Skip to main content
SpringerLink
Log in

Compactness bound of Buchdahl–Vaidya–Tikekar anisotropic star in \(D\ge 4\) dimensional spacetime

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

Abstract

We study the higher dimensional scenario of an anisotropic compact star using the Buchdahl–Vaidya–Tikekar metric ansatz. In our formalism, the anisotropy is assumed in such a way that, in the absence of it, the solution reduces to Schwarzschild’s interior solution in \(D \ge 4\) dimensions. The model is so developed that it correlates anisotropy to the curvature parameter K which characterizes a departure from spherical geometry of the \(t=\) constant hypersurface of the associated spacetime when embedded in a 4 dimensional Euclidean space. Due to the particular choice of anisotropy, the pressure balancing equation for hydrostatic equilibrium continues to have the same form in higher dimensions. Consequently, our approach permits extending a four-dimensional solution to a higher dimensional spacetime without deforming the sphericity of the configuration. Making use of the model, we propose a higher dimensional anisotropic analogue of the Buchdahl bound on compactness. We show that additional dimension as well as anisotropy reduce the compactification limit. Our technique helps to regain the original Buchdahl limit in \(D=4\) dimensions and also, in the absence of anisotropy, the compactification limit in higher dimensions obtained earlier by Leon and Cruz (Gen Relativ Gravit 32:1207–1216, 2000. https://doi.org/10.1023/A:1001982402392). It turns out that the maximum achievable dimension remains model dependent through the causality condition and the compactification limit. We scrutinize the model under all the requisite physical conditions for a relativistic anisotropic fluid sphere which might serve as the internal structure of a compact star in higher dimensions. We analyze the consequences of the departure from homogeneous spherical distribution and dimensionality on the physical behaviour of the star. The EOS becomes stiffer in higher dimensions and comparatively lower anisotropic stress. Our calculation shows that the central density reduces as we move towards higher dimensions and inclusion of anisotropy increases the rate of fall of the density profile. We also note that the two pressures get reduced considerably in higher dimensions. We show that, for a given curvature parameter specifying the sphericity, an extra dimension is analogous to moving towards a homogeneous distribution of an anisotropic star.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20

Similar content being viewed by others

Availability of data and materials

This article’s data is accessible within the public domain as specified and duly referenced in the citations.

References

  1. Delgaty, M.S.R., Lake, K.: Physical acceptability of isolated, static, spherically symmetric, perfect fluid solutions of Einstein’s equations. Comput. Phys. Commun. 115, 395–415 (1998). https://doi.org/10.1016/S0010-4655(98)00130-1

    Article  ADS  MathSciNet  Google Scholar 

  2. Randall, L., Sundrum, S.: An alternative to compactification. Phys. Rev. Lett. 83, 4690 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  3. Lovelock, D.: The Einstein tensor and its generalizations. J. Math. Phys. 12, 498–501 (1971)

    Article  ADS  MathSciNet  Google Scholar 

  4. Penrose, R.: Gravitational collapse: the role of general relativity. Riv. Nuovo Cimento 1, 252 (1969)

    Google Scholar 

  5. Kaluza, T.: Zum Unitätsproblem in der Physik. Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1921, 966–972 (1921)

    Google Scholar 

  6. Klein, O.: Quantentheorie und fünfdimensionale Relativitätstheorie. Z. Physik 37, 895–906 (1926)

    Article  ADS  Google Scholar 

  7. Yoshimura, M.: Classification of the static vacuum metric with Ricci-flat compactification. Phys. Rev. D 34, 1021 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  8. Koikawa, T.: More about exact solutions of the higher-dimensional Einstein equation in the presence of matter. Phys. Lett. A 117, 279 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  9. Koikawa, T., Yoshimura, M.: An exact solution of higher dimensional Einstein equation in presence of matter. Prog. Theor. Phys. 75, 977 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  10. Myers, R.C., Perry, M.J.: Black holes in higher dimensional space-times. Ann. Phys. (N.Y.) 172, 304–347 (1986)

    Article  ADS  MathSciNet  Google Scholar 

  11. Krori, K.D., Borgohain, P., Das, K.: Interior Schwarzschild-like solution in higher dimensions. Phys. Lett. A 132, 321 (1988)

    Article  ADS  MathSciNet  Google Scholar 

  12. Krori, K.D., Borgohain, P., Das, K.: Exact interior solutions in higher dimensions. Can. J. Phys. 67, 25 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  13. Tikekar, R.: A class of exact solutions of Einstein equations in higher dimensional space-times. Indian Math. Soc. 61, 37 (1995)

    MathSciNet  Google Scholar 

  14. Maharaj, S.D., Patel, L.K.: A note on exact spherically symmetric interior solutions in higher dimensions. Nuovo Cimento B 111, 1005–1010 (1996). https://doi.org/10.1007/BF02743296

    Article  ADS  MathSciNet  Google Scholar 

  15. Patel, L.K., Mehta, N.P., Maharaj, S.D.: Higher-dimensional relativistic-fluid spheres. Nuovo Cimento Soc. Ital. Fis. 112, 1037 (1997)

    Google Scholar 

  16. Liddle, A.R., et al.: Neutron stars and extra dimensions. Class. Quantum Grav. 7, 1009 (1990)

    Article  ADS  MathSciNet  Google Scholar 

  17. Ponce de Leon, J., Cruz, N.: Hydrostatic equilibrium of a perfect fluid sphere with exterior higher-dimensional Schwarzschild spacetime. Gen. Relativ. Gravit. 32, 1207–1216 (2000). https://doi.org/10.1023/A:1001982402392

    Article  ADS  MathSciNet  Google Scholar 

  18. Yamada, Y., Shinkai, H.A.: Formation of naked singularities in five-dimensional space-time. Phys. Rev. D 83(6), 064006 (2011)

    Article  ADS  Google Scholar 

  19. Bhar, P., Rahaman, F., Ray, S., Chatterjee, V.: Possibility of higher-dimensional anisotropic compact star. Eur. Phys. J. C 75, 190 (2015). https://doi.org/10.1140/epjc/s10052-015-3375-z

    Article  ADS  Google Scholar 

  20. Bagchi, M.: A study of neutron stars in \(D\ge 4\) dimensions (2020). Preprint at arxiv:2010.08928

  21. Akmal, A., Pandharipande, V.R., Ravenhall, D.G.: Equation of state of nucleon matter and neutron star structure. Phys. Rev. C 58, 1804–1828 (1998)

    Article  ADS  Google Scholar 

  22. Abbott, B.P., et al.: Observation of a 50-solar-mass binary black hole coalescence at redshift 0.2. Phys. Rev. Lett. 118, 221101 (2017). https://doi.org/10.1103/PhysRevLett.118.221101

    Article  ADS  Google Scholar 

  23. Burikham, P., Cheamsawat, K., Harko, T., et al.: The minimum mass of a charged spherically symmetric object in D dimensions, its implications for fundamental particles, and holography. Eur. Phys. J. C 76, 106 (2016). https://doi.org/10.1140/epjc/s10052-016-3948-5

    Article  ADS  Google Scholar 

  24. Arbañil, J.D.V., Malheiro, M.: Radial pulsation of a compact object in d dimensions. J. Phys. Conf. Ser. 1558, 012003 (2020)

    Article  Google Scholar 

  25. Maharaj, S.D., Brassel, B.P.: Junction conditions for composite matter in higher dimensions. Class. Quantum Grav. 38, 195006 (2021)

    Article  ADS  MathSciNet  Google Scholar 

  26. Brassel, B.P., Maharaj, S.D., Goswami, R.: Higher-dimensional inhomogeneous composite fluids: energy conditions. Progress Theor. Exp. Phys. 2021, 103E01 (2021). https://doi.org/10.1093/ptep/ptab116

    Article  Google Scholar 

  27. Naidoo, N., Maharaj, S.D., Govinder, K.S.: Radiating stars and Riccati equations in higher dimensions. Eur. Phys. J. C 83, 160 (2023). https://doi.org/10.1140/epjc/s10052-023-11296-2

    Article  ADS  Google Scholar 

  28. Khugaev, A., Dadhich, N., Molina, A.: Higher dimensional generalization of the Buchdahl–Vaidya–Tikekar model for a supercompact star. Phys. Rev. D 94, 064065 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  29. Vaidya, P.C., Tikekar, R.: Exact relativistic model for a superdense star. J. Astrophys. Astron. 3, 325–334 (1982). https://doi.org/10.1007/BF02714870

    Article  ADS  Google Scholar 

  30. Ruderman, M.: Pulsars: structure and dynamics. Ann. Rev. Astron. Astrophys. 10, 427 (1972)

    Article  ADS  Google Scholar 

  31. Herrera, L.: Cracking of self-gravitating compact objects. Phys. Lett. A 165, 206 (1992)

    Article  ADS  Google Scholar 

  32. Herrera, L., Santos, N.O.: Local anisotropy in self-gravitating systems. Phys. Rep. 286, 53–130 (1997)

    Article  ADS  MathSciNet  Google Scholar 

  33. Dev, K., Gleiser, M.: Anisotropic stars: exact solutions. Gen. Relativ. Gravit. 34, 1793 (2002)

    Article  MathSciNet  Google Scholar 

  34. Yazadjiev, S.S.: Relativistic models of magnetars: nonperturbative analytical approach. Phys. Rev. D 85(4), 044030 (2012)

    Article  ADS  Google Scholar 

  35. Sawyer, R.F.: Condensed \(\pi ^-\) phase in neutron-star matter. Erratum Phys. Rev. Lett. 29, 823 (1972)

    Article  ADS  Google Scholar 

  36. Maurya, S.K., Al Aamri, A.M., Al Aamri, A.K., et al.: Spherically symmetric anisotropic charged solution under complete geometric deformation approach. Eur. Phys. J. C 81, 701 (2021)

    Article  ADS  Google Scholar 

  37. Sokolov, A.I.: Phase transitions in a superfluid neutron liquid. JETP 79, 1137 (1980)

    Google Scholar 

  38. Kippenhahn, R., Weigert, A.: Steller Structure and Evolution. Springer, Berlin (1990). https://doi.org/10.1007/978-3-642-61523-8

    Book  Google Scholar 

  39. Herrera, L.: Stability of the isotropic pressure condition. Phys. Rev. D 101, 104024 (2020)

    Article  ADS  MathSciNet  Google Scholar 

  40. Sharma, R., Mukherjee, S.: HER X-1: a quark–diquark star? Mod. Phys. Lett. A 16, 1049 (2001)

    Article  ADS  Google Scholar 

  41. Sharma, R., Mukherjee, S., Dey, M., Dey, J.: A general relativistic model for SAX J1808.4-3658. Mod. Phys. Lett. A 17, 827 (2002)

    Article  ADS  Google Scholar 

  42. Sharma, R., Maharaj, S.D.: A class of relativistic stars with a linear equation of state. Mon. Not. R. Astron. Soc. 375, 1265–1268 (2007). https://doi.org/10.1111/j.1365-2966.2006.11355.x

    Article  ADS  Google Scholar 

  43. Karmarkar, S., Mukherjee, S., Sharma, R., Maharaj, S.D.: The role of pressure anisotropy on the maximum mass of cold compact stars. Pramana-J. Phys. 68, 881–889 (2007). https://doi.org/10.1007/s12043-007-0088-3

    Article  ADS  Google Scholar 

  44. Komathiraj, K., Maharaj, S.D.: Tikekar superdense stars in electric fields. J. Math. Phys. 48, 042501 (2007). https://doi.org/10.1063/1.2716204

    Article  ADS  MathSciNet  Google Scholar 

  45. Kumar, J., Gupta, Y.K.: A class of well-behaved generalized charged analogues of Vaidya–Tikekar type fluid sphere in general relativity. Astrophys. Space Sci. 351, 243–250 (2014). https://doi.org/10.1007/s10509-013-1772-z

    Article  ADS  Google Scholar 

  46. Chattopadhyay, P.K., Deb, R., Paul, B.C.: Relativistic solution for a class of static compact charged star in pseudo-spheroidal spacetime. Int. J. Mod. Phys. D 21, 1250071 (2012). https://doi.org/10.1142/S021827181250071X

    Article  ADS  MathSciNet  Google Scholar 

  47. Paul, B.C., Deb, R.: Relativistic solutions of anisotropic compact objects. Astrophys. Space Sci. 354, 421–430 (2014). https://doi.org/10.1007/s10509-014-2097-2

    Article  ADS  Google Scholar 

  48. Sharma, R., Tikekar, R.: Space-time inhomogeneity, anisotropy and gravitational collapse. Gen. Relativ. Gravit. 44, 2503–2520 (2012). https://doi.org/10.1007/s10714-012-1406-8

    Article  ADS  MathSciNet  Google Scholar 

  49. Vaidya, P.C., Patel, L.K.: A spherically symmetric gravitational collapse-field with radiation. Pramana-J. Phys. 46, 341–348 (1996). https://doi.org/10.1007/BF02847008

    Article  ADS  Google Scholar 

  50. Sarwe, S., Tikekar, R.: Non-adiabatic gravitational collapse of a superdense star. Int. J. Mod. Phys. D 19, 1889–1904 (2010)

    Article  ADS  Google Scholar 

  51. Sharma, R., Tikekar, R.: Non-adiabatic radiative collapse of a relativistic star under different initial conditions. Pramana-J. Phys. 79, 501–509 (2012). https://doi.org/10.1007/s12043-012-0323-4

    Article  ADS  Google Scholar 

  52. Maurya, S.K., Maharaj, S.D., Kumar, J., et al.: Effect of pressure anisotropy on Buchdahl-type relativistic compact stars. Gen. Relativ. Gravit. 51, 86 (2019). https://doi.org/10.1007/s10714-019-2570-x

    Article  ADS  MathSciNet  Google Scholar 

  53. Maurya, S.K., et al.: Vaidya–Tikekar type anisotropic fluid model by gravitational decoupling. Phys. Scr. 97, 105002 (2022). https://doi.org/10.1088/1402-4896/ac8d39

    Article  ADS  Google Scholar 

  54. Sharma, R., Dadhich, N., Das, S., et al.: An electromagnetic extension of the Schwarzschild interior solution and the corresponding Buchdahl limit. Eur. Phys. J. C 81, 79 (2021). https://doi.org/10.1140/epjc/s10052-021-08894-3

    Article  ADS  Google Scholar 

  55. Naicker, S., Maharaj, S.D., Brassel, B.P.: Charged fluids in higher order gravity. Eur. Phys. J. C 83, 343 (2023). https://doi.org/10.1140/epjc/s10052-023-11483-1

    Article  ADS  Google Scholar 

  56. Lemaître, G.: L’Univers en expansion. Annales de la Société Scientifique de Bruxelles A 53, 51 (1933)

    ADS  Google Scholar 

  57. Buchdahl, H.A.: General relativistic fluid spheres. Phys. Rev. 116(4), 1027–1034 (1959). https://doi.org/10.1103/PhysRev.116.1027

    Article  ADS  MathSciNet  Google Scholar 

  58. Tangherlini, F.R.: Schwarzschild field inn dimensions and the dimensionality of space problem. Nuovo Cim. 27, 636–651 (1963). https://doi.org/10.1007/BF02784569

    Article  ADS  MathSciNet  Google Scholar 

  59. Finch, M.R., Skea, J.E.F.: A realistic stellar model based on an ansatz of Duorah and Ray. Class. Quantum Grav. 6, 467 (1989)

    Article  ADS  MathSciNet  Google Scholar 

  60. Maharaj, S.D., et al.: A family of Finch and Skea relativistic stars. Int. J. Mod. Phys. D 26, 1750014 (2016)

    Article  ADS  MathSciNet  Google Scholar 

  61. Sharma, R., Ratanpal, B.S.: Relativistic stellar model admitting a quadratic equation of state. Int. J. Mod. Phys. D 13, 1350074 (2013)

    Article  Google Scholar 

  62. Pandya, D.M., et al.: Modified Finch and Skea stellar model compatible with observational data. Astrophys. Space Sci. 356, 285 (2015)

    Article  ADS  Google Scholar 

  63. Bhar, P., et al.: Exact solution of a (2+1)-dimensional anisotropic star in Finch and Skea spacetime. Commun. Theor. Phys. 62, 221 (2014)

    Article  MathSciNet  Google Scholar 

  64. Miller, M.C., et al.: The radius of PSR J0740+6620 from NICER and XMM-Newton data. ApJL 918, L28 (2021). https://doi.org/10.3847/2041-8213/ac089b

    Article  ADS  Google Scholar 

  65. Abbott, B.P., et al.: GW151226: observation of gravitational waves from a 22-solar-mass binary black hole coalescence. Phys. Rev. Lett. 116, 241103 (2016). https://doi.org/10.1103/PhysRevLett.116.241103

    Article  ADS  Google Scholar 

  66. Abbott, B.P., et al.: Binary black hole mergers in the first advanced LIGO observing run. Phys. Rev. X 6, 041015 (2016). https://doi.org/10.1103/PhysRevX.6.041015

    Article  Google Scholar 

  67. Abbott, B.P., et al.: GW170608: observation of a 19-solar-mass binary black hole coalescence. ApJL 851, L35 (2017). https://doi.org/10.3847/2041-8213/aa9f0c

    Article  ADS  Google Scholar 

  68. Abbott, B.P., et al.: GW170814: a three-detector observation of gravitational waves from a binary black hole coalescence. Phys. Rev. Lett. 119, 141101 (2017). https://doi.org/10.1103/PhysRevLett.119.141101

    Article  ADS  Google Scholar 

  69. Abbott, B.P., et al.: GWTC-1: a gravitationalwave transient catalog of compact binary mergers observed by LIGO and virgo during the first and second observing runs. Phys. Rev. X 9, 031040 (2019). https://doi.org/10.1103/PhysRevX.9.031040

    Article  Google Scholar 

  70. Abbott, B.P., et al.: GW170817: observation of gravitational waves from a binary neutron star inspiral. Phys. Rev. Lett. 119, 161101 (2017). https://doi.org/10.1103/PhysRevLett.119.161101

    Article  ADS  Google Scholar 

Download references

Acknowledgements

RS gratefully acknowledges support from the Inter-University Centre for Astronomy and Astrophysics (IUCAA), Pune, India, under its Visiting Research Associateship Programme. We are also thankful to the anonymous referee for useful suggestions.

Funding

Not applicable.

Author information

Author notes

  1. Samstuti Chanda

    Present address: Department of Physics, Islampur College, Islampur, West Bengal, 733202, India

Authors and Affiliations

  1. Department of Physics, IUCAA Centre for Astronomy Research and Development, Cooch Behar Panchanan Barma University, Cooch Behar, West Bengal, 736101, India

    Samstuti Chanda & Ranjan Sharma

Authors
  1. Samstuti Chanda
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ranjan Sharma
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

Samstuti Chanda and Ranjan Sharma have contributed equally to the manuscript.

Corresponding author

Correspondence to Ranjan Sharma.

Ethics declarations

Conflict of interest

The authors declare no conflict of interest.

Ethics approval and consent to participate

All authors have read and agreed to the published version of the manuscript.

Consent for publication

All authors have read and agreed to the published version of the manuscript.

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

Chanda, S., Sharma, R. Compactness bound of Buchdahl–Vaidya–Tikekar anisotropic star in \(D\ge 4\) dimensional spacetime. Gen Relativ Gravit 56, 41 (2024). https://doi.org/10.1007/s10714-024-03231-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10714-024-03231-x

Keywords

Search

Navigation