Skip to main content
SpringerLink
Log in

Spin and Quadrupole Contributions to the Motion of Astrophysical Binaries

  • Chapter
  • First Online:
Equations of Motion in Relativistic Gravity

Part of the book series: Fundamental Theories of Physics ((FTPH,volume 179))

Abstract

Compact objects in general relativity approximately move along geodesics of spacetime. It is shown that the corrections to geodesic motion due to spin (dipole), quadrupole, and higher multipoles can be modeled by an extension of the point mass action. The quadrupole contributions are discussed in detail for astrophysical objects like neutron stars or black holes. Implications for binaries are analyzed for a small mass ratio situation. There quadrupole effects can encode information about the internal structure of the compact object, e.g., in principle they allow a distinction between black holes and neutron stars, and also different equations of state for the latter. Furthermore, a connection between the relativistic oscillation modes of the object and a dynamical quadrupole evolution is established.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. M. Mathisson, Neue Mechanik materieller Systeme. Acta Phys. Pol. 6, 163–200 (1937)

    MATH  Google Scholar 

  2. M. Mathisson, Republication of: new mechanics of material systems. Gen. Relativ. Gravit. 42, 1011–1048 (2010)

    ADS  MathSciNet  MATH  Google Scholar 

  3. W.G. Dixon, Extended bodies in general relativity: their description and motion, in Proceedings of the International School of Physics Enrico Fermi LXVII, ed. by J. Ehlers (North Holland, Amsterdam, 1979), pp. 156–219

    Google Scholar 

  4. A. Papapetrou, Spinning test-particles in general relativity. I. Proc. R. Soc. A 209, 248–258 (1951)

    ADS  MathSciNet  MATH  Google Scholar 

  5. K. Westpfahl, Relativistische Bewegungsprobleme. VI. Rotator-Spinteilchen und allgemeine Relativitätstheorie. Ann. Phys. (Berlin) 477, 361–371 (1969)

    ADS  Google Scholar 

  6. I. Bailey, W. Israel, Lagrangian dynamics of spinning particles and polarized media in general relativity. Commun. Math. Phys. 42, 65–82 (1975)

    ADS  MathSciNet  Google Scholar 

  7. T. Damour, Gravitational radiation and the motion of compact bodies, in Gravitational Radiation, ed. by N. Deruelle, T. Piran (North Holland, Amsterdam, 1983), pp. 59–144

    Google Scholar 

  8. W.D. Goldberger, I.Z. Rothstein, An effective field theory of gravity for extended objects. Phys. Rev. D 73, 104029 (2006)

    ADS  MathSciNet  Google Scholar 

  9. R.A. Porto, Post-Newtonian corrections to the motion of spinning bodies in nonrelativistic general relativity. Phys. Rev. D 73, 104031 (2006)

    ADS  Google Scholar 

  10. J. Steinhoff, G. Schäfer, Canonical formulation of self-gravitating spinning-object systems. Europhys. Lett. 87, 50004 (2009)

    ADS  Google Scholar 

  11. J. Steinhoff, Canonical formulation of spin in general relativity. Ann. Phys. (Berlin) 523, 296–353 (2011)

    ADS  MathSciNet  MATH  Google Scholar 

  12. B.S. DeWitt, Bryce DeWitt’s Lectures on Gravitation, vol. 826, 1st edn. Lecture Notes in Physics (Springer, Berlin, 2011)

    Google Scholar 

  13. L. Blanchet, A. Buonanno, A. Le Tiec, First law of mechanics for black hole binaries with spins. Phys. Rev. D 87, 024030 (2013)

    ADS  Google Scholar 

  14. W.G. Laarakkers, E. Poisson, Quadrupole moments of rotating neutron stars. Astrophys. J. 512, 282–287 (1999)

    ADS  Google Scholar 

  15. T. Damour, A. Nagar, Relativistic tidal properties of neutron stars. Phys. Rev. D 80, 084035 (2009)

    ADS  Google Scholar 

  16. T. Binnington, E. Poisson, Relativistic theory of tidal love numbers. Phys. Rev. D 80, 084018 (2009)

    ADS  Google Scholar 

  17. T. Hinderer, Tidal love numbers of neutron stars. Astrophys. J. 677, 1216–1220 (2008)

    ADS  Google Scholar 

  18. S. Chakrabarti, T. Delsate, J. Steinhoff. New perspectives on neutron star and black hole spectroscopy and dynamic tides (2013)

    Google Scholar 

  19. S. Chakrabarti, T. Delsate, J. Steinhoff, Effective action and linear response of compact objects in Newtonian gravity. Phys. Rev. D 88, 084038 (2013)

    ADS  Google Scholar 

  20. H. Goenner, K. Westpfahl, Relativistische Bewegungsprobleme. II. Der starre Rotator. Ann. Phys. (Berlin) 475, 230–240 (1967)

    ADS  Google Scholar 

  21. H. Römer, K. Westpfahl, Relativistische Bewegungsprobleme. IV. Rotator-Spinteilchen in schwachen Gravitationsfeldern. Ann. Phys. (Berlin) 477, 264–276 (1969)

    ADS  Google Scholar 

  22. A.J. Hanson, T. Regge, The relativistic spherical top. Ann. Phys. (N.Y.) 87, 498–566 (1974)

    ADS  MathSciNet  Google Scholar 

  23. M. Leclerc, Mathisson-Papapetrou equations in metric and gauge theories of gravity in a Lagrangian formulation. Class. Quantum Gravity 22, 3203–3222 (2005)

    ADS  MathSciNet  MATH  Google Scholar 

  24. J. Natario, Tangent Euler top in general relativity. Commun. Math. Phys. 281, 387–400 (2008)

    ADS  MathSciNet  MATH  Google Scholar 

  25. B.S. DeWitt. Dynamical theory of groups and fields, in Relativity, Groups, and Topology, Les Houches 1963 (Gordon and Breach, New York, 1964)

    Google Scholar 

  26. T. Damour, G. Schäfer, Redefinition of position variables and the reduction of higher order Lagrangians. J. Math. Phys. 32, 127–134 (1991)

    ADS  MathSciNet  MATH  Google Scholar 

  27. W.M. Tulczyjew, Motion of multipole particles in general relativity theory. Acta Phys. Pol. 18, 393–409 (1959)

    MathSciNet  MATH  Google Scholar 

  28. J. Steinhoff, D. Puetzfeld, Multipolar equations of motion for extended test bodies in general relativity. Phys. Rev. D 81, 044019 (2010)

    ADS  Google Scholar 

  29. A.I. Harte, Mechanics of extended masses in general relativity. Class. Quantum Gravity 29, 055012 (2012)

    ADS  MathSciNet  MATH  Google Scholar 

  30. E. Noether, Invariante Variationsprobleme. Nachr. Akad. Wiss. Gött. 235–257 (1918)

    Google Scholar 

  31. J. Ehlers, E. Rudolph, Dynamics of extended bodies in general relativity—center-of-mass description and quasirigidity. Gen. Relativ. Gravit. 8, 197–217 (1977)

    ADS  MathSciNet  MATH  Google Scholar 

  32. K. Yee, M. Bander, Equations of motion for spinning particles in external electromagnetic and gravitational fields. Phys. Rev. D 48, 2797–2799 (1993)

    ADS  MathSciNet  Google Scholar 

  33. R.A. Porto, I.Z. Rothstein, Spin(1)spin(2) effects in the motion of inspiralling compact binaries at third order in the post-Newtonian expansion. Phys. Rev. D 78, 044012 (2008)

    ADS  Google Scholar 

  34. R.A. Porto, I.Z. Rothstein, Next to leading order spin(1)spin(1) effects in the motion of inspiralling compact binaries. Phys. Rev. D 78, 044013 (2008)

    ADS  Google Scholar 

  35. L. Rosenfeld, Zur Quantelung der Wellenfelder. Ann. Phys. (Berlin) 397, 113–152 (1930)

    ADS  MATH  Google Scholar 

  36. P.A.M. Dirac, Generalized Hamiltonian dynamics. Can. J. Math. 2, 129–148 (1950)

    MathSciNet  MATH  Google Scholar 

  37. W. Beiglböck, The center-of-mass in Einsteins theory of gravitation. Commun. Math. Phys. 5, 106–130 (1967)

    ADS  MathSciNet  MATH  Google Scholar 

  38. R. Schattner, The center of mass in general relativity. Gen. Relativ. Gravit. 10, 377–393 (1979)

    ADS  MathSciNet  Google Scholar 

  39. R. Schattner, The uniqueness of the center of mass in general relativity. Gen. Relativ. Gravit. 10, 395–399 (1979)

    ADS  MathSciNet  Google Scholar 

  40. K. Kyrian, O. Semerák, Spinning test particles in a Kerr field – II. Mon. Not. R. Astron. Soc. 382, 1922–1932 (2007)

    ADS  Google Scholar 

  41. C.W. Misner, K.S. Thorne, J.A. Wheeler, Gravitation (W. H. Freeman and Company, New York, 1973)

    Google Scholar 

  42. N. Stergioulas, S. Morsink, RNS code

    Google Scholar 

  43. N. Stergioulas, J.L. Friedman, Comparing models of rapidly rotating relativistic stars constructed by two numerical methods. Astrophys. J. 444, 306–311 (1995)

    ADS  Google Scholar 

  44. H.P. Künzle, Canonical dynamics of spinning particles in gravitational and electromagnetic fields. J. Math. Phys. 13, 739–744 (1972)

    ADS  MathSciNet  MATH  Google Scholar 

  45. W.D. Goldberger, I.Z. Rothstein, Dissipative effects in the worldline approach to black hole dynamics. Phys. Rev. D 73, 104030 (2006)

    ADS  MathSciNet  Google Scholar 

  46. T. Damour, G. Esposito-Farèse, Gravitational-wave versus binary-pulsar tests of strong-field gravity. Phys. Rev. D 58, 042001 (1998)

    ADS  Google Scholar 

  47. K.S. Thorne, Multipole expansions of gravitational radiation. Rev. Mod. Phys. 52, 299–339 (1980)

    ADS  MathSciNet  Google Scholar 

  48. D. Bini, A. Geralico, Deviation of quadrupolar bodies from geodesic motion in a Kerr spacetime. Phys. Rev. D 89, 044013 (2014)

    ADS  Google Scholar 

  49. G. Pappas, T.A. Apostolatos, Revising the multipole moments of numerical spacetimes, and its consequences. Phys. Rev. Lett. 108, 231104 (2012)

    ADS  Google Scholar 

  50. S. Chakrabarti, T. Delsate, N. Gürlebeck, J. Steinhoff, The I-Q relation for rapidly rotating neutron stars (2013)

    Google Scholar 

  51. S.N. Rasband, Black holes and spinning test bodies. Phys. Rev. Lett. 30, 111–114 (1973)

    ADS  Google Scholar 

  52. J. Steinhoff, D. Puetzfeld, Influence of internal structure on the motion of test bodies in extreme mass ratio situations. Phys. Rev. D 86, 044033 (2012)

    ADS  Google Scholar 

  53. D. Bini, A. Geralico, Dynamics of quadrupolar bodies in a Schwarzschild spacetime. Phys. Rev. D 87, 024028 (2013)

    ADS  Google Scholar 

  54. A. Le Tiec, E. Barausse, A. Buonanno, Gravitational self-force correction to the binding energy of compact binary systems. Phys. Rev. Lett. 108, 131103 (2012)

    ADS  Google Scholar 

  55. W.H. Press, S.A. Teukolsky, On formation of close binaries by two-body tidal capture. Astrophys. J. 213, 183–192 (1977)

    ADS  Google Scholar 

  56. M.E. Alexander, Tidal resonances in binary star systems. Mon. Not. R. Astron. Soc. 227, 843–861 (1987)

    ADS  MATH  Google Scholar 

  57. Y. Rathore, A.E. Broderick, R. Blandford, A variational formalism for tidal excitation: Non-rotating, homentropic stars. Mon. Not. R. Astron. Soc. 339, 25–32 (2003)

    ADS  Google Scholar 

  58. É.E. Flanagan, É. Racine, Gravitomagnetic resonant excitation of Rossby modes in coalescing neutron star binaries. Phys. Rev. D 75, 044001 (2007)

    ADS  Google Scholar 

  59. W.D. Goldberger, A. Ross, Gravitational radiative corrections from effective field theory. Phys. Rev. D 81, 124015 (2010)

    ADS  Google Scholar 

  60. F.J. Zerilli, Effective potential for even parity Regge-Wheeler gravitational perturbation equations. Phys. Rev. Lett. 24, 737–738 (1970)

    ADS  Google Scholar 

  61. T. Regge, J.A. Wheeler, Stability of a Schwarzschild singularity. Phys. Rev. 108, 1063–1069 (1957)

    ADS  MathSciNet  MATH  Google Scholar 

  62. S. Chandrasekhar, On the equations governing the perturbations of the Schwarzschild black hole. Proc. R. Soc. A 343, 289–298 (1975)

    ADS  MathSciNet  Google Scholar 

  63. D. Bini, T. Damour, G. Faye, Effective action approach to higher-order relativistic tidal interactions in binary systems and their effective one body description. Phys. Rev. D 85, 124034 (2012)

    ADS  Google Scholar 

  64. B. Kol, M. Smolkin, Black hole stereotyping: induced gravito-static polarization. JHEP 1202, 010 (2012)

    ADS  MATH  Google Scholar 

  65. S. Mano, E. Takasugi, Analytic solutions of the Teukolsky equation and their properties. Prog. Theor. Phys. 97, 213–232 (1997)

    ADS  MathSciNet  Google Scholar 

  66. K. Yagi, N. Yunes, I-Love-Q: Unexpected universal relations for neutron stars and quark stars. Science 341, 365–368 (2013)

    ADS  Google Scholar 

  67. K. Yagi, N. Yunes, I-Love-Q relations in neutron stars and their applications to astrophysics, gravitational waves and fundamental physics. Phys. Rev. D 88, 023009 (2013)

    ADS  Google Scholar 

  68. A. Maselli, V. Cardoso, V. Ferrari, L. Gualtieri, P. Pani, Equation-of-state-independent relations in neutron stars. Phys. Rev. D 88, 023007 (2013)

    ADS  Google Scholar 

  69. M. Bauböck, E. Berti, D. Psaltis, F. Özel, Relations between neutron-star parameters in the Hartle-Thorne approximation. Astrophys. J. 777, 68 (2013)

    ADS  Google Scholar 

  70. B. Haskell, R. Ciolfi, F. Pannarale, L. Rezzolla, On the universality of I-Love-Q relations in magnetized neutron stars. Mon. Not. R. Astron. Soc. Lett. 438, L71–L75 (2014)

    ADS  Google Scholar 

  71. D.D. Doneva, S.S. Yazadjiev, N. Stergioulas, K.D. Kokkotas, Breakdown of I-Love-Q universality in rapidly rotating relativistic stars. Astrophys. J. Lett. 781, L6 (2014)

    ADS  Google Scholar 

  72. G. Pappas, T.A. Apostolatos, Universal behavior of rotating neutron stars in GR: even simpler than their Newtonian counterparts (2013)

    Google Scholar 

  73. K. Yagi, Multipole love relations (2013)

    Google Scholar 

Download references

Acknowledgments

I am indebted to all of my collaborators contributing directly or indirectly to the material presented here: Sayan Chakrabarti, Trence Delsate, Norman Gürlebeck, Johannes Hartung, Steven Hergt, Dirk Puetzfeld, Gerhard Schäfer, and Manuel Tessmer. This work was supported by DFG (Germany) through projects STE 2017/1-1 and STE 2017/2-1, and by FCT (Portugal) through projects SFRH/BI/52132/2013 and PCOFUND-GA-2009-246542 (co-funded by Marie Curie Actions).

Author information

Authors and Affiliations

  1. Centro Multidisciplinar de Astrofísica (CENTRA), Instituto Superior Técnico (IST), Avenida Rovisco Pais 1, 1049-001, Lisbon, Portugal

    Jan Steinhoff

Authors
  1. Jan Steinhoff
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jan Steinhoff .

Editor information

Editors and Affiliations

  1. ZARM University of Bremen, Bremen, Germany

    Dirk Puetzfeld

  2. ZARM University of Bremen, Bremen, Germany

    Claus Lämmerzahl

  3. Max-Planck-Institut für Gravitationsphysik, Golm, Brandenburg, Germany

    Bernard Schutz

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Steinhoff, J. (2015). Spin and Quadrupole Contributions to the Motion of Astrophysical Binaries. In: Puetzfeld, D., Lämmerzahl, C., Schutz, B. (eds) Equations of Motion in Relativistic Gravity. Fundamental Theories of Physics, vol 179. Springer, Cham. https://doi.org/10.1007/978-3-319-18335-0_19

Download citation

Publish with us

Policies and ethics

Search

Navigation