Metric of a Slow Rotating Body with Quadrupole Moment from the Erez-Rosen Metric

Author(s)
Francisco Frutos-Alfaro,
Edwin Retana-Montenegro,
Iván Cordero-García,
Javier Bonatti-González

ABSTRACT

A metric representing a slowly rotating object with quadrupole moment is obtained using a perturbation method to include rotation into the weak limit of the Erez-Rosen metric. This metric is intended to tackle relativistic astrometry and gravitational lensing problems in which a quadrupole moment has to be taken into account.

Cite this paper

F. Frutos-Alfaro, E. Retana-Montenegro, I. Cordero-García and J. Bonatti-González, "Metric of a Slow Rotating Body with Quadrupole Moment from the Erez-Rosen Metric,"*International Journal of Astronomy and Astrophysics*, Vol. 3 No. 4, 2013, pp. 431-437. doi: 10.4236/ijaa.2013.34051.

F. Frutos-Alfaro, E. Retana-Montenegro, I. Cordero-García and J. Bonatti-González, "Metric of a Slow Rotating Body with Quadrupole Moment from the Erez-Rosen Metric,"

References

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[2] A. G. Doroshkevich, Ya. B. Zel’dovich and I. D. Novikov, “Gravitational Collapse of Nonsymmetric and Rotating Masses,” Journal of Experimental and Theoretical Physics (Soviet Physics JETP), Vol. 22, No. 1, 1966, pp. 122-130. http://www.jetp.ac.ru/cgi-bin/e/index/e/22/1/p122?a=list

[3] J. Winicour, A. I. Janis and E. T. Newman, “Static, Axially Symmetric Point Horizons,” Physical Review, Vol. 176, No. 5, 1968, pp. 1507-1513. http://dx.doi.org/10.1103/PhysRev.176.1507

[4] J. H. Young and C. A. Coulter, “Exact Metric for a Nonrotating Mass with a Quadrupole Moment,” Physical Review, Vol. 184, No. 5, 1969, pp. 1313-1315. http://dx.doi.org/10.1103/PhysRev.184.1313

[5] H. Quevedo, “Class of Stationary Axisymmetric Solutions of Einstein’s Equations in Empty Space,” Physical Review D, Vol. 33, No. 2, 1986, pp. 324-327. http://dx.doi.org/10.1103/PhysRevD.33.324

[6] H. Quevedo, “General Static Axisymmetric Solution of Einstein’s Vacuum Field Equations in Prolate Spheroidal Coordinates,” Physical Review D, Vol. 39, No. 10, 1989, pp. 2904-2911. http://dx.doi.org/ 10.1103/PhysRevD.39.2904

[7] H. Quevedo and B. Mashhoon, “Generalization of Kerr Spacetime,” Physical Review D, Vol. 43, No. 12, 1991, pp. 3902-3906. http://dx.doi.org/10.1103/PhysRevD.43.3902

[8] J. Castejon-Amenedo and V. S. Manko, “Superposition of the Kerr Metric with the Generalized Erez-Rosen Solution,” Physical Review D, Vol. 41, No. 6, 1990, pp. 2018-2020. http://dx.doi.org/10. 1103/PhysRevD.41.2018

[9] C. Hoenselaers, W. Kinnersley and B. C. Xanthopoulos, “Symmetries of the Stationary Einstein-Maxwell Equations. VI. Transformations which generate Asymptotically Flat Spacetimes with Arbitrary Multipole Moments,” Journal of Mathematical Physics, Vol. 20, No. 12, 1979, pp. 2530-2536. http://dx.doi.org/10.1063/1.524058

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[11] K. Boshkayev, H. Quevedo and R. Ruffini, “Gravitational Field of Compact Objects in General Relativity,” Physical Review D, Vol. 86, No. 6, 2012, Article ID: 064043. http://dx.doi.org/10. 1103/PhysRevD.86.064043

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[14] M. Carmeli, “Classical Fields,” World Scientific Publishing, Singapore, 2001.

http://www.worldscientific.com/worldscibooks/10.1142/4843#t=toc

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http://rspa.royalsocietypublishing.org/content/136/829

[17] R. P. Kerr, “Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics,” Physical Review Letter, Vol. 11, No. 5, 1963, pp. 237-238. http://dx.doi.org/10.1103/PhysRev Lett.11.237

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[20] E. Berti, F. White, A. Maniopoulou and M. Bruni, “Rotating Neutron Stars: an Invariant Comparison of Approximate and Numerical Spacetime Models,” Monthly Notices of the Royal Astronomical Society, Vol. 358, No. 3, 2005, pp. 923-938. http://dx.doi.org/10.1111/j.1365-2966.2005.08812.x

[21] M. A. Abramowicz, G. J. E. Almergren, W. Kluzniak and A. V. Thampan, “Circular Geodesics in the Hartle-Thorne Metric,” ArXiv (gr-qc/0312070), 2003.

[22] H. Asada, “Effects of a Deformation of a Star on the Gravitational Lensing,” Monthly Notices of the Royal Astronomical Society, Vol. 356, No. 4, 2005, pp. 1249-1255. http://dx.doi.org/10.1111/j.1365-2966.2004.08486.x

[23] M. H. Soffel, “Relativity in Astrometry, Celestial Mechanics and Geodesy (Astronomy and Astrophysics Library),” Springer-Verlag, Berlin, 1989. http://dx.doi.org/10.1007/978-3-642-73406-9

[24] M. H. Soffel, J. Schastok, H. Ruder and M. Schneider, “Relativistic Astrometry,” Astrophysics and Space Science, Vol. 110, No. 1, 1985, pp. 95-101. http://dx.doi.org/10.1007/BF00660610

[25] J. Páez and F. Frutos, “Astrometry in the Kerr Field in PPN Approximation,” Astrophysics and Space Science, Vol. 214, No. 1-2, 1994, pp. 71-87. http://dx.doi.org/10.1007/BF00982325

[26] H. Asada, M. Kasai and T. Yamamoto, “Separability of Rotational Effects on a Gravitational Lens,” Physical Review D, Vol. 67, No. 4, 2003, Article ID: 043006. http://dx.doi.org/10.1103/Phys RevD.67.043006

[27] K. Yamada and H. Asada, “Post-Newtonian Effects of Planetary Gravity on the Perihelion Shift,” Monthly Notices of the Royal Astronomical Society, Vol. 423, No. 4, 2012, pp. 3540-3544.

http://dx.doi.org/10.1111/j.1365-2966.2012.21150.x

[1] G. Erez and N. Rosen, “The Gravitational Field of a Particle Possessing a Multipole Moment,” Bulletin of the Research Council of Israel, Vol. 8F, 1959, pp. 47-50.

[2] A. G. Doroshkevich, Ya. B. Zel’dovich and I. D. Novikov, “Gravitational Collapse of Nonsymmetric and Rotating Masses,” Journal of Experimental and Theoretical Physics (Soviet Physics JETP), Vol. 22, No. 1, 1966, pp. 122-130. http://www.jetp.ac.ru/cgi-bin/e/index/e/22/1/p122?a=list

[3] J. Winicour, A. I. Janis and E. T. Newman, “Static, Axially Symmetric Point Horizons,” Physical Review, Vol. 176, No. 5, 1968, pp. 1507-1513. http://dx.doi.org/10.1103/PhysRev.176.1507

[4] J. H. Young and C. A. Coulter, “Exact Metric for a Nonrotating Mass with a Quadrupole Moment,” Physical Review, Vol. 184, No. 5, 1969, pp. 1313-1315. http://dx.doi.org/10.1103/PhysRev.184.1313

[5] H. Quevedo, “Class of Stationary Axisymmetric Solutions of Einstein’s Equations in Empty Space,” Physical Review D, Vol. 33, No. 2, 1986, pp. 324-327. http://dx.doi.org/10.1103/PhysRevD.33.324

[6] H. Quevedo, “General Static Axisymmetric Solution of Einstein’s Vacuum Field Equations in Prolate Spheroidal Coordinates,” Physical Review D, Vol. 39, No. 10, 1989, pp. 2904-2911. http://dx.doi.org/ 10.1103/PhysRevD.39.2904

[7] H. Quevedo and B. Mashhoon, “Generalization of Kerr Spacetime,” Physical Review D, Vol. 43, No. 12, 1991, pp. 3902-3906. http://dx.doi.org/10.1103/PhysRevD.43.3902

[8] J. Castejon-Amenedo and V. S. Manko, “Superposition of the Kerr Metric with the Generalized Erez-Rosen Solution,” Physical Review D, Vol. 41, No. 6, 1990, pp. 2018-2020. http://dx.doi.org/10. 1103/PhysRevD.41.2018

[9] C. Hoenselaers, W. Kinnersley and B. C. Xanthopoulos, “Symmetries of the Stationary Einstein-Maxwell Equations. VI. Transformations which generate Asymptotically Flat Spacetimes with Arbitrary Multipole Moments,” Journal of Mathematical Physics, Vol. 20, No. 12, 1979, pp. 2530-2536. http://dx.doi.org/10.1063/1.524058

[10] F. J. Ernst, “New Formulation of the Axially Symmetric Gravitational Field Problem,” Physical Review, Vol. 167, No. 5, 1968, pp. 1175-1177. http://dx.doi.org/10.1103/PhysRev.167.1175

[11] K. Boshkayev, H. Quevedo and R. Ruffini, “Gravitational Field of Compact Objects in General Relativity,” Physical Review D, Vol. 86, No. 6, 2012, Article ID: 064043. http://dx.doi.org/10. 1103/PhysRevD.86.064043

[12] A. C. Hearn, “REDUCE (User’s and Contributed Packages Manual),” Konrad-Zuse-Zentrum für Informationstechnik, Berlin, 1999. http://reduce-algebra.com/

[13] J. B. Hartle and K. S. Thorne, “Slowly Rotating Relativistic Stars. II. Models for Neutron Stars and Supermassive Stars,” Astrophysical Journal, Vol. 153, 1968, pp. 807-834. http://dx.doi.org/10. 1086/149707

[14] M. Carmeli, “Classical Fields,” World Scientific Publishing, Singapore, 2001.

http://www.worldscientific.com/worldscibooks/10.1142/4843#t=toc

[15] Ya. B. Zel’dovich and I. D. Novikov, “Stars and Relativity,” Dover Publications, Mineola, 2011.

[16] T. Lewis, “Some Special Solutions of the Equations of Axially Symmetric Gravitational Fields,” Proceedings of the Royal Society London A, Vol. 136, No. 829, 1932, pp. 176-192.

http://rspa.royalsocietypublishing.org/content/136/829

[17] R. P. Kerr, “Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics,” Physical Review Letter, Vol. 11, No. 5, 1963, pp. 237-238. http://dx.doi.org/10.1103/PhysRev Lett.11.237

[18] S. Chandrasekhar, “The Mathematical Theory of Black Holes,” Oxford University Press, Oxford, 2000.

[19] S. Weinberg, “Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity,” John Wiley & Sons, Inc., Hoboken, 1972.

[20] E. Berti, F. White, A. Maniopoulou and M. Bruni, “Rotating Neutron Stars: an Invariant Comparison of Approximate and Numerical Spacetime Models,” Monthly Notices of the Royal Astronomical Society, Vol. 358, No. 3, 2005, pp. 923-938. http://dx.doi.org/10.1111/j.1365-2966.2005.08812.x

[21] M. A. Abramowicz, G. J. E. Almergren, W. Kluzniak and A. V. Thampan, “Circular Geodesics in the Hartle-Thorne Metric,” ArXiv (gr-qc/0312070), 2003.

[22] H. Asada, “Effects of a Deformation of a Star on the Gravitational Lensing,” Monthly Notices of the Royal Astronomical Society, Vol. 356, No. 4, 2005, pp. 1249-1255. http://dx.doi.org/10.1111/j.1365-2966.2004.08486.x

[23] M. H. Soffel, “Relativity in Astrometry, Celestial Mechanics and Geodesy (Astronomy and Astrophysics Library),” Springer-Verlag, Berlin, 1989. http://dx.doi.org/10.1007/978-3-642-73406-9

[24] M. H. Soffel, J. Schastok, H. Ruder and M. Schneider, “Relativistic Astrometry,” Astrophysics and Space Science, Vol. 110, No. 1, 1985, pp. 95-101. http://dx.doi.org/10.1007/BF00660610

[25] J. Páez and F. Frutos, “Astrometry in the Kerr Field in PPN Approximation,” Astrophysics and Space Science, Vol. 214, No. 1-2, 1994, pp. 71-87. http://dx.doi.org/10.1007/BF00982325

[26] H. Asada, M. Kasai and T. Yamamoto, “Separability of Rotational Effects on a Gravitational Lens,” Physical Review D, Vol. 67, No. 4, 2003, Article ID: 043006. http://dx.doi.org/10.1103/Phys RevD.67.043006

[27] K. Yamada and H. Asada, “Post-Newtonian Effects of Planetary Gravity on the Perihelion Shift,” Monthly Notices of the Royal Astronomical Society, Vol. 423, No. 4, 2012, pp. 3540-3544.

http://dx.doi.org/10.1111/j.1365-2966.2012.21150.x