Approximate Metric for a Rotating Deformed Mass


A new Kerr-like metric with quadrupole moment is obtained by means of perturbing the Kerr spacetime. The form of this new metric is simple as the Kerr metric. By comparison with the exterior Hartle-Thorne metric, it is shown that it could be matched to an interior solution. This approximate metric may represent the spacetime of a real astrophysical object with any Kerr rotation parameter a and slightly deformed.

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Frutos-Alfaro, F. , Montero-Camacho, P. , Araya, M. and Bonatti-González, J. (2015) Approximate Metric for a Rotating Deformed Mass. International Journal of Astronomy and Astrophysics, 5, 1-10. doi: 10.4236/ijaa.2015.51001.

Conflicts of Interest

The authors declare no conflicts of interest.


[1] Kerr, R.P. (1963) Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics. Physical Review Letters, 11, 237-238.
[2] Hernandez, W. (1967) Material Sources for the Kerr Metric. Physical Review, 159, 1070-1072.
[3] Thorne, K.S. (1969) Relativistic Stars, Black Holes and Gravitational Waves. Sachs, B.K., Ed., General Relativity and Cosmology, Proceedings of the International School of Physics Enrico Fermi, Course XLVII, Academic Press, Waltham, 237-283.
[4] Marsh, G.E. (2014) Rigid Rotation and the Kerr Metric.
[5] Boshkayev, K., Quevedo, H. and Ruffini, R. (2012) Gravitational Field of Compact Objects in General Relativity. Physical Review D, 86, Article ID: 064043.
[6] Cuchí, J.E., Molina, A. and Ruiz, E. (2011) Double Shell Stars as Source of the Kerr Metric in the CMMR Approximation. Journal of Physics: Conference Series, 314, Article ID: 012070.
[7] Krisch, J.P. and Glass, E.N. (2009) Counter-Rotating Kerr Manifolds Separated by a Fluid Shell. Classical and Quantum Gravity, 26, Article ID: 175010.
[8] Haggag, S. and Marek, J. (1981) A Nearly-Perfect-Fluid Source for the Kerr metric. Il Nuovo Cimento B, 62, 273-282.
[9] Haggag, S. (1990) A Fluid Source for the Kerr Metric. Il Nuovo Cimento B, 105, 365-370.
[10] Haggag, S. (1990) A Static Axisymmetric Anisotropic Fluid Solution in General Relativity. Astrophysics and Space Science, 173, 47-51.
[11] Krasiński, A. (1980) A Newtonian Model of the Source of the Kerr Metric. Physics Letters, 80A, 238-242.
[12] Ramadan, A. (2004) Fluid Sources for the Kerr Metric. Il Nuovo Cimento, 119B, 123-129.
[13] Krasiński, A. (1978) Ellipsoidal Space-Times, Sources for the Kerr Metric. Annals of Physics, 112, 22-40.
[14] Drake, S.P. and Turolla, R. (1997) The Application of the Newman-Janis Algorithm in Obtaining Interior Solutions of the Kerr Metric. Classical and Quantum Gravity, 14, 1883-1897.
[15] Viaggiu, S. (2006) Interior Kerr Solutions with the Newman-Janis Algorithm Starting with Physically Reasonable Space-Times. International Journal of Modern Physics D, 15, 1441-1453.
[16] Castejon-Amenedo, J. and Manko, V.S. (1990) Superposition of the Kerr Metric with the Generalized Erez-Rosen Solution. Physical Review D, 41, 2018-2020.
[17] Manko, V.S. and Novikov, I.D. (1992) Generalizations of the Kerr and Kerr-Newman Metrics Possessing an Arbitrary Set of Mass-Multipole Moments. Classical and Quantum Gravity, 9, 2477-2487.
[18] Manko, V.S., Mielke, E.W. and Sanabria-Gomez, J.D. (2000) Exact Solution for the Exterior Field of a Rotating Neutron Star. Physical Review D, 61, Article ID: 081501.
[19] Pachon, L.A., Rueda, J.A. and Sanabria-Gomez, J.D. (2006) Realistic Exact Solution for the Exterior Field of a Rotating Neutron Star. Physical Review D, 73, Article ID: 104038.
[20] Quevedo, H. (1986) Class of Stationary Axisymmetric Solutions of Einsteins Equations in Empty Space. Physical Review D, 33, 324-327.
[21] Quevedo, H. (1989) General Static Axisymmetric Solution of Einsteins Vacuum Field Equations in Prolate Spheroidal Coordinates. Physical Review D, 39, 2904-2911.
[22] Quevedo, H. and Mashhoon, B. (1991) Generalization of Kerr Spacetime. Physical Review D, 43, 3902-3906.
[23] Quevedo, H. (2011) Exterior and Interior Metrics with Quadrupole Moment. General Relativity and Gravitation, 43, 1141-1152.
[24] Ernst, F.J. (1968) New Formulation of the Axially Symmetric Gravitational Field Problem. Physical Review, 167, 1175-1177.
[25] Hoenselaers, C., Kinnersley, W. and Xanthopoulos, B.C. (1979) Symmetries of the Stationary Einstein-Maxwell Equations. VI. Transformations Which Generate Asymptotically Flat Spacetimes with Arbitrary Multipole Moments. Journal of Mathematical Physics, 20, 2530-2536.
[26] Quevedo, H. (2012) Matching Conditions in Relativistic Astrophysics. In: Damour, T., Jantzen, R.T. and Ruffini, R., Eds., Proceedings of the Twelfth Marcel Grossmann Meeting on General Relativity, World Scientific, Singapore.
[27] Andersson, N. and Comer, G.L. (2001) Slowly Rotating General Relativistic Superfluid Neutron Stars. Classical and Quantum Gravity, 18, 969-1002.
[28] Stergioulas, N. (2003) Rotating Stars in Relativity. Living Reviews in Relativity, 6.
[29] Fragile, P.C., Blaes, O.M., Anninos, P. and Salmonson, J.D. (2007) Global General Relativistic Magnetohydrodynamic Simulation of a Tilted Black Hole Accretion Disk. Astrophysical Journal, 668, 417-429.
[30] Hawley, J.F. (2009) MHD Simulations of Accretion Disks and Jets: Strengths and Limitations. Astrophysics and Space Science, 320, 107-114.
[31] Fendt, C. and Memola, E. (2008) Formation of Relativistic MHD Jets: Stationary State Solutions and Numerical Simulations. International Journal of Modern Physics, D17, 1677-1686.
[32] Frutos-Alfaro, F., Retana-Montenegro, E., Cordero-Garca, I. and Bonatti-Gonzalez, J. (2013) Metric of a Slow Rotating Body with Quadrupole Moment from the Erez-Rosen Metric. International Journal of Astronomy and Astrophysics, 3, 431-437.
[33] Soffel, M.H. (1989) Relativity in Astrometry, Celestial Mechanics and Geodesy (Astronomy and Astrophysics Library). Springer-Verlag, Berlin.
[34] Frutos-Alfaro, F. (2001) A Computer Program to Visualize Gravitational Lenses. American Journal of Physics, 69, 218-222.
[35] Dexter, J. and Algol, E. (2009) A Fast New Public Code for Computing Photon Orbits in a Kerr Spacetime. Astrophysical Journal, 696, 1616-1629.
[36] Frutos-Alfaro, F., Grave, F., Muller, T. and Adis, D. (2012) Wavefronts and Light Cones for Kerr Spacetimes. Journal of Modern Physics, 3, 1882-1890.
[37] Vincent, F.H., Paumard, T., Gourgoulhon, E. and Perrin, G. (2011) GYOTO: A New General Relativistic Ray-Tracing Code. Classical and Quantum Gravity, 28, Article ID: 225011.
[38] Hearn, A.C. (1999) REDUCE (User’s and Contributed Packages Manual). Konrad-Zuse-Zentrum fur Informationstechnik, Berlin.
[39] Carmeli, M. (2001) Classical Fields: General Relativity and Gauge Theory. World Scientific Publishing, Singapore.
[40] Winicour, J., Janis, A.I. and Newman, E.T. (1968) Static, Axially Symmetric Point Horizons. Physical Review, 176, 1507-1513.
[41] Young, J.H. and Coulter, C.A. (1969) Exact Metric for a Nonrotating Mass with a Quadrupole Moment. Physical Review, 184, 1313-1315.
[42] Zel’dovich, Y.B. and Novikov, I.D. (2011) Stars and Relativity. Dover Publications, New York.
[43] Lewis, T. (1932) Some Special Solutions of the Equations of Axially Symmetric Gravitational Fields. Proceedings of the Royal Society of London A, 136, 176-192.
[44] Chandrasekhar, S. (2000) The Mathematical Theory of Black Holes. Oxford University Press, Oxford.
[45] Hartle, J.B. and Thorne, K.S. (1968) Slowly Rotating Relativistic Stars. II. Models for Neutron Stars and Supermassive Stars. Astrophysical Journal, 153, 807-834.
[46] Berti, E., White, F., Maniopoulou, A. and Bruni, M. (2005) Rotating Neutron Stars: An Invariant Comparison of Approximate and Numerical Spacetime Models. Monthly Notices of the Royal Astronomical Society, 358, 923-938.
[47] Sato, H. and Tomimatsu, A. (1973) Gravitational Field of Slowly Rotating Deformed Masses. Progress of Theoretical Physics, 49, 790-799.
[48] Hernandez-Pastora, J.L. (2006) Approximate Gravitational Field of a Rotating Deformed Mass. General Relativity and Gravitation, 38, 871-884.

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