitational radiation at null infinity. The Maxwell NP scalar 𝜙2 is proportional to the complex electric field 𝐸 𝜃 + 𝑖𝐸 𝜙 . In an axisymmetric background magnetic field, 𝐸 𝜃 corresponds to the Alfvénic modes and 𝐸 𝜙 to the fast magnetosonic modes. 102 Inclined magnetic field (𝜃 𝐵 = 30◦ , 60◦ ) — Both simulations show 𝜏𝜙NP = 31𝑟 𝑔 /𝑐, which is the same as their magnetic flux decaying timescale 𝜏Φ . Periodic oscillations of 𝜙2 for 𝑡 − 𝑡merger ≳ 1.5 ms are coming from the rotation of current sheets (see Fig. 4.6), which has a half-period of 2𝜋/Ω𝐻 ≈ 1.8 ms. From the fact that the measured decay timescales 𝜏Φ and 𝜏𝜙NP not only agree with each other but also being disparate from the QNM frequency, we deduce that 𝜏Φ = 31𝑟 𝑔 /𝑐 is the the flux decay timescale of the BH pulsar due to magnetic reconnection in our setup, and the time decay of 𝜙2 is simply a consequence of the declining magnetic field strength. Aligned magnetic field (𝜃 𝐵 = 0◦ ) — In the early phase of the ringdown (𝑡 − 𝑡merger ≤ 5 ms), 𝜙2 exhibits a rapid decay with 𝜏𝜙NP ≈ 𝜏𝜓NP . The period of the oscillations in 𝜙2 , which lasts about 2.5 cycles, is measured to be 0.9ms and shows a good agreement with the QNM frequency (0.94ms). This indicates that the evolution of the post-merger magnetosphere is dominated by the ringdown of the BH, which rapidly sheds off magnetic fluxes from the horizon. We show 𝜙2 in the meridional (𝑥𝑧) plane in Fig. 4.9. The magnetic flux shedding driven by QNMs induces episodes of quasi-periodic modulations in the magnetosphere, which leads to a more rapid reconnection of the field lines on the equatorial plane. The same process has been also observed by Most, Beloborodov, and Ripperda (2024) for the gravitational collapse of a NS with its spin axis aligned with the magnetic moment. On the other hand, both 𝜏Φ and 𝜏𝜙NP are slowed down to ≈ 31𝑟 𝑔 /𝑐 later in 𝑡 − 𝑡merger ≥ 6 ms, implying that the balding process of the BH begins to be affected more by resistivity. The magnetic flux shedding by QNMs becomes subdominant as gravitational perturbations fade out, then the flux decay is governed by magnetic reconnection afterwards. We caution that this observed behavior may change at higher numerical resolutions, which will exhibit a better scale separation between plasma and gravitational effects. 4.4.4 Striped wind The rotation of an inclined split-monopole magnetosphere on a BH leads to a striped wind (Selvi et al., 2024) which appears to be similar to those from oblique pulsars (e.g. Michel, 1982; Petri, 2012; Tchekhovskoy and Spitkovsky, 2013; Cerutti and Philippov, 2017), albeit without the presence of a closed zone. We illustrate this in Fig. 4.10 for the 𝜃 𝐵 = 30◦ simulation, where the sign change in the toroidal magnetic field is clearly visible. Different from a stationary pulsar solution with 𝐵 𝜙 ∼ 1/𝑟 (Michel, 1982; Bogovalov, 1999), the magnetic field here decays over time. While this will not affect the geometry of the striped wind, its amplitude will naturally 103 Figure 4.10: Pulsar-like striped wind from the remnant black hole at 𝑡 − 𝑡merger = 7.0 ms from 𝜃 𝐵 = 30◦ simulation. We show the toroidal magnetic field 𝐵 𝜙 with the magnetic field lines on the equatorial (𝑥𝑦) plane in both panels. The remnant black hole is shown with a black circle and spinning counter clockwise in this figure. In the right panel, we show the stagnation surface (𝑢𝑟 = 0) with a black dashed line. become a function of retarded time, 𝐵 𝜙 = 𝐵 𝜙 (𝑟, 𝜃, 𝑡 − 𝑟/𝑐), as can be seen from the left panel of Fig. 4.10. A stagnation surface at which 𝑢𝑟 = 0, separating the inflow and outflow region of the plasma, appears in the vicinity of the BH (Bransgrove, Ripperda, and Philippov, 2021); we show it on the right panel of Fig. 4.10. Interestingly, the stagnation surface is discontinuous across the rotating current sheet, with its radius being greater at the upstream of the current sheet (trailing part of the striped wind), appearing as a half-split spheroid with an offset along the current sheet. We also find that the rotation angular velocity of the magnetic field lines is slower (faster) at the upstream (downstream) of the rotating current sheet, exhibiting a symmetric deviation from Ω𝐹 = Ω𝐻 /2. We reserve a more detailed analysis of these near-horizon dynamics of oblique BH pulsars for future work. An analytic model of the toroidal magnetic field 𝐵 𝜙 in the wind can be developed as follows. For a nearly force-free wind from a rotating split-monopole, 𝐵 𝜙 can be approximated as (Michel, 1982; Bogovalov, 1999; Tchekhovskoy, Philippov, and 104 10−5 |B φ |/B∗ t − tmerger = 9.3ms 10−6 10−7 10−5 |B φ |/B∗ t − tmerger = 11.5ms 10−6 10−7 0 500 1000 1500 2000 r [km] Figure 4.11: Toroidal magnetic field of the striped wind |𝐵 𝜙 (𝑟)| on the equatorial plane along the 𝑥ˆ axis. Alternating signs (polaritires) of 𝐵 𝜙 in each stripes are denoted with different colors. The dashed line shows the fit with Eq. (4.11). Spitkovsky, 2016) Ω𝑟 sin 𝜃 Ω 𝐵∗𝑟 ∗2 sin 𝜃 |𝐵𝑟 | = , (4.10) 𝑐 𝑐 𝑟 where Ω is the rotation angular velocity, 𝐵∗ is the surface magnetic field strength, and 𝑟 ∗ is the radius of the rotator. For a BH pulsar, we can replace the angular velocity Ω with Ω𝐹 = Ω𝐻 /2, the radius 𝑟 ∗ with 𝑟 𝐻 , and the surface magnetic field 𝐵∗ with 𝐵 𝐻 (𝑡) = 𝐵 𝐻,0 𝑒 −𝑡/𝜏Φ . The resulting extension of Eq. (4.10) for a BH pulsar is 2 𝑒 −(𝑡−𝑟/𝑐)/𝜏Φ sin 𝜃 Ω𝐻 𝐵 𝐻,0𝑟 𝐻 |𝐵 𝜙 (𝑟, 𝜃, 𝑡)| = , (4.11) 2𝑐 𝑟 where (𝑡 − 𝑟/𝑐) accounts for a retarded time. |𝐵 𝜙 (𝑟, 𝜃)| ≈ Fig. 4.11 compares the 𝜃 𝐵 = 30◦ simulation data with Eq. (4.11) on the equatorial plane, using 𝜏Φ = 31𝑟 𝑔 /𝑐 measured from the balding process (Sec. 4.4.3) and 105 shifting 𝑡 → 𝑡 − 𝑡merger . Our approximate analytic model shows a good agreement with the simulation result. The value of 𝐵 𝐻,0 fitted from the simulation data is 1.5 × 10−2 𝐵∗ , revealing that the split-monopole BH pulsar inherits about 1% of the magnetic field strength from the companion NS. A separate estimate from the 2𝐵 BH magnetic flux Φ𝐵 = 2𝜋𝑟 𝐻 𝐻,0 (top panel of Fig. 4.8) yields almost the same value of 𝐵 𝐻,0 , reassuring the validity of the analytic model Eq. (4.11) as well as the measured value of 𝐵 𝐻,0 . 4.4.5 Energetics The wind from the BH pulsar is powered by the energy extracted from the remnant BH through the Blandford-Znajek (BZ) process (Blandford and Znajek, 1977), leading to a spin-down of the BH. The spin-down power of an aligned split-monopole magnetosphere, to a leading order of the BH spin,8 is given as (Tchekhovskoy, Narayan, and McKinney, 2010) (Φ2𝐵 /4𝜋)Ω2𝐻 𝑃BZ = . 6𝜋𝑐 (4.12) The BZ power (4.12) can be written into a form more commonly used in the pulsar literature 2 4 , (4.13) 𝐿 = Ω2𝐹 𝐵2𝐻 𝑟 𝐻 3𝑐 2𝐵 . with Ω𝐹 = Ω𝐻 /2 and Φ𝐵 = 2𝜋𝑟 𝐻 𝐻 This spin-down power is carried by the electromagnetic Poynting flux, which is not a direct observable. It is the dissipation in the current sheets which converts the electromagnetic field energy of the wind into kinetic energy of particles and subsequent electromagnetic emissions (Philippov, Uzdensky, et al., 2019). In a steady pulsar magnetosphere, about 10–20 percent of the spin-down power can be dissipated within 10 light cylinder radii (e.g., Parfrey, Beloborodov, and Hui, 2012; A. Y. Chen and Beloborodov, 2014; Philippov, Spitkovsky, and Cerutti, 2015; see also Cerutti and Beloborodov, 2017 for a review). Here we develop a toy model for the dissipation luminosity of a BH pulsar, closely following the approach by Cerutti, Philippov, and Dubus (2020). From here we will use 𝑡 to denote the time after the formation of the split monopole i.e. (𝑡 −𝑡merger ) → 𝑡. 8 The relative correction from the next order term ∝ (Ω 𝐻) 4 is less than 10 −3 in our case. Tchekhovskoy, Narayan, and McKinney (2010) for the expansion formula up to (Ω 𝐻 ) 6 . See 106 The total dissipation luminosity is given by a volume integral ∫ 𝐿𝐷 = (J · E) 𝑟 2 sin 𝜃𝑑𝑟𝑑𝜃𝑑𝜙 ∫ 𝑐𝛽rec = (𝐵 𝜙 ) 2 𝑟 sin 𝜃𝑑𝑟𝑑𝜃, 𝜋 (4.14) where 𝛽rec is the dimensionless reconnection rate (Uzdensky and Spitkovsky, 2014). A primary difference of our toy model from that of Cerutti, Philippov, and Dubus (2020) is the exponential damping term in the Eq. (4.11) associated with the flux decay of the BH. Substituting the expression (4.11) into (4.14) and performing angular integration, ∫ 2𝛽rec 𝐿 0 −2𝑡/𝜏Φ 𝑟max 𝑒 2𝑟/𝑐𝜏Φ 𝑒 𝑑𝑟, (4.15) 𝐿𝐷 = 𝜋 𝑟 𝑟 min 4 is an instantaneous BZ power of the BH pulsar at where 𝐿 0 = (2/3𝑐)Ω2𝐹 𝐵2𝐻,0𝑟 𝐻 𝑡 = 0. The upper and lower bounds of the integral in Eq. (4.15) correspond to the radial extent of the striped wind, 𝑟 min = 𝑟 𝐻 and 𝑟 max ≈ 𝑐𝑡, which gives      2𝑡 2𝛽rec 𝐿 0 −2𝑡/𝜏Φ 2𝑟 𝐻 𝑒 Ei 𝐿 𝐷 (𝑡) = − Ei , (4.16) 𝜋 𝜏Φ 𝑐𝜏Φ ∫𝑥 where Ei(𝑥) = −∞ (𝑒 𝑡 /𝑡)𝑑𝑡 is the exponential intergral. We apply our toy model to the 𝜃 𝐵 = 30◦ simulation. The initial spin-down power 𝐿 0 can be computed from the mass and spin of the remnant BH, and using 𝐵 𝐻,0 /𝐵∗ = 1.5% fitted from the simulation result (see Sec. 4.4.4).9 The flux decay timescale 𝜏Φ = 31𝑟 𝑔 /𝑐 from our simulation is dominated by unphysical numerical resistivity, therefore we consider 𝜏Φ = 100𝑟 𝑔 /𝑐 and 𝜏Φ = 500𝑟 𝑔 /𝑐 motivated from the highresolution (kinetic) simulations of Bransgrove, Ripperda, and Philippov (2021) as a more realistic input for assessing light curves. The reconnection rate is fixed to 𝛽rec = 0.1 from kinetic plasma simulations (Sironi and Spitkovsky, 2014). In Fig. 4.12, we show the modelled dissipation luminosity 𝐿 𝐷 (𝑡) scaled with the initial NS magnetic field strength. The time curve of the dissipation luminosity exhibits a rapid rise to its peak value within a few milliseconds, followed by exponential then power-law decay over tens of milliseconds. A magnetar can power a burst with the luminosity ∼ 1047 erg s−1 , while a NS with 𝐵∗ ∼ 1012 G will emit a 9 Note that this ratio 𝐵 𝐻,0 /𝐵∗ , namely the portion of the magnetic flux that a nascent BH pulsar inherits from the swallowed NS, can only be probed with a full numerical relativity merger simulation as performed here. 107 LD,43 /(B∗,13 )2 τΦ = 100rg /c 1.0 τΦ = 500rg /c 0.8 0.6 0.4 0.2 0.0 0 10 20 30 40 50 t − tmerger [ms] Figure 4.12: The dissipation luminosity from a BH pulsar 𝐿 𝐷 (𝑡) computed with an analytic model developed in Sec. 4.4.5, normalized with 𝐿 𝐷,43 ≡ 𝐿 𝐷 /(1043 erg s−1 ) and 𝐵∗,13 ≡ 𝐵∗ /(1013 G). Due to a high (unphysical) numerical resistivity in our simulation, we construct the light curves using 𝜏Φ = 100𝑟 𝑔 /𝑐 (blue solid line) and 𝜏Φ = 500𝑟 𝑔 /𝑐 (orange solid line) consistent with bounds from high-resolution kinetic simulations of Bransgrove, Ripperda, and Philippov (2021). relatively faint one with ∼ 1041 erg s−1 . The exponential factor 𝑒 2𝑟/𝑐𝜏Φ in Eq. (4.15) suggests that the region 𝑟 ≈ 𝑟 max is predominantly contributing to the total integral, implying the forefront of the expanding striped wind with a thickness Δ𝑟 ≈ 𝑐𝜏Φ is mainly powering the total dissipation luminosity. ∫ The total dissipated energy 𝐸 = 𝐿 𝐷 (𝑡)𝑑𝑡 does not converge due to a 𝑡 −1 asymptotic decay of 𝐿 𝐷 (𝑡). Realistically, dissipation in the current sheets would introduce a faster decrease of 𝐵 𝜙 in radius, and the decay of 𝐵 𝐻 (𝑡) below a certain threshold can halt the pair production around the BH, turning off the BH pulsar. Naively setting the end time of the burst as when 𝐿 𝐷 (𝑡) drops down to 1/10 of its peak value, the burst lasts about 15 ms (60 ms) for 𝜏Φ = 100𝑟 𝑔 /𝑐 (500𝑟 𝑔 /𝑐), with the average luminosity 2.6 × 1043 erg s−1 (4.2 × 1043 erg s−1 ) for 𝐵∗ = 1013 G. 108 4.5 4.5.1 Electromagnetic transient Radio burst The power dissipated in monster shocks at small radii is immediately radiated in X-rays (Beloborodov, 2023). Later, when the shock expands to larger radii, it can become a bright source of radio emission and emit a powerful fast radio burst (FRB). Magnetized shocks emit a radio precursor by the synchrotron maser mechanism; it was initially proposed for termination shocks of pulsar winds (Hoshino et al., 1992; Lyubarsky, 2014) and then for internal shocks in magnetized 𝑒 ± outflows to explain repeating FRBs (Beloborodov, 2017). Consider first the monster shock at small radii 𝑟 ∼ 107 –108 cm. Kinetic plasma simulations of magnetized shocks (Sironi, Plotnikov, et al., 2021; Vanthieghem and ˜ 𝑒 𝑐, Levinson, 2025) show precursor emission with frequency 𝜔pre ∼ 3𝜔˜ 𝐵 = 3𝑒 𝐵/𝑚 where 𝐵˜ is the upstream magnetic field measured in the plasma rest frame, 𝑒 is the elementary charge, and 𝑚 𝑒 is the electron mass. 𝐵˜ is reduced from the background value 𝐵bg by the strong expansion of the plasma ahead of the monster shock (Beloborodov, 2023): 𝜔𝑟 𝐵bg , (4.17) 𝐵˜ ≈ 2𝑐𝜎bg 2 /4𝜋𝑛 𝑚 𝑐 2 is the background magnetization parameter, and 𝜔 is where 𝜎bg = 𝐵bg bg 𝑒 the frequency of the magnetospheric perturbation that led to shock formation (our simulation shows 𝜔𝑟/𝑐 ∼ 10). Density 𝑛bg can be parameterized by multiplicity M ≡ 𝑛bg /𝑛0 , where 𝑛0 = ∇ · 𝑬/4𝜋𝑒 ∼ Ω𝐵bg /2𝜋𝑒𝑐 is the minimum density required to support the magnetospheric rotation with drift speed ∼ Ω𝑟 (Goldreich and Julian, 1969). This gives 𝜔pre ∼ (𝑟𝜔/𝑐)MΩ ∼ 104 M rad/s. This simple estimate is, however, deficient because it neglects the deceleration of the upstream flow by strong radiative losses. Losses dramatically change the radio precursor from monster shocks at small radii by increasing its frequency and suppressing its power (Beloborodov, in preparation). Powerful radio emission is expected from the relativistic shock when it expands far into the 𝑒 ± outflow, reaching 𝑟 ∼ 1013 –1014 cm (Beloborodov, 2017; Beloborodov, 2020). Then, a fraction ∼ 10−4 of the blast wave power is expected to convert to radio waves, whose frequency decreases with time (proportionally to the local 𝐵) and passes through the GHz band, best for radio observations. In this study, we do not follow the outflow dynamics with shocks at large radii; however, this may become possible for future MHD simulations. Our simulation shows that shocks launched 109 from NS-BH mergers are asymmetric, but not strongly collimated. Therefore, they can produce FRBs observable for a broad range of line of sights. Note that no baryonic ejecta are expected from BH swallowing a NS, so nothing should block the FRB from observers. 4.5.2 Gamma-ray burst The X-ray transient expected from the simulated merger is powered by dissipation of magnetospheric energy. Two dissipation mechanisms are observed in the simulation: shocks and magnetic reconnection in the split-monopole current sheet around the BH after the merger. Dissipation occurs at small radii, which correspond to a large compactness parameter ℓ = 𝜎T 𝐿/𝑟𝑚 𝑒 𝑐3 , where 𝐿 is the dissipation power and 𝜎T is the Thompson cross section. Note that 𝐿 and ℓ scale as 𝐵2 . For a strongly magnetized NS, e.g. with 𝐵 ∼ 1014 G, the huge ℓ implies that the dissipated energy becomes immediately thermalized. Thus, the merger ejects a hot “fireball” – a thermalized, magnetically dominated 𝑒 ± outflow. As the outflow expands to larger radii, it adiabatically cools, 𝑒 ± annihilate and release a burst of quasi-thermal radiation similar to the GRB from the magnetar collapse described in Most, Beloborodov, and Ripperda (2024). An additional dissipation mechanism is expected to operate in the outflow at large radii, and can add a nonthermal tail to the GRB spectrum. It is caused by the striped structure of the outflow, similar to the striped winds from pulsars. The stripes develop current sheets where magnetic reconnection gradually dissipates the alternating magnetic flux (Lyubarsky and Kirk, 2001; Cerutti, Philippov, and Dubus, 2020). A similar mechanism was previously proposed to operate in canonical GRBs (Drenkhahn and Spruit, 2002). It will release energy after the outflow becomes optically thin (which happens quickly in the baryon-free outflow from BH–NS merger). Therefore, it can generate energetic particles, emitting a nonthermal component of the GRB. 4.6 Conclusions We have presented a detailed numerical investigation into the magnetospheric dynamics of BH–NS mergers without tidal disruption. Using GRMHD simulations capable of probing the near force-free limit, we identify two mechanisms for generating an electromagnetic transient. First, we observe that fast magnetosonic waves are launched into the magnetosphere of the NS before it plunges into the BH. These waves, as expanding outward with 110 almost the speed of light, develop into monster shocks due to a more rapidly decaying ambient magnetic field (Beloborodov, 2023). The full MHD simulation is essential for tracking this effect, so it could not be captured by earlier vacuum or force-free simulations. The launched shocks are expected to emit a bright radio transient when they expand to large radii. When the BH swallows the NS together with its magnetic dipole moment, its external magnetosphere quickly rearranges itself into a split-monopole configuration with a large-scale current sheet. Then, the BH gradually loses the acquired “magnetic hair.” This balding is assisted by magnetic reconnection and gravitational effects (QNMs). The relative importance of these two processes varies over time and depends on the misalignment between the magnetic dipole moment and the BH spin. The split monopole is dragged into rotation by the BH and forms a transient BH pulsar which can power a post-merger EM signal in the X-ray and 𝛾-ray band. The monster shocks and the balding BH pulsar were previously studied in symmetric setups with a single compact object (Bransgrove, Ripperda, and Philippov, 2021; Beloborodov, 2023; Most, Beloborodov, and Ripperda, 2024; Selvi et al., 2024). Our ab-initio simulations demonstrate how both phenomena naturally occur in the complex dynamical spacetime of the BH–NS merger. 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(2005). “Accurate black hole evolutions by fourth-order numerical relativity”. In: Physical Review D 72, p. 024021. doi: 10.1103/PhysRevD. 72.024021. arXiv: gr-qc/0505055. 141 Appendix A SPHERICAL KERR-SCHILD COORDINATES The line element of the Kerr spacetime (see e.g., Teukolsky, 2015b for a review) in the Kerr-Schild coordinates is given as 𝑔𝑎𝑏 = 𝜂𝑎𝑏 + 2𝐻𝑙 𝑎 𝑙 𝑏 , (A.1) with  𝑟𝑥 + 𝑎𝑦 𝑟 𝑦 − 𝑎𝑥 𝑧  𝑀𝑟 3 , 𝑙 = 1, 2 , , . 𝑎 𝑟 4 + 𝑎2 𝑧2 𝑟 + 𝑎2 𝑟 2 + 𝑎2 𝑟 Here, 𝑀 is the mass and 𝑎 ≡ 𝐽/𝑀 is the rotational parameter of the black hole, where 𝐽 is its angular momentum.1 The coordinate variable 𝑟 is defined via 𝐻= 𝑥 2 + 𝑦2 𝑧2 + = 1. 𝑟 2 + 𝑎2 𝑟 2 (A.2) For 𝑎 = 0, we see that 𝑟 2 = 𝑥 2 + 𝑦 2 + 𝑧2 is the usual radial coordinate. Explicitly writing out the metric, we have 𝑑𝑠2 = −𝑑𝑡 2 + 𝑑𝑥 2 + 𝑑𝑦 2 + 𝑑𝑧 2  2 2𝑀𝑟 3 𝑟 (𝑥𝑑𝑥 + 𝑦𝑑𝑦) + 𝑎(𝑦𝑑𝑥 − 𝑥𝑑𝑦) 𝑧𝑑𝑧 + 4 , 𝑑𝑡 + + 𝑟 𝑟 + 𝑎2 𝑧2 𝑟 2 + 𝑎2 (A.3) which is often called ‘Cartesian’ Kerr-Schild coordinates. A nice feature of the Kerr-Schild form (A.3) is that the determinant of the spacetime metric is unity, i.e. √ −𝑔 = 1, often greatly simplifying algebra for some calculations. The singularity is at 𝑟 = 0, which is a ring 𝑥 2 + 𝑦 2 = 𝑎 2 , 𝑧 = 0. Inner and outer horizon radii are √︁ 𝑟± = 𝑀 ± 𝑀 2 − 𝑎2 (A.4) In the spherical Kerr-Schild coordinates, the Kerr metric has a form 𝑑𝑠2 = − (1 − 𝐵) 𝑑𝑡 2 + (1 + 𝐵) 𝑑𝑟 2 + Σ 𝑑𝜃 2 + (𝑟 2 + 𝑎 2 + 𝐵𝑎 2 sin2 𝜃) sin2 𝜃 𝑑𝜙2 + 2𝐵 𝑑𝑡𝑑𝑟 − 2𝑎𝐵 sin2 𝜃 𝑑𝑡𝑑𝜙 − 2𝑎(1 + 𝐵) sin2 𝜃𝑑𝑟𝑑𝜙 1 The dimensionless spin 𝑎 = 𝐽/𝑀 2 is also frequently used in literature. (A.5) 142 where Σ = 𝑟 2 + 𝑎 2 cos2 𝜃 and 𝐵 = 2𝑀𝑟/Σ. The locations of inner and outer horizon are given same as (A.4). Horizons appear as exact spheres in the spherical KerrSchild coordinates, which is often helpful for implementing a computational domain for numerical simulations. The coordinate transformation between the Kerr-Schild (A.3) and the spherical Kerr-Schild coordinates (A.5) is 𝑥 = (𝑟 cos 𝜙 − 𝑎 sin 𝜙) sin 𝜃, (A.6a) 𝑦 = (𝑟 sin 𝜙 + 𝑎 cos 𝜙) sin 𝜃, (A.6b) 𝑧 = 𝑟 cos 𝜃. (A.6c) 143 Appendix B THE WALD MAGNETOSPHERE SOLUTION Wald (1974) derived an exact axisymmetric solution of Maxwell equations in the Kerr spacetime, describing a magnetosphere with an asymptotically uniform magnetic field 𝐵0 aligned with the spin axis of the black hole. The original solution given in terms of the vector potential is 𝐴𝑏 = 𝐵0 [(𝜕𝜙 ) 𝑏 + 2𝑎(𝜕𝑡 ) 𝑏 ], 2 (B.1) where 𝜕𝜙 and 𝜕𝑡 are Killing vector fields of the Kerr spacetime in 𝜙 and 𝑡 directions, and 𝑎 = 𝐽/𝑀 is the rotational parameter of the black hole. In the spherical Kerr-Schild coordinates (see Appendix A), each components of the Wald solution (B.1) are 𝐵0 (𝑔𝑡𝜙 + 2𝑎𝑔𝑡𝑡 ), 2 𝐵0 (𝑔𝑟 𝜙 + 2𝑎𝑔𝑡𝑟 ), 𝐴𝑟 = 2 𝐴𝑡 = 𝐴𝜃 = 0, 𝐴𝜙 = 𝐵0 (𝑔𝜙𝜙 + 2𝑎𝑔𝑡𝜙 ). 2 Computing magnetic fields from the vector potential (B.2), we get    𝑟 4 − 𝑎4 𝑎 2 2𝑀 𝑟 2 ˜ 𝐵 = 𝐵0𝑟 sin 𝜃 cos 𝜃 1 + 2 + −1 𝑟 (𝑟 2 + 𝑎 2 cos2 𝜃) 2 𝑟 𝑎 2 𝑀 𝐵0 sin2 𝜃 𝐵˜ 𝜃 = −𝐵0𝑟 sin2 𝜃 − 2 (𝑟 2 − 𝑎 2 cos2 𝜃)(2 − sin2 𝜃) (𝑟 + 𝑎 2 cos2 𝜃) 2 (B.2a) (B.2b) (B.2c) (B.2d) (B.3a) (B.3b)  2𝑀𝑟 (𝑟 2 − 𝑎 2 ) (B.3c) 𝐵 = 𝑎𝐵0 sin 𝜃 cos 𝜃 1 + 2 . (𝑟 + 𝑎 2 cos2 𝜃) 2 √ √ √ Here 𝐵˜ 𝑖 = 𝛾𝐵𝑖 where 𝛾 = 1 + 𝐵 Σ sin 𝜃 is the square root of the determinant of the spatial metric in the spherical Kerr-Schild coordinates. ˜𝜙  144 We also write out 𝐵˜ 𝑖 in the Cartesian representation 𝑥¯ = 𝑟 sin 𝜃 cos 𝜙, (B.4a) 𝑦¯ = 𝑟 sin 𝜃 sin 𝜙, (B.4b) 𝑧¯ = 𝑟 cos 𝜃, (B.4c) which are often used for representing tensor quantities in the code (e.g. in SpECTRE). Resulting expressions are:    1 2𝑀𝑟 (𝑟 2 − 𝑎 2 ) 𝑥¯ ˜ 𝐵 = 𝑎𝐵0 𝑧¯ (𝑎 𝑥¯ − 𝑟 𝑦¯ ) 4 + 𝑟 (𝑟 4 + 𝑎 2 𝑧2 ) 2 (B.5a)   𝑟 2 − 𝑧2 4(𝑟 2 + 𝑧2 ) − + 𝑎𝑀𝑟 𝑥¯ 4 4 𝑟 (𝑟 + 𝑎 2 𝑧2 ) (𝑟 4 + 𝑎 2 𝑧2 ) 2    1 2𝑀𝑟 (𝑟 2 − 𝑎 2 ) 𝑦¯ ˜ 𝐵 = 𝑎𝐵0 𝑧¯ (𝑟 𝑥¯ + 𝑎 𝑦¯ ) 4 + 𝑟 (𝑟 4 + 𝑎 2 𝑧2 ) 2   𝑟 2 − 𝑧2 4(𝑟 2 + 𝑧2 ) + 𝑎𝑀𝑟 𝑦¯ 4 4 − 𝑟 (𝑟 + 𝑎 2 𝑧2 ) (𝑟 4 + 𝑎 2 𝑧2 ) 2 (B.5b)   𝑎 2 𝑧2 𝑀𝑎 2 𝑧2 (𝑎 2 + 𝑧2 )(5𝑟 4 + 𝑎 2 𝑧 2 ) 𝐵 = 𝐵0 1 + 4 + 3 1 − . 𝑟 𝑟 (𝑟 4 + 𝑎 2 𝑧2 ) 2 (B.5c) ˜𝑧  from which one can check that 𝐵˜ 𝑖 → (0, 0, 𝐵0 ) for 𝑎 → 0. We use the expressions (B.5) for initializing the densitized magnetic fields 𝐵˜ 𝑖 in Chapter 2. Note that the barred coordinates 𝑥, ¯ 𝑦¯ are not equal to the 𝑥, 𝑦 coordinates appearing in the original Kerr-Schild form (A.3), whereas 𝑧¯ = 𝑧. Barred coordinates 𝑥, ¯ 𝑦¯ , 𝑧¯ are simply Cartesian projections of the spherical Kerr-Schild coordinates (A.5), where they are related with the Kerr-Schild coordinates by 𝑥 𝑥¯ =√ , 𝑟 𝑟 2 + 𝑎2 𝑦¯ 𝑦 =√ , 𝑟 𝑟 2 + 𝑎2 (B.6a) (B.6b) 𝑧¯ = 𝑧, (B.6c) 𝑟 2 = 𝑥¯ 2 + 𝑦¯ 2 + 𝑧¯2 . (B.6d)