Skip to main content
SpringerLink
Log in

The space–time line element for static ellipsoidal objects

  • Research Article
  • Published:
General Relativity and Gravitation Aims and scope Submit manuscript

A Correction to this article was published on 24 February 2024

This article has been updated

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Data availability statement

Since this is a theoretical work data availability is not applicable and no data has been used or analysed throughout the work.

Change history

Notes

  1. In literature q-metric is also known as the Zipoy–Voorhees metric, \(\delta \)-metric and \(\gamma \)-metric.

  2. 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 \((+,-,-,-)\).

  3. We denoted the prolate coordinates by \((t, \sigma , \tau , \phi )\) instead of \((t, x, y, \phi )\) used in [6] in order to avoid confusion with the Cartesian coordinates in Eq. (2).

  4. The notations \(\gamma \) and q used in reference [16] are respectively denoted by \((1+q)\) and p in the present work.

References

  1. 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)

    ADS  MathSciNet  Google Scholar 

  2. 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

    Article  ADS  Google Scholar 

  3. 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

    Article  ADS  Google Scholar 

  4. Papapetrou, A.: Eine rotationssymmetrische lösung in der allgemeinen relativitätstheorie. Ann. Phys. (Berlin) 4474–6, 309–315 (1953). https://doi.org/10.1002/andp.19534470412

    Article  ADS  Google Scholar 

  5. 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

    Article  ADS  MathSciNet  Google Scholar 

  6. 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

    Article  ADS  PubMed  PubMed Central  Google Scholar 

  7. 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)

    MathSciNet  Google Scholar 

  8. 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)

    ADS  Google Scholar 

  9. 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

    Article  ADS  Google Scholar 

  10. 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

    Article  ADS  Google Scholar 

  11. Quevedo, H., Parkes, L.: Geodesies in the Erez–Rosen space–time. Gen. Relativ. Gravit. 2110, 1047–1072 (1989). https://doi.org/10.1007/BF00774088

    Article  ADS  Google Scholar 

  12. 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

    Book  Google Scholar 

  13. 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

    Article  ADS  Google Scholar 

  14. Voorhees, B.H.: Static axially symmetric gravitational fields. Phys. Rev. D 2, 2119–2122 (1970). https://doi.org/10.1103/PhysRevD.2.2119

    Article  ADS  Google Scholar 

  15. Zipoy, D.M.: Topology of some spheroidal metrics. J. Math. Phys. 76, 1137–1143 (1966). https://doi.org/10.1063/1.1705005

    Article  ADS  MathSciNet  Google Scholar 

  16. 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

    Article  ADS  MathSciNet  Google Scholar 

  17. 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

    Article  MathSciNet  Google Scholar 

  18. 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

    Article  ADS  Google Scholar 

  19. Nikouravan, B.: Schwarzschild-like solution for ellipsoidal celestial objects. Int. J. Phys. Sci. 66, 1426–1430 (2011)

    Google Scholar 

  20. Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields, vol. 2. Pergamon Press, Oxford (1971)

    Google Scholar 

  21. Heiskanen, W.A., Moritz, H.: Physical Geodesy. W.H. Freeman and Company, San Francisco (1967)

    Google Scholar 

  22. Wiltshire, D.L., Visser, M., Scott, S.M. (eds.): The Kerr Spacetime: Rotating Black Holes in General Relativity. Cambridge University Press, Cambridge (2009)

    Google Scholar 

  23. 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

    Article  ADS  MathSciNet  Google Scholar 

  24. 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

    Article  MathSciNet  Google Scholar 

  25. Chandrasekhar, S.: The Mathematical Theory of Black Holes. Claredon press, Oxford (1998)

    Book  Google Scholar 

  26. 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)

    Google Scholar 

  27. 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

    Article  ADS  Google Scholar 

  28. Geroch, R.: Multipole moments. II. Curved space. J. Math. Phys. 118, 2580–2588 (1970). https://doi.org/10.1063/1.1665427

    Article  ADS  MathSciNet  Google Scholar 

  29. 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

    Article  ADS  MathSciNet  CAS  Google Scholar 

  30. 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

    Article  ADS  MathSciNet  Google Scholar 

  31. 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

    Article  ADS  MathSciNet  Google Scholar 

  32. Faraji, S.: Circular geodesics in a new generalization of q-metric. Universe 8, 195 (2022). https://doi.org/10.3390/universe8030195

    Article  ADS  Google Scholar 

  33. Laarakkers, W.G., Poisson, E.: Quadrupole moments of rotating neutron stars. Astrophys. J. 5121, 282 (1999)

    Article  ADS  Google Scholar 

  34. 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

    Article  CAS  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Department of Physics, Assam University, Silchar, Assam, 788011, India

    Ranchhaigiri Brahma & A. K. Sen

Authors
  1. Ranchhaigiri Brahma
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. A. K. Sen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Ranchhaigiri Brahma.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10714-023-03078-8

Keywords

Search

Navigation