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Shadow of a noncommutative geometry inspired Ayón Beato García black hole

Abstract

We introduce the noncommutative geometry inspired Ayón Beato García black hole metric and study various properties of this metric by which we try to probe the allowed values of the noncommutative parameter \(\vartheta \) under certain conditions. We then construct the shadow (apparent shape) cast by this black hole. We derive the corresponding photon orbits and explore the effects of noncommutative spacetime on them. We then study the effects of noncommutative parameter \(\vartheta \), smeared mass m(r), smeared charge q(r) on the silhouette of the shadow analytically and present the results graphically. We then discuss the deformation which arises in the shape of the shadow under various conditions. Finally, we introduce a plasma background and observe how the shadow behaves in this scenario.

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References

  1. 1.

    Synge, J.L.: Mon. Not. R. Astron. Soc. 131, 463 (1966)

  2. 2.

    Luminet, J.P.: Astron. Astrophys. 75, 228 (1979)

  3. 3.

    Bardeen, J. M.: In: “Black Holes” (Les Astres Occulus), edited by C. Dewitt and B. S. Dewitt (Gordon and Breach, 1973), pp. 215–239

  4. 4.

    Virbhadra, K.S., Ellis, G.F.R.: Phys. Rev. D 62, 084003 (2000)

  5. 5.

    Virbhadra, K.S.: Phys. Rev. D 79, 083004 (2008)

  6. 6.

    Hioki, K., Maeda, K.I.: Phys. Rev. D 80, 024042 (2009)

  7. 7.

    Zakharov, A.F., De Paolis, F., Ingrosso, G., Nucita, A.A.: Astron. Astrophys. 442, 795 (2005)

  8. 8.

    Zakharov, A.F.: Phys. Rev. D 90, 062007 (2014)

  9. 9.

    de Vries, A.: Class. Quantum Gravity 17, 123 (2000)

  10. 10.

    Takahashi, R.: Publ. Astron. Soc. Jpn. 57, 273 (2005)

  11. 11.

    Hioki, K., Miyamoto, U.: Phys. Rev. D 78, 044007 (2008)

  12. 12.

    Abdujabbarov, A., Atmurotov, F., Kucukacka, Y., Ahmedov, B., Camci, U.: Astrophys. Space Sci. 344, 429 (2013)

  13. 13.

    Grenzebach, A., Perlick, V., Lammerzahl, C.: Phys. Rev. D 89, 124004 (2014)

  14. 14.

    Atamurotov, F., Andujabbarov, A., Ahmedov, B.: Phys. Rev. D 88, 064004 (2013)

  15. 15.

    Amarilla, L., Eiroa, E.F.: Phys. Rev. D 85, 064019 (2012)

  16. 16.

    Shipley, J.O., Dolan, S.R.: Class. Quantum Gravity 33, 175001 (2016)

  17. 17.

    Amir, M., Ghosh, S.G.: Phys. Rev. D 94, 024054 (2016)

  18. 18.

    Papnoi, U., Atamurotov, F., Ghosh, S.G., Ahmedov, B.: Phys. Rev. D 90, 024073 (2014)

  19. 19.

    The Event Horizon Telescope. www.eventhorizontelescope.org

  20. 20.

    Doeleman, S.S., et al.: Nature 455, 78 (2008)

  21. 21.

    Doeleman, S.S., et al.: Science 338, 355 (2012)

  22. 22.

    Nicolini, P.: Int. J. Mod. Phys. A 24, 1229 (2009)

  23. 23.

    Banerjee, R., Gangopadhyay, S., Modak, S.K.: Phys. Lett. B 686, 181–187 (2010)

  24. 24.

    Gangopadhyay, S.: J. Phys. Conf. Ser. 405, 012014 (2012)

  25. 25.

    Maggiore, M.: Phys. Lett. B 304, 65 (1993)

  26. 26.

    Sharif, M., Iftikar, S.: Eur. Phys. J. C 76, 630 (2016)

  27. 27.

    Ayón-Beato, E., García, A.: Phys. Rev. Lett. 80, 5056 (1998)

  28. 28.

    Toshmatov, B., Ahmedov, B., Abdujabbarov, A., Stuchlik, Z.: Phys. Rev. D 89, 104017 (2014)

  29. 29.

    Azreg-Aïnou, M.: Phys. Rev. D 90, 064041 (2014)

  30. 30.

    Drake, Szekeres: Gen. Relativ. Gravit. 32, 445 (2000)

  31. 31.

    Spallucci, E., Smailagic, A., Nicolini, P.: Phys. Lett. B 670, 449 (2009)

  32. 32.

    Ahmadjon, A., Amir, M., Ahmedov, B., Ghosh, S.G.: Phys. Rev. D 93, 104004 (2016)

  33. 33.

    Raine, D., Thomas, E.: Black Holes An Introduction, 2nd edn. Imperial College Press, London (2005)

  34. 34.

    Carter, B.: Phys. Rev. 174, 1559 (1968)

  35. 35.

    Chandrasekhar, S.: The Mathematical Theory of Black Holes. Oxford University Press, Oxford (1998)

  36. 36.

    Vazquez, A.E., Esteban, E.P.: Nuovo Cim. B 119, 489 (2004)

  37. 37.

    Synge, J.L.: Relativity: The General Theory. North Holland, Amsterdam (1960)

  38. 38.

    Rogers, A.: Mon. Not. R. Astron. Soc. 451, 4536 (2015)

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Acknowledgements

AS acknowledges the support by Council of Scientific and Industrial Research (CSIR, Govt. of India) for Junior Research Fellowship. SG acknowledges the support by DST-SERB under Start Up Research Grant (Young Scientist), File No. YSS/2014/000180. SG also acknowledges the support of IUCAA, Pune for the Visting Associateship Programme. The authors would like to thank the referees for very useful comments.

Author information

Correspondence to Ashis Saha.

Appendix

Appendix

In this “Appendix”, we present the explicit expressions for \(\xi \) and \(\eta \). These can be obtained by substituting Eq. 31 in Eqs. (28) and (29). The expressions read

$$\begin{aligned} \xi = \frac{\mathcal {X}}{a\mathcal {D}} \qquad \eta = \frac{\mathcal {G}}{\mathcal {D}} \end{aligned}$$
(52)

where

$$\begin{aligned} \mathcal {X}= & {} (r^2+a^2)\left( -\frac{2m^\prime (r)r^4}{(r^2+q(r)^2)^{3/2}} - \frac{4m(r)r^3}{(r^2+q(r)^2)^{3/2}} + \frac{6m(r)r^4}{(r^2+q(r)^2)^{5/2}}(r+q(r)q^\prime (r))\right. \nonumber \\&+\, \frac{2q(r)q^\prime (r)r^4}{(q(r)^2+r^2)^2} + \frac{2q(r)^2r^3}{(q(r)^2+r^2)^2} - (q(r)q^\prime (r)+r)\frac{4q(r)^2r^4}{(q(r)^2+r^2)^3}+2r-\frac{4m(r)r^3}{(r^2+q(r)^2)^\frac{3}{2}}\nonumber \\&\left. +\,\frac{2q(r)^2r^3}{(r^2+q(r)^2)^2}\right) -4r^3 +\frac{8m(r)r^5}{(r^2+q(r)^2)^\frac{3}{2}} -\frac{4q(r)^2r^5}{(r^2+q(r)^2)^2} \end{aligned}$$
(53)
$$\begin{aligned} \mathcal {D}= & {} -\frac{2m^\prime (r)r^4}{(r^2+q(r)^2)^{3/2}} - \frac{4m(r)r^3}{(r^2+q(r)^2)^{3/2}}+\, \frac{6m(r)r^4}{(r^2+q(r)^2)^{5/2}}(r+q(r)q^\prime (r)) + \frac{2q(r)q^\prime (r)r^4}{(q(r)^2+r^2)^2}\nonumber \\&+\, \frac{2q(r)^2r^3}{(q(r)^2+r^2)^2} - (q(r)q^\prime (r)+r)\frac{4q(r)^2r^4}{(q(r)^2+r^2)^3}+\,2r-\frac{4m(r)r^3}{(r^2+q(r)^2)^\frac{3}{2}} +\frac{2q(r)^2r^3}{(r^2+q(r)^2)^2} \nonumber \\ \end{aligned}$$
(54)
$$\begin{aligned} \mathcal {G}= & {} 4r^3+4a^2r+2a\xi \left( -\frac{2m^\prime (r)r^4}{(r^2+q(r)^2)^{3/2}} - \frac{4m(r)r^3}{(r^2+q(r)^2)^{3/2}}\right. +\frac{6m(r)r^4}{(r^2+q(r)^2)^{5/2}}(r+q(r)q^\prime (r)) \nonumber \\&+\, \frac{2q(r)q^\prime (r)r^4}{(q(r)^2+r^2)^2}+\frac{2q(r)^2r^3}{(q(r)^2+r^2)^2} - (q(r)q^\prime (r)+r)\frac{4q(r)^2r^4}{(q(r)^2+r^2)^3}+2r-\frac{4m(r)r^3}{(r^2+q(r)^2)^\frac{3}{2}}\nonumber \\&\left. +\,\frac{2q(r)^2r^3}{(r^2+q(r)^2)^2}-r\right) -(a^2+\xi ^2)\left( -\frac{2m^\prime (r)r^4}{(r^2+q(r)^2)^{3/2}} - \frac{4m(r)r^3}{(r^2+q(r)^2)^{3/2}} \right. \nonumber \\&+\, \frac{6m(r)r^4}{(r^2+q(r)^2)^{5/2}}(r+q(r)q^\prime (r)) + \frac{2q(r)q^\prime (r)r^4}{(q(r)^2+r^2)^2} + \frac{2q(r)^2r^3}{(q(r)^2+r^2)^2}\nonumber \\&\left. -\, (q(r)q^\prime (r)+r)\frac{4q(r)^2r^4}{(q(r)^2+r^2)^3}+2r-\frac{4m(r)r^3}{(r^2+q(r)^2)^\frac{3}{2}}+\frac{2q(r)^2r^3}{(r^2+q(r)^2)^2}\right) .\nonumber \\ \end{aligned}$$
(55)

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Saha, A., Modumudi, S.M. & Gangopadhyay, S. Shadow of a noncommutative geometry inspired Ayón Beato García black hole. Gen Relativ Gravit 50, 103 (2018). https://doi.org/10.1007/s10714-018-2423-z

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Keywords

  • Black hole shadow
  • Noncommutativity
  • Plasma