A Comparison between Second-Order Post-Newtonian Hamiltonian and Coherent Post-Newtonian Lagrangian in Spinning Compact Binaries
Abstract
:1. Introduction
2. PN Lagrangian Formulation and PN Hamiltonian Formulation
2.1. PN Lagrangian Formulation
2.1.1. Approximate Equations of Motion for the PN Lagrangian
2.1.2. Coherent Equations of Motion for the PN Lagrangian
2.2. PN Hamiltonian Formulation
3. Numerical Comparisons
3.1. Chaos Indicators
3.2. The Effects of Varying the Mass Ratio on Chaos
4. Summary
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
- Abbott, B.P. Observation of Gravitational Waves from a Binary Black Hole Merger. Phys. Rev. Lett. 2016, 116, 061102. [Google Scholar] [CrossRef]
- Buonanno, A.; Chen, Y. Quantum noise in second generation, signal-recycled laser interferometric gravitational-wave detectors. Phys. Rev. D 2001, 64, 042006. [Google Scholar] [CrossRef] [Green Version]
- Kidder, L.E. Coalescing binary systems of compact objects to (post)5/2-Newtonian order. V. Spin effects. Phys. Rev. D 1995, 52, 821–847. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Thorne, K.S.; Hartle, J.B. Laws of motion and precession for black holes and other bodies. Phys. Rev. D 1985, 31, 1815–1837. [Google Scholar] [CrossRef] [Green Version]
- Cervantes-Cota, C.L.; Galindo-Uribarri, S.; Smoot, G.F. A brief history of gravitational waves. Universe 2016, 2, 22. [Google Scholar] [CrossRef] [Green Version]
- Asada, H.; Futamase, T. Post-Newtonian ApproximationIts Foundation and Applications. Prog. Theor. Phys. Suppl. 1997, 128, 123–181. [Google Scholar] [CrossRef] [Green Version]
- Blanchet, L. Post-Newtonian theory and its application. arXiv 2003, arXiv:gr-qc/0304014. [Google Scholar]
- Will, C.M. On the unreasonable effectiveness of the post-Newtonian approximation in gravitational physics. Proc. Natl. Acad. Sci. USA 2011, 108, 5938–5945. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Levi, M. Binary dynamics from spin1-spin2 coupling at fourth post-Newtonian order. Phys. Rev. D 2012, 85, 064043. [Google Scholar] [CrossRef] [Green Version]
- Damour, T.; Jaranowski, P.; Schäfer, G. Nonlocal-in-time action for the fourth post-Newtonian conservative dynamics of two-body systems. Phys. Rev. D 2014, 89, 064058. [Google Scholar] [CrossRef] [Green Version]
- Levi, M.; Steinhoff, J. Equivalence of ADM Hamiltonian and Effective Field Theory approaches at next-to-next-to-leading order spin1-spin2 coupling of binary inspirals. J. Cosmol. Astropart. Phys. 2014, 2014, 003. [Google Scholar] [CrossRef] [Green Version]
- Levi, M.; Steinhoff, J. Spinning gravitating objects in the effective field theory in the post-Newtonian scheme. J. High Energy Phys. 2015, 2015, 219. [Google Scholar] [CrossRef] [Green Version]
- Levi, M.; Steinhoff, J. Leading order finite size effects with spins for inspiralling compact binaries. J. High Energy Phys. 2015, 2015, 59. [Google Scholar] [CrossRef] [Green Version]
- Levi, M.; Steinhoff, J. Next-to-next-to-leading order gravitational spin-squared potential via the effective field theory for spinning objects in the post-Newtonian scheme. J. Cosmol. Astropart. Phys. 2016, 2016, 008. [Google Scholar] [CrossRef] [Green Version]
- Varvoglis, H.; Papadopoulos, D. Chaotic interaction of charged particles with a gravitational wave. Astron. Astrophys. 1992, 261, 664–670. [Google Scholar]
- Letelier, P.; Vieira, W. Chaos in black holes surrounded by gravitational waves. Class. Quantum Gravity 1997, 14, 1249. [Google Scholar] [CrossRef]
- Veselỳ, K.; Podolskỳ, J. Chaos in a modified Hénon–Heiles system describing geodesics in gravitational waves. Phys. Lett. A 2000, 271, 368–376. [Google Scholar] [CrossRef] [Green Version]
- Kiuchi, K.; Koyama, H.; Maeda, K.I. Gravitational wave signals from a chaotic system: A point mass with a disk. Phys. Rev. D 2007, 76, 024018. [Google Scholar] [CrossRef] [Green Version]
- Wang, Y.; Wu, X. Gravitational Waves from a Pseudo-Newtonian Kerr Field with Halos. Commun. Theor. Phys. 2011, 56, 1045–1051. [Google Scholar] [CrossRef]
- Wang, Y.; Wu, X.; Zhong, S.Y. Gravitational wave signatures of rotating dense binaries. Acta Phys. Sin. 2012, 61, 16. [Google Scholar]
- Cornish, N.J. Chaos and gravitational waves. Phys. Rev. D 2001, 64, 084011. [Google Scholar] [CrossRef] [Green Version]
- Kiuchi, K.; Maeda, K.I. Gravitational waves from a chaotic dynamical system. Phys. Rev. D 2004, 70, 064036. [Google Scholar] [CrossRef] [Green Version]
- Levin, J. Gravity waves, chaos, and spinning compact binaries. Phys. Rev. Lett. 2000, 84, 3515. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Schnittman, J.D.; Rasio, F.A. Ruling out chaos in compact binary systems. Phys. Rev. Lett. 2001, 87, 121101. [Google Scholar] [CrossRef] [PubMed] [Green Version]
- Zhong, S.Y.; Wu, X. Manifold corrections on spinning compact binaries. Phys. Rev. D 2010, 81, 104037. [Google Scholar] [CrossRef]
- Zhong, S.Y.; Wu, X.; Liu, S.Q.; Deng, X.F. Global symplectic structure-preserving integrators for spinning compact binaries. Phys. Rev. D 2010, 82, 124040. [Google Scholar] [CrossRef]
- Wang, Y.; Wu, X. Next-order spin–orbit contributions to chaos in compact binaries. Class. Quantum Gravity 2011, 28, 025010. [Google Scholar]
- Mei, L.; Ju, M.; Wu, X.; Liu, S. Dynamics of spin effects of compact binaries. Mon. Not. R. Astron. Soc. 2013, 435, 2246–2255. [Google Scholar] [CrossRef] [Green Version]
- Huang, G.; Ni, X.; Wu, X. Chaos in two black holes with next-to-leading order spin–spin interactions. Eur. Phys. J. C 2014, 74, 1–8. [Google Scholar] [CrossRef] [Green Version]
- Wu, X.; Huang, G. Ruling out chaos in comparable mass compact binary systems with one body spinning. Mon. Not. R. Astron. Soc. 2015, 452, 3167–3178. [Google Scholar] [CrossRef] [Green Version]
- Ibrahim, D.; Abergel, F. Non-linear filtering and optimal investment under partial information for stochastic volatility models. Math. Methods Oper. Res. 2018, 87, 311–346. [Google Scholar] [CrossRef] [Green Version]
- De Vecchi, F. Lie Symmetry Analysis and Geometrical Methods for Finite and Infinite Dimensional Stochastic Differential Equations. Ph.D. Thesis, Università Degli Studi di Milano, Milano, Italy, 2018. [Google Scholar]
- Germ, F. Estimation for Linear and Semi-Linear Infinite-Dimensional Systems. Master’s Thesis, University of Waterloo, Waterloo, ON, Canada, 2019. [Google Scholar]
- Mirebeau, J.M.; Portegies, J. Hamiltonian fast marching: A numerical solver for anisotropic and non-holonomic eikonal PDEs. Image Process. Line 2019, 9, 47–93. [Google Scholar] [CrossRef]
- Holler, M.; Weinmann, A. Non-smooth variational regularization for processing manifold-valued data. In Handbook of Variational Methods for Nonlinear Geometric Data; Springer: Berlin/Heidelberg, Germany, 2020; pp. 51–93. [Google Scholar]
- Chen, C.; Cohen, D.; D’Ambrosio, R.; Lang, A. Drift-preserving numerical integrators for stochastic Hamiltonian systems. Adv. Comput. Math. 2020, 46, 1–22. [Google Scholar] [CrossRef] [Green Version]
- Sun, W.; Qiu, M.; Lv, X. H filter design for a class of delayed Hamiltonian systems with fading channel and sensor saturation. AIMS Math. 2020, 5, 2909–2922. [Google Scholar] [CrossRef]
- Treanţă, S. Constrained variational problems governed by second-order Lagrangians. Appl. Anal. 2020, 99, 1467–1484. [Google Scholar] [CrossRef]
- Treanţă, S. On a modified optimal control problem with first-order PDE constraints and the associated saddle-point optimality criterion. Eur. J. Control 2020, 51, 1–9. [Google Scholar] [CrossRef]
- Udriste, C.; Pitea, A. Optimization problems via second order Lagrangians. Balk. J. Geom. Appl. 2011, 16, 174–185. [Google Scholar]
- Machado, L.; Abrunheiro, L.; Martins, N. Variational and Optimal Control Approaches for the Second-Order Herglotz Problem on Spheres. J. Optim. Theory Appl. 2019, 182, 965–983. [Google Scholar] [CrossRef] [Green Version]
- Damour, T.; Jaranowski, P.; Schäfer, G. Equivalence between the ADM-Hamiltonian and the harmonic-coordinates approaches to the third post-Newtonian dynamics of compact binaries. Phys. Rev. D 2001, 63, 044021. [Google Scholar] [CrossRef] [Green Version]
- Cornish, N.J.; Levin, J. Comment on “Ruling out chaos in compact binary systems”. Phys. Rev. Lett. 2002, 89, 179001. [Google Scholar] [CrossRef] [Green Version]
- Cornish, N.J.; Levin, J. Chaos and damping in the post-Newtonian description of spinning compact binaries. Phys. Rev. D 2003, 68, 024004. [Google Scholar] [CrossRef]
- Wu, X.; Xie, Y. Revisit on “Ruling out chaos in compact binary systems”. Phys. Rev. D 2007, 76, 124004. [Google Scholar] [CrossRef] [Green Version]
- Levin, J. Fate of chaotic binaries. Phys. Rev. D 2003, 67, 044013. [Google Scholar] [CrossRef] [Green Version]
- Hartl, M.D.; Buonanno, A. Dynamics of precessing binary black holes using the post-Newtonian approximation. Phys. Rev. D 2005, 71, 024027. [Google Scholar] [CrossRef] [Green Version]
- Wu, X.; Xie, Y. Resurvey of order and chaos in spinning compact binaries. Phys. Rev. D 2008, 77, 103012. [Google Scholar] [CrossRef] [Green Version]
- de Andrade, V.C.; Blanchet, L.; Faye, G. Third post-Newtonian dynamics of compact binaries: Noetherian conserved quantities and equivalence between the harmonic-coordinate and ADM-Hamiltonian formalisms. Class. Quantum Gravity 2001, 18, 753. [Google Scholar] [CrossRef]
- Wu, X.; Mei, L.; Huang, G.; Liu, S. Analytical and numerical studies on differences between Lagrangian and Hamiltonian approaches at the same post-Newtonian order. Phys. Rev. D 2015, 91, 024042. [Google Scholar] [CrossRef]
- Königsdörffer, C.; Gopakumar, A. Post-Newtonian accurate parametric solution to the dynamics of spinning compact binaries in eccentric orbits: The leading order spin-orbit interaction. Phys. Rev. D 2005, 71, 024039. [Google Scholar] [CrossRef] [Green Version]
- Gopakumar, A.; Königsdörffer, C. Deterministic nature of conservative post-Newtonian accurate dynamics of compact binaries with leading order spin-orbit interaction. Phys. Rev. D 2005, 72, 121501. [Google Scholar] [CrossRef] [Green Version]
- Chen, R.C.; Wu, X. A note on the equivalence of post-Newtonian Lagrangian and Hamiltonian formulations. Commun. Theor. Phys. 2016, 65, 321. [Google Scholar] [CrossRef] [Green Version]
- Wu, X.; Xie, Y. Symplectic structure of post-Newtonian Hamiltonian for spinning compact binaries. Phys. Rev. D 2010, 81, 084045. [Google Scholar] [CrossRef] [Green Version]
- Wang, H.; Huang, G.Q. The Effect of Spin-Orbit Coupling and Spin-Spin Coupling of Compact Binaries on Chaos. Commun. Theor. Phys. 2015, 64, 159–165. [Google Scholar] [CrossRef]
- Huang, L.; Wu, X.; Ma, D. Second post-Newtonian Lagrangian dynamics of spinning compact binaries. Eur. Phys. J. C 2016, 76, 1–10. [Google Scholar] [CrossRef] [Green Version]
- Huang, L.; Wu, X.; Mei, L.J.; Huang, G.Q. Dynamics of High-Order Spin-Orbit Couplings about Linear Momenta in Compact Binary Systems. Commun. Theor. Phys. 2017, 68, 375. [Google Scholar] [CrossRef]
- Iorio, L. Revisiting the 2PN Pericenter Precession in View of Possible Future Measurements. Universe 2020, 6, 53. [Google Scholar] [CrossRef]
- Iorio, L. On the 2PN Pericentre Precession in the General Theory of Relativity and the Recently Discovered Fast-Orbiting S-Stars in Sgr A. Universe 2021, 7, 37. [Google Scholar] [CrossRef]
- Li, D.; Wang, Y.; Deng, C.; Wu, X. Coherent post-Newtonian Lagrangian equations of motion. Eur. Phys. J. Plus 2020, 135, 390. [Google Scholar] [CrossRef]
- Li, D.; Wu, X.; Liang, E. Effect of the Quadrupole–Monopole Interaction on Chaos in Compact Binaries. Ann. Der Phys. 2019, 531, 1900136. [Google Scholar] [CrossRef]
- Nagar, A. Effective one-body Hamiltonian of two spinning black holes with next-to-next-to-leading order spin-orbit coupling. Phys. Rev. D 2011, 84, 084028. [Google Scholar] [CrossRef] [Green Version]
- Tancredi, G.; Sánchez, A.; Roig, F. A comparison between methods to compute Lyapunov exponents. Astron. J. 2001, 121, 1171. [Google Scholar] [CrossRef]
- Froeschlé, C.; Lega, E. On the structure of symplectic mappings. The fast Lyapunov indicator: A very sensitive tool. In New Developments in the Dynamics of Planetary Systems; Springer: Berlin/Heidelberg, Germany, 2001; pp. 167–195. [Google Scholar]
- Wu, X.; Huang, T.Y.; Zhang, H. Lyapunov indices with two nearby trajectories in a curved spacetime. Phys. Rev. D 2006, 74, 083001. [Google Scholar] [CrossRef] [Green Version]
- Wu, X. A new interpretation of zero Lyapunov exponents in BKL time for Mixmaster cosmology. Res. Astron. Astrophys. 2010, 10, 211. [Google Scholar] [CrossRef]
- Huang, G.; Wu, X. Dynamics of a test particle around two massive bodies in decay circular orbits. Gen. Relativ. Gravit. 2014, 46, 1798. [Google Scholar] [CrossRef]
Post-Newtonian Form | Mass Ratio |
---|---|
ø | |
0.89, [1.22,1.40] | |
[0.20,0.88], 0.94, 0.96, 0.97, [1.00,1.06], 1.09, [1.13,1.40] | |
[0.20,0.99], [1.01,1.40] | |
[0.20,0.46], 0.47, 0.48, 0.50, 0.54, 0.55, 0.56, 0.60 | |
[0.62,1.40] |
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Cheng, X.-H.; Huang, G.-Q. A Comparison between Second-Order Post-Newtonian Hamiltonian and Coherent Post-Newtonian Lagrangian in Spinning Compact Binaries. Symmetry 2021, 13, 584. https://doi.org/10.3390/sym13040584
Cheng X-H, Huang G-Q. A Comparison between Second-Order Post-Newtonian Hamiltonian and Coherent Post-Newtonian Lagrangian in Spinning Compact Binaries. Symmetry. 2021; 13(4):584. https://doi.org/10.3390/sym13040584
Chicago/Turabian StyleCheng, Xu-Hui, and Guo-Qing Huang. 2021. "A Comparison between Second-Order Post-Newtonian Hamiltonian and Coherent Post-Newtonian Lagrangian in Spinning Compact Binaries" Symmetry 13, no. 4: 584. https://doi.org/10.3390/sym13040584
APA StyleCheng, X. -H., & Huang, G. -Q. (2021). A Comparison between Second-Order Post-Newtonian Hamiltonian and Coherent Post-Newtonian Lagrangian in Spinning Compact Binaries. Symmetry, 13(4), 584. https://doi.org/10.3390/sym13040584