On the post-linear quadrupole-quadrupole metric

• Autores: Francisco Frutos Alfaro, Michael Soffel
• Localización: Revista de Matemática: Teoría y Aplicaciones, ISSN 2215-3373, ISSN-e 2215-3373, Vol. 24, Nº. 2, 2017, págs. 239-255
• Idioma: inglés
• DOI: 10.15517/rmta.v24i2.29856
• Títulos paralelos:
  • Sobre la métrica post-lineal cuadrupolo-cuadrupolo
• Enlaces
  • Texto completo (pdf)
• Resumen
  • español

    La métrica de Hartle-Thorne define un espacio-tiempo confiable para la mayoría de propósitos astrofísicos, por ejemplo simulaciones de estrellas girando lentamente. Añadimos términos de segundo orden en el momento cuadripolar a su versión post-lineal al resolver las ecuaciones decampo de Einstein. La solución encontrada es comparada con la encontrada por Blanchet en el marco post-Minkowskiano. Primero derivamos la métrica de Hartle-Thorne en coordenadas armónicas y luego mostramos la concordancia con la correspondiente métrica post-lineal de Blanchet. También encontramos una transformación de coordenadas de la métrica de Erez-Rosen post-lineal al espacio-tiempo de Hartle-Thorne obtenido. Es bien sabido que la solución de Hartle-Thorne puede estar suavemente acoplada con una solución de fluido interior perfecto con propiedades físicas apropiadas. Una comparación entre estas soluciones proporciona una validación de las mismas. Está claro que para representar soluciones realistas de distribuciones de materia auto-gravitantes (axialmente simétricas) de fluido perfecto, el momento cuadripolar tiene que serincluido como un parámetro físico

  • English

    The Hartle-Thorne metric defines a reliable spacetime for most astrophysical purposes, for instance simulations of slowly rotating stars. Solving the Einstein field equations, we added terms of second order in the quadrupole moment to its post-linear version in order to compare it with solutions found by Blanchet in the multi-polar post-Minkowskian framework. We first derived the extended Hartle-Thorne metric in harmonic coordinates and then showed agreement with the corresponding post-linear metric from Blanchet. We also found a coordinate transformation from the post-linear Erez-Rosen metric to our extended Hartle-Thorne spacetime. It is well known that the Hartle-Thorne solution can be smoothly matched with an interior perfect fluid solution with appropriate physical properties. A comparison among these solutions provides a validation of them. It is clear that inorder to represent realistic solutions of self-gravitating (axially symmetric) matter distributions of perfect fluid, the quadrupole moment has to be included as a physical parameter.

• Referencias bibliográficas
  • Berti, E.; White, F.; Maniopoulou, A.; Bruni, M. (2005) “Rotating neutron stars: An invariant comparison of approximate and numerical space-time...
  • Blanchet, L.; Damour, T. (1986) “Radiative gravitational fields in general relativity I. General structure of the field outside the source”,...
  • Blanchet, L. (1998) “Quadrupole-quadrupole gravitational waves”, Classical and Quantum Gravity 15(1): 89–111. http://dx.doi.org/10.1088/0264-9381/15/1/008
  • Blanchet, L. (2014) “Gravitational radiation from Post-Newtonian sources and inspiralling compact binaries”, Living Reviews in Relativity...
  • Carmeli, M. (2001) Classical Fields: General Relativity and Gauge Theory. World Scientific Publishing, Singapore. http://www.worldscientific.com/worldscibooks/10.1142/4843
  • Cook, G.B.; Shapiro, S.L.; Teukolsky, S A. (1994) “Rapidly rotation neutron stars in general relativity: Realistic equations of state”, Astrophysical...
  • Damour, T.; Iyer, B. (1991) “Multipole analysis for electromagnetism and linearized gravity with irreducible Cartesian tensors”, Physical...
  • Doroshkevich, A.G.; Zel’dovich, Ya.B.; Novikov, I.D. (1966) “Gravitational collapse of nonsymmetric and rotating masses”, Journal of Experimental...
  • Ernst. F.J. (1968) “New formulation of the axially symmetric gravitational field problem”, Physical Review 167(5): 1175–1177. http://dx.doi.org/10.1103/PhysRev.167.1175
  • Fodor, G.; Hoenselaer, C.; Perjés, Z. (1989) “Multipole moments of axisymmetric systems in relativity”, Journal of Mathematical Physics 30(10):...
  • Frutos-Alfaro, F.; Retana-Montenegro, E.; Cordero-García, I.; Bonatti- González, J. (2013) “Metric of a slow rotating body with quadrupole...
  • Frutos-Alfaro, F.; Montero-Camacho, P.; Araya, M.; Bonatti-González, J. (2015) “Approximate metric for a rotating deformed mass”, International...
  • Geroch, R. (1970) “Multipole moments. II. Curved space”, Journal of Mathematical Physics 11(8): 2580–2588. http://dx.doi.org/10.1063/1.1665427
  • Gürsel, Y. (1983) “Multipole moments for stationary systems: The equivalence of the Geroch-Hansen formulation and the Thorne formulation”,...
  • Hansen, R.O. (1974) “Multipole moments of stationary space-times”, Journal of Mathematical Physics 15(1): 46–52. http://dx.doi.org/10.1063/1.1666501
  • Hartle, J.B.; Thorne K.S. (1968) “Slowly rotating relativistic stars. II. Models for neutron stars and supermassive stars”, Astrophysical...
  • Hearn, A.C. (1999) REDUCE (User’s and Contributed Packages Manual). Konrad-Zuse-Zentrum für Informationstechnik, Berlin. http://www.reduce-algebra.com/docs/reduce.pdf
  • Hernández-Pastora, J.L.; Martín, J. (1994) “Monopole-quadrupole static axisymmetric solutions of Einstein field equations”, General Relativity...
  • Hoenselaers, C.; Perjés, Z. (1990) “Multipole moments of axisymmetric electrovacuum spacetimes”, Classical and Quantum Gravity 7(10): 1819–1825....
  • Manko, V.S.; Mielke, E.W.; Sanabria-Gómez, J.D. (2000) “Exact solution for the exterior field of a rotating neutron star”, Physical Review...
  • Manko, V.S.; Sanabria-Gómez, J.D.; Manko, O.V. (2000) “Nine-parameter electrovac metric involving rational functions”, Physical Review D 62,...
  • Pachón, L.A.; Rueda, J.A.; Sanabria-Gómez, J.D. (2006) “Realistic exact solution for the exterior field of a rotating neutron star”, Physical...
  • Pappas, G.; Apostolatos, T.A. (2012) “Revising the multipole moments of numerical spacetimes and its consequences”, Physical Review Letters...
  • Poisson, E.; Will, C.M. (2014) Gravity (Newtonian, Post-Newtonian, Relativistic). Cambridge University Press, Cambridge. http://www.cambridge.org/us/academic/subjects/physics/
  • Quevedo, H.; Mashhoon, B. (1991) “Generalization of Kerr space-time”, Physical Review 43(12): 3902–3906. http://dx.doi.org/10.1103/PhysRevD.43.3902
  • Quevedo, H. (2011) “Exterior and interior metrics with quadrupole moment”, General Relativity and Gravitation, 43(4): 1141–1152. http://dx.doi.org/10.1007/s10714-010-0940-5
  • Ryan, F.D. (1995) “Gravitational waves from the inspiral of a compact object into a massive, axisymmetric body with arbitrary multipole moments”,...
  • Simon, W.; Beig, R. (1983) “The multipole structure of stationary space-times”, Journal of Mathematical Physics 24(5): 1163–1171. http://dx.doi.org/10.1063/1.525846
  • Thorne, K.S. (1980) “Multipole expansions of gravitational radiation”, Reviews on Modern Physics 52(2): 299–340. http://dx.doi.org/10.1103/RevModPhys.52.299
  • Winicour, J.; Janis, A.I.; Newman, E.T. (1968) “Static, axially symmetric point horizons”, Physical Review 176: 1507–1513. http://dx.doi.org/10.1103/PhysRev.176.1507
  • Young, J.H.; Coulter; C.A. (1969) “Exact metric for a nonrotating mass with a quadrupole moment”, Physical Review 184: 1313–1315. http://dx.doi.org/10.1103/PhysRev.184.1313
  • Zel’dovich, Ya.B.; Novikov, I.D. (2011) Stars and Relativity. Dover Publications, New York. http://store.doverpublications.com/0486694240.html