Abstract
In this paper, we solved the Einstein’s field equation and obtained a line element for static, ellipsoidal objects characterized by the linear eccentricity (\(\eta \)) instead of quadrupole parameter (q). This line element recovers the Schwarzschild line element when \(\eta \) is zero. In addition to that it also reduces to the Schwarzschild line element, if we neglect terms of the order of \(r^{-2}\) or higher which are present within the expressions for metric elements for large distances. Furthermore, as the ellipsoidal character of the derived line element is maintained by the linear eccentricity (\(\eta \)), which is an easily measurable parameter, this line element could be more suitable for various analytical as well as observational studies.
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Since this is a theoretical work data availability is not applicable and no data has been used or analysed throughout the work.
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Notes
In literature q-metric is also known as the Zipoy–Voorhees metric, \(\delta \)-metric and \(\gamma \)-metric.
Note: The English alphabet indices (\(i,j, k\ldots \)) denotes to run (0, 1, 2, 3) that represents ct, u, \(\theta \) and \(\phi \) co-ordinates respectively. Metric sign convention is \((+,-,-,-)\).
The notations \(\gamma \) and q used in reference [16] are respectively denoted by \((1+q)\) and p in the present work.
References
Schwarzschild, K.: On the gravitational field of a mass point according to Einstein’s theory. Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1916, 189–196 (1916)
Weyl, H.: Republication of: 3. On the theory of gravitation. Gen. Relativ. Gravit. 44, 779–810 (2012). https://doi.org/10.1007/s10714-011-1310-7
Lewis, T.: Some special solutions of the equations of axially symmetric gravitational fields. Proc. R. Soc. Lond. A 136829, 176–192 (1932). https://doi.org/10.1098/rspa.1932.0073
Papapetrou, A.: Eine rotationssymmetrische lösung in der allgemeinen relativitätstheorie. Ann. Phys. (Berlin) 4474–6, 309–315 (1953). https://doi.org/10.1002/andp.19534470412
Sloane, A.: The axially symmetric stationary vacuum field equations in Einstein’s Theory of general relativity. Aust. J. Phys. 31, 427–438 (1978). https://doi.org/10.1071/PH780427
Frutos-Alfaro, F., Quevedo, H., Sanchez, P.A.: Comparison of vacuum static quadrupolar metrics. R. Soc. Open Sci. 5, 170826 (2018). https://doi.org/10.1098/rsos.170826
Erez, G., Rosen, N.: The gravitational field of a particle possessing a multipole moment. Bull. Res. Council Israel Vol. Sect. F 8, 47–50 (1959)
Doroshkevich, A.G., Zel’Dovich, Y.B., Novikov, I.D.: Gravitational collapse of nonsymmetric and rotating masses. J. Exp. Theor. Phys. (Sov. Phys. JETP) 22, 122 (1966)
Winicour, J., Janis, A.I., Newman, E.T.: Static, axially symmetric point horizons. Phys. Rev. 176, 1507–1513 (1968). https://doi.org/10.1103/PhysRev.176.1507
Young, J.H., Coulter, C.A.: Exact metric for a nonrotating mass with a quadrupole moment. Phys. Rev. 184, 1313–1315 (1969). https://doi.org/10.1103/PhysRev.184.1313
Quevedo, H., Parkes, L.: Geodesies in the Erez–Rosen space–time. Gen. Relativ. Gravit. 2110, 1047–1072 (1989). https://doi.org/10.1007/BF00774088
Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., Herlt, E.: Exact Solutions of Einstein’s Field Equations. Cambridge Monographs on Mathematical Physics, 2nd edn. Cambridge University Press, Cambridge (2003). https://doi.org/10.1017/CBO9780511535185
Quevedo, H.: Mass quadrupole as a source of naked singularities. Int. J. Mod. Phys. D 2010, 1779–1787 (2011). https://doi.org/10.1142/s0218271811019852
Voorhees, B.H.: Static axially symmetric gravitational fields. Phys. Rev. D 2, 2119–2122 (1970). https://doi.org/10.1103/PhysRevD.2.2119
Zipoy, D.M.: Topology of some spheroidal metrics. J. Math. Phys. 76, 1137–1143 (1966). https://doi.org/10.1063/1.1705005
Herrera, L., Pastora, J.H.: Measuring multipole moments of Weyl metrics by means of gyroscopes. J. Math. Phys. 4111, 7544–7555 (2000). https://doi.org/10.1063/1.1319517
Quevedo, H.: Multipole moments in general relativity—static and stationary vacuum solutions. Fortschr. Phys. 38, 733–840 (1990). https://doi.org/10.1002/prop.2190381002
Pireaux, S.: Solar quadrupole moment and purely relativistic gravitation contributions to mercury’s perihelion advance. Astrophys. Space Sci. 2844, 1159–1194 (2003). https://doi.org/10.1023/a:1023673227013
Nikouravan, B.: Schwarzschild-like solution for ellipsoidal celestial objects. Int. J. Phys. Sci. 66, 1426–1430 (2011)
Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields, vol. 2. Pergamon Press, Oxford (1971)
Heiskanen, W.A., Moritz, H.: Physical Geodesy. W.H. Freeman and Company, San Francisco (1967)
Wiltshire, D.L., Visser, M., Scott, S.M. (eds.): The Kerr Spacetime: Rotating Black Holes in General Relativity. Cambridge University Press, Cambridge (2009)
Zsigrai, J.: Ellipsoidal shapes in general relativity: general definitions and an application. Class. Quantum Gravity 2013, 2855–2870 (2003). https://doi.org/10.1088/0264-9381/20/13/330
Racz, I.: Note on stationary-axisymmetric vacuum spacetimes with inside ellipsoidal symmetry. Class. Quantum Gravity 98, 93–98 (1992). https://doi.org/10.1088/0264-9381/9/8/004
Chandrasekhar, S.: The Mathematical Theory of Black Holes. Claredon press, Oxford (1998)
Podolsky, J., Semerak, O., Zofka, M. (eds.): Gravitation: Following the Prague Inspiration: A Volume in Celebration of the 60th Birthday of Jiri Bicak. World Scientific, Singapore (2002)
Carter, B.: Global structure of the Kerr family of gravitational fields. Phys. Rev. 174, 1559–1571 (1968). https://doi.org/10.1103/PhysRev.174.1559
Geroch, R.: Multipole moments. II. Curved space. J. Math. Phys. 118, 2580–2588 (1970). https://doi.org/10.1063/1.1665427
Boshkayev, K., Gasperín, E., Gutiérrez-Piñeres, A.C., Quevedo, H., Toktarbay, S.: Motion of test particles in the field of a naked singularity. Phys. Rev. D 93, 024024 (2016). https://doi.org/10.1103/PhysRevD.93.024024
Mejía, I.M., Manko, V.S., Ruiz, E.: Simplest static and stationary vacuum quadrupolar metrics. Phys. Rev. D 100, 124021 (2019). https://doi.org/10.1103/PhysRevD.100.124021
Neznamov, V.P., Shemarulin, V.E.: Motion of spin-half particles in the axially symmetric field of naked singularities of the static q-metric. Gravit. Cosmol. 23, 149 (2017). https://doi.org/10.1134/S0202289317020050
Faraji, S.: Circular geodesics in a new generalization of q-metric. Universe 8, 195 (2022). https://doi.org/10.3390/universe8030195
Laarakkers, W.G., Poisson, E.: Quadrupole moments of rotating neutron stars. Astrophys. J. 5121, 282 (1999)
Toktarbay, S., Quevedo, H., Abishev, M., Muratkhan, A.: Gravitational field of slightly deformed naked singularities. Eur. Phys. J. C 82(4), 1–8 (2022). https://doi.org/10.1140/epjc/s10052-022-10230-2
Acknowledgements
The author Ranchhaigiri Brahma acknowledged the Ministry of Tribal Affairs, Govt. of India for supporting to carryout research work via NFST fellowship (201920-NFST-ASS-00678). We thank Prof. B. Indrajit Singh, Head, Dept. of Physics, Assam University for his encouragement and support to do this work. Finally, we thank the anonymous referees of this paper for their valuable comments, which we feel have improved the quality of the paper.
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Brahma, R., Sen, A.K. The space–time line element for static ellipsoidal objects. Gen Relativ Gravit 55, 24 (2023). https://doi.org/10.1007/s10714-023-03078-8
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DOI: https://doi.org/10.1007/s10714-023-03078-8