Project A4;     (2003 - 2014)

 Project List   Abstracts   All Publications 

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Analytical Approximation Methods

    Principal Investigators: Gerhard Schäfer

Project A4 aims at developing schemes and constructing approximate analytical solutions of the equations of motion resulting from the higher order post-Newtonian (PN) Hamiltonians of spinning binary systems derived in recent years. Another aim is the construction of approximate analytical or semi-analytical solutions of damped inspiraling orbits.

Current Numerical Relativity is fast progressing in the determination of the orbital motions and spin precessions of spinning compact binaries. Compared with those results much analytical work has still to be done to analytically balance, complement, and support Numerical Relativity. A well-known analytical scheme is the Effective One Body (EOB) method. Although the PI of Project A4 has done some work within that scheme, the aim of A4 is different from EOB which, in higher order approximations, needs much input from Numerical Relativity. In some sense, the advanced EOB approach can be regarded as efficient analytical representation of Numerical Relativity results obtained through matching. Project A4, however, seeks solutions of the equations of motion resulting from the higher PN spinning binary Hamiltonians derived in the PI's research group including quadrupole terms from rotational deformation (spin-squared terms). Approximate analytical solutions are known in special cases only, like near-equal masses on circular orbits, equal masses or special values and orientations of the spins on general orbits, all in leading order PN approximations in the spin interactions. Only quite recently, in the PI's research group, an analytical solution of the eccentric motion of compact binaries with aligned spins and orbital angular momentum was obtained under the next-to-leading order spin-orbit coupling, see Project B4.

The motion of spinless binaries was solved analytically up to the highest known conservative level reading 3PN, the first time by the PI and his collaborators. Many forms of motions resulting from explicitly given binary Hamiltonians with spin are not known analytically but need urgent construction. Though previous those calculations have been performed within former Project B4, the proposed aim is much more involved, including developments of analytical approximation tools, so the topic seems to better fit into Project A4 than B4.

Treating spinning binaries as spin-extended ordinary binary systems, it is proposed to construct a more complete set of approximate analytical solutions of the orbital and spin motions of spinning binaries developing on most recent results in the theory of orbits in celestial mechanics and satellite geodesy as reported in, e.g., [M. Schneider and C. Cui, Deutsche Geodätische Komission, Reihe A 121 (2005); E. Mai, M. Schneider, and C. Cui, Deutsche Geodätische Komission, Reihe A 122 (2008)], where the power of Lie-series methods in solving N-body equations of motion with quadratically convergent series is clearly shown. Although these references are not concerned with spin, in a most recent paper in B4, [M. Tessmer, Phys. Rev. D 80:124034, (2009)], for a much simplified system, a first successful attempt has been made in applying Lie-series methods for solving equations of motion of spinning binaries.

It is evident that explicit solutions of the conservative equations of motion are very convenient starting points for the analytical treatment of dissipative effects from gravitational radiation emission as well as for the construction of related gravitational wave forms. The construction of damped solutions will be the subject of Project A4 too.

Researchers

  Thibault Damour   Advisor, 2003
  Steven Hergt   Postdoc, 2011
  Piotr Jaranowski   Advisor, 2003
  Gerhard Schaefer   Professor, PI 2003
  Jan Sperrhake   Student, 2010
  Jan Steinhoff   Postdoc, 2010
  Manuel Tessmer   PhD Student, 2011
  Norbert Wex   Advisor, 2011

Former Associates
  Yvonne Blum   Student, 2004-2005
  David Brizuela   Postdoc, 2009-2010
  Michael Brügmann   PhD Student, 2004-2004
  Guillaume Faye   Advisor, 2006-2010
  Frank Herrmann   PhD Student, 2004-2004
  Raoul-Martin Memmesheimer   Student, 2003-2004
  Tilman Rothe   Student, 2008-2009
  Jan Steinhoff   PhD Student, 2009-2010
  Christian Thierfelder   Student, 2004-2005
  Han Wang   Postdoc, 2007-2009

Publications

[1] Nonlocal-in-time action for the fourth post-Newtonian conservative dynamics of two-body systems
Thibault Damour, Piotr Jaranowski, Gerhard Schäfer, Phys. Rev. D 89:064058 (2014)

[2] Aligned spins: Orbital elements, decaying orbits, and last stable circular orbit to high post-Newtonian orders
Manuel Tessmer, Johannes Hartung, Gerhard Schäfer, Class. Quantum Grav. 30, 015007 (2013)

[3] Canonical angles in a compact binary star system with spinning components: Approximative solution through next-to-leading-order spin-orbit interaction for circular orbits
Manuel Tessmer, Jan Steinhoff, Gerhard Schäfer, Phys. Rev. D87:064035 (2013)

[4] Next-to-next-to-leading order post-Newtonian linear-in-spin binary Hamiltonians
Johannes Hartung, Jan Steinhoff, Gerhard Schäfer, Ann. Phys. (Berlin) 525:359 (2013)

[5] Dimensional regularization of local singularities in the fourth post-Newtonian two-point-mass Hamiltonian
Piotr Jaranowski, Gerhard Schäfer, Phys. Rev. D87:081503(R) (2013)

[6] Observables of a test-mass along an inclined orbit in a post-Newtonian approximated Kerr spacetime to leading-order-quadratic-in-spin
Steven Hergt, Abhay Shah, Gerhard Schäfer, Phys. Rev. Lett. 111:021101 (2013)

[7] Towards the fourth post-Newtonian Hamiltonian for two-point-mass systems
Piotr Jaranowski, Gerhard Schäfer, Phys. Rev. D 86:061503(R) (2012)

[8] Influence of internal structure on the motion of test bodies in extreme mass ratio situations
Jan Steinhoff, Dirk Puetzfeld, Phys. Rev. D 86:044033 (2012)

[9] Elimination of the spin supplementary condition in the effective field theory approach to the post-Newtonian approximation
Steven Hergt, Jan Steinhoff, Gerhard Schäfer, Ann. Phys (N.Y.) 327:1494 (2012)

[10] Gravitational wave phasing for spinning compact binaries in inspiraling eccentric orbits
Achamveedu Gopakumar, Gerhard Schäfer, Phys. Rev. D 84:124007 (2011)

[11] Leading-order spin-orbit and spin(1)-spin(2) radiation-reaction Hamiltonians
Han Wang, Jan Steinhoff, Jing Zeng, Gerhard Schäfer, Phys. Rev. D 84:124005 (2011)

[12] Next-to-next-to-leading order post-Newtonian spin(1)-spin(2) Hamiltonian for self-gravitating binaries
Johannes Hartung, Jan Steinhoff, Ann. Phys. (Berlin) 523:919 (2011)

[13] Next-to-next-to-leading order post-Newtonian spin-orbit Hamiltonian for self-gravitating binaries
Johannes Hartung, Jan Steinhoff, Ann. Phys. (Berlin) 523:783 (2011)

[14] Canonical formulation of spin in general relativity
Jan Steinhoff, Ann. Phys. (Berlin) 523:296 (2011) [Dissertation]

[15] Next-to-leading order spin-orbit and spin(a)-spin(b) Hamiltonians for n gravitating spinning compact objects
Johannes Hartung, Jan Steinhoff, Phys. Rev. D 83:044008 (2011)

[16] Post-Minkowskian Hamiltonian for Gravitating N-Body Systems
Tomas Ledvinka, Gerhard Schäfer, Jiri Bicak, to be published in Proceedings of the 12th Marcel Grossmann Meeting on General Relativity

[17] Binary spinning black hole Hamiltonian in canonical center-of-mass and rest-frame coordinates through higher post-Newtonian order
Tilman J. Rothe, Gerhard Schäfer, J. Math. Phys. 51:082501 (2010)

[18] High-order perturbations of a spherical collapsing star
David Brizuela, José M. Martín-García, Ulrich Sperhake, Kostas D. Kokkotas, Phys. Rev. D 82:104039 (2010)

[19] Mode coupling of Schwarzschild perturbations: Ringdown frequencies
Enrique Pazos, David Brizuela, José M. Martín-García, Manuel Tiglio, Phys. Rev. D 82:104028 (2010)

[20] Interacting holographic tachyon model of dark energy
Alberto Rozas-Fernández, David Brizuela, Norman Cruz, Int. J. Mod. Phys. D 19:573 (2010)

[21] Fourth-post-Newtonian exact approximation to general relativity
David Brizuela, Gerhard Schäfer, Phys. Rev. D 81:084014 (2010)

[22] ADM canonical formulation with spin and application to post-Newtonian approximations
Jan Steinhoff, Steven Hergt, Gerhard Schäfer, to be published in Proceedings of the 12th Marcel Grossmann Meeting on General Relativity, edited by T. Damour, R.T. Jantzen and R. Ruffini, World Scientific, Singapore, 2010

[23] Canonical formulation of gravitating spinning objects at 3.5 post-Newtonian order
Jan Steinhoff, Han Wang, Phys. Rev. D 81:024022 (2010)

[24] Comment on two recent papers regarding next-to-leading order spin-spin effects in gravitational interaction
Jan Steinhoff, Gerhard Schäfer, Phys. Rev. D 80:088501 (2009)

[25] Canonical formulation of self-gravitating spinning-object systems
Jan Steinhoff, Gerhard Schäfer, EPL 87:50004 (2009)

[26] Spin-squared Hamiltonian of next-to-leading order gravitational interaction
Jan Steinhoff, Steven Hergt, Gerhard Schäfer, Phys. Rev. D 78:101503 (2008)

[27] ADM canonical formalism for gravitating spinning objects
Jan Steinhoff, Gerhard Schäfer, Steven Hergt, Phys. Rev. D 77:104018 (2008)

[28] Higher-order-in-spin interaction Hamiltonians for binary black holes from Poincaré invariance
Steven Hergt, Gerhard Schäfer, Phys. Rev. D 78:124004 (2008)

[29] Relativistic closed-form Hamiltonian for many-body gravitating systems
Tomas Ledvinka, Gerhard Schäfer, Jiri Bicak, Phys. Rev. Lett. 100:251101 (2008)

[30] The next-to-leading order gravitational spin(1)-spin(2) dynamics in Hamiltonian form
Jan Steinhoff, Steven Hergt, Gerhard Schäfer, Phys. Rev. D 77:081501 (2008)

[31] Higher-order-in-spin interaction Hamiltonians for binary black holes from source terms of Kerr geomtery in approximate ADM coordinates
Steven Hergt, Gerhard Schäfer, Phys. Rev. D 77:104001 (2008)

[32] Hamiltonian of two spinning compact bodies with next-to-leading order gravitational spin-orbit coupling
Thibault Damour, Piotr Jaranowski, Gerhard Schäfer, Phys. Rev. D 77:064032 (2008)

[33] Binary black hole coalescence in semi-analytic puncture evolution
Achamveedu Gopakumar, Gerhard Schäfer, Phys. Rev. D 77:104023 (2008)

[34] Gravitational recoil during binary black-hole coalescence using the effective one-body approach
T. Damour and A. Gopakumar, Phys. Rev. D 73:125006 (2006)

[35] Third post-Newtonian constrained canonical dynamics for binary point masses in harmonic coordinates
R.-M. Memmesheimer and G. Schäfer, Phys. Rev. D 71:044021 (2005)

[36] CFC+: Improved dynamics and gravitational waveforms from relativistic core collapse simulations
P. Cerda-Duran, G. Faye, H. Dimmelmeier, J. A. Font, J.-M. Ibanez, E. Müller and G. Schäfer, Astronomy & Astrophysics 439, 1033 (2005)

[37] Light deflection in the post-linear gravitational field of bounded pointlike masses
M. H. Brügmann, Phys. Rev. D 72:024012 (2005)

[38] A minimal no-radiation approximation to Einstein's field equations
G. Schäfer, A. Gopakumar, Phys. Rev. D 69:021501(R) (2004)

[39] A skeleton approximate solution of the Einstein field equations for multiple black-hole systems
G. Faye, P. Jaranowski, and G. Schäfer, Phys. Rev. D 69:124029 (2004)

Theses

[40] Motion and gravitational wave emission of spinning compact binaries
Manuel Tessmer, PhD thesis (2011)

[41] Hamiltonische Formulierung und Behandlung nichtlinearer Eigendrehimpulsbeiträge in Binärsystemen der Allgemeinen Relativitätstheorie
Steven Hergt, PhD thesis (2011)

[42] Canonical Formulation of Spin in General Relativity
Jan Steinhoff, PhD thesis (2010)

[43] Grenzfälle des post-Newtonschen Dreikörperproblems
Tilman J. Rothe, Diploma thesis (2009)

[44] Zur ADM-Eichung der Kerr-Metrik
Steven Hergt, Diploma thesis (2007)

[45] Dynamik geladener Teilchen mit Spin in post-Coulombscher Näherung höherer Ordnung
Y. Blum, Diploma thesis (2005)

[46] Kanonische Formulierung der post-Newtonschen / post-Coulombschen Punktteilchen-Dynamik höherer Ordnung in harmonischer / Lorentz-Eichung
R.-M. Memmesheimer, Diploma thesis (2004)

[47] Untersuchungen zur Skeleton-Näherungslösung der Einsteinschen Feldgleichungen für Binärsysteme
Christian Thierfelder, Diploma thesis (2004)

[48] Light deflection in the post-linear gravitational field of bounded point-like masses
Michael H. Brügmann, PhD thesis (2006)