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Shadows and strong gravitational lensing: a brief review

  • Pedro V. P. Cunha
  • Carlos A. R. Herdeiro
Editor’s Choice (Invited Review: State of the Field)
Part of the following topical collections:
  1. Testing the Kerr spacetime with gravitational-wave and electromagnetic observations

Abstract

For ultra compact objects, light rings and fundamental photon orbits (FPOs) play a pivotal role in the theoretical analysis of strong gravitational lensing effects, and of BH shadows in particular. In this short review, specific models are considered to illustrate how FPOs can be useful in order to understand some non-trivial gravitational lensing effects. This paper aims at briefly overviewing the theoretical foundations of these effects, touching also some of the related phenomenology, both in general relativity and alternative theories of gravity, hopefully providing some intuition and new insights for the underlying physics, which might be critical when testing the Kerr black hole hypothesis.

Keywords

Strong gravitational lensing Black hole shadow 

Notes

Acknowledgements

We would like to thank E. Berti, J. Grover, E. Radu, H. Rúnarsson, A. Wittig for collaboration on some of the work reviewed in this paper. We would also like to thank all the participants in the Gravitational lensing and black hole shadows workshop that took place in Aveiro, Portugal, in November 2016, for many stimulating discussions on these topics. P.C. is supported by Grant No. PD/BD/114071/2015 under the FCT-IDPASC Portugal Ph.D. program. C.H. acknowledges funding from the FCT-IF programme. This work was partially supported by the H2020-MSCA-RISE-2015 Grant No. StronGrHEP-690904, the H2020-MSCA-RISE-2017 Grant No. FunFiCO-777740 and by the CIDMA Project UID/MAT/04106/2013 The authors would like to acknowledge networking support by the COST Action CA16104.

References

  1. 1.
    Eddington, A.: Space, Time and Gravitation. Cambridge University Press, Cambridge (1920)Google Scholar
  2. 2.
    Chwolson, O.: Über eine mögliche Form fiktiver Doppelsterne. Astron. Nachr. 221, 329 (1924)ADSCrossRefGoogle Scholar
  3. 3.
    Einstein, A.: Lens-like action of a star by the deviation of light in the gravitational field. Science 84, 506–507 (1936)ADSzbMATHCrossRefGoogle Scholar
  4. 4.
    Renn, J., Sauer, T., Stachel, J.: The origin of gravitational lensing: a postscript to Einstein’s 1936 science paper. Science 275, 184–186 (1997)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Schmidt, M.: 3C 273: a star-like object with large red-shift. Nature 197, 1040 (1963)ADSCrossRefGoogle Scholar
  6. 6.
    Walsh, D., Carswell, R.F., Weymann, R.J.: 0957 + 561 A, B—twin quasistellar objects or gravitational lens. Nature 279, 381–384 (1979)ADSCrossRefGoogle Scholar
  7. 7.
  8. 8.
    Inada, N., Oguri, M., Pindor, B., Hennawi, J.F., Chiu, K., Zheng, W., Ichikawa, S.-I., Gregg, M.D., Becker, R.H., Suto, Y., Strauss, M.A., Turner, E.L., Keeton, C.R., Annis, J., Castander, F.J., Eisenstein, D.J., Frieman, J.A., Fukugita, M., Gunn, J.E., Johnston, D.E., Kent, S.M., Nichol, R.C., Richards, G.T., Rix, H.-W., Sheldon, E.S., Bahcall, N.A., Brinkmann, J., Ivezić, Ž., Lamb, D.Q., McKay, T.A., Schneider, D.P., York, D.G.: A gravitationally lensed quasar with quadruple images separated by 14.62 arcseconds. Nature 426, 810–812 (2003)ADSCrossRefGoogle Scholar
  9. 9.
    Abbott, B.P., et al.: Observation of gravitational waves from a binary black hole merger. Phys. Rev. Lett. 116(6), 061102 (2016)ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    Abbott, B.P., et al.: GW151226: observation of gravitational waves from a 22-solar-mass binary black hole coalescence. Phys. Rev. Lett. 116(24), 241103 (2016)ADSCrossRefGoogle Scholar
  11. 11.
    Abbott, B.P., et al.: GW170104: observation of a 50-solar-mass binary black hole coalescence at redshift 0.2. Phys. Rev. Lett. 118(22), 221101 (2017)ADSMathSciNetCrossRefGoogle Scholar
  12. 12.
    Abbott, B.P., et al.: GW170814: a three-detector observation of gravitational waves from a binary black hole coalescence. Phys. Rev. Lett. 119(14), 141101 (2017)ADSCrossRefGoogle Scholar
  13. 13.
    Abbott, B.P., et al.: GW170608: observation of a 19-solar-mass binary black hole coalescence. Astrophys. J. 851(2), L35 (2017)ADSCrossRefGoogle Scholar
  14. 14.
    Cardoso, V., Franzin, E., Pani, P.: Is the gravitational-wave ringdown a probe of the event horizon? Phys. Rev. Lett. 116(17), 171101 (2016). [Erratum: Phys. Rev. Lett.117,no.8,089902(2016)]ADSCrossRefGoogle Scholar
  15. 15.
    Penrose, R.: Gravitational collapse and space–time singularities. Phys. Rev. Lett. 14, 57–59 (1965)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    Penrose, R.: Gravitational collapse: the role of general relativity. Riv. Nuovo Cim. 1, 252–276 (1969). [Gen. Relat. Gravit. 34, 1141 (2002)]Google Scholar
  17. 17.
    Cunha, P.V.P., Berti, E., Herdeiro, C.A.R.: Light ring stability in ultra-compact objects. Phys. Rev. Lett. 119(25), 251102 (2017)ADSMathSciNetCrossRefGoogle Scholar
  18. 18.
    Broderick, A.E., Johannsen, T., Loeb, A., Psaltis, D.: Testing the no-hair theorem with event horizon telescope observations of Sagittarius A*. Astrophys. J. 784, 7 (2014)ADSCrossRefGoogle Scholar
  19. 19.
    Bardeen, J.M.: Timelike and null geodesics in the Kerr metric. In: Dewitt, C., Dewitt, B.S. (eds.) Black Holes (Les Astres Occlus), pp. 215–239. Gordon and Breach, New York (1973)Google Scholar
  20. 20.
    Synge, J.L.: The escape of photons from gravitationally intense stars. Mon. Not. R. Astron. Soc. 131(3), 463–466 (1966)ADSCrossRefGoogle Scholar
  21. 21.
    Johannsen, T.: Photon rings around Kerr and Kerr-like black holes. Astrophys. J. 777, 170 (2013)ADSCrossRefGoogle Scholar
  22. 22.
    Riazuelo, A.: Seeing relativity—I. Basics of a raytracing code in a Schwarzschild metric (2015). arXiv:1511.06025
  23. 23.
    Grandclément, P.: Light rings and light points of boson stars. Phys. Rev. D95(8), 084011 (2017)ADSGoogle Scholar
  24. 24.
    Cunha, P.V.P., Grover, J., Herdeiro, C., Radu, E., Runarsson, H., Wittig, A.: Chaotic lensing around boson stars and Kerr black holes with scalar hair. Phys. Rev. D94(10), 104023 (2016)ADSGoogle Scholar
  25. 25.
    Cunha, P.V.P., Font, J.A., Herdeiro, C., Radu, E., Sanchis-Gual, N., Zilhão, M.: Lensing and dynamics of ultracompact bosonic stars. Phys. Rev. D96(10), 104040 (2017)ADSGoogle Scholar
  26. 26.
    Keir, J.: Slowly decaying waves on spherically symmetric spacetimes and ultracompact neutron stars. Class. Quantum Gravity 33(13), 135009 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Carter, B.: Global structure of the Kerr family of gravitational fields. Phys. Rev. 174, 1559–1571 (1968)ADSzbMATHCrossRefGoogle Scholar
  28. 28.
    Teo, E.: Spherical photon orbits around a Kerr black hole. Gen. Relativ. Gravit. 35(11), 1909–1926 (2003)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  29. 29.
    Arnowitt, R., Deser, S., Misner, C.W.: Dynamical structure and definition of energy in general relativity. Phys. Rev. 116, 1322–1330 (1959)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  30. 30.
    Bardeen, J.M., Press, W.H., Teukolsky, S.A.: Rotating black holes: locally nonrotating frames, energy extraction, and scalar synchrotron radiation. Astrophys. J. 178, 347 (1972)ADSCrossRefGoogle Scholar
  31. 31.
    Grover, J., Wittig, A.: Black hole shadows and invariant phase space structures. Phys. Rev. D96(2), 024045 (2017)ADSGoogle Scholar
  32. 32.
    Cunha, P.V.P., Herdeiro, C.A.R., Radu, E.: Fundamental photon orbits: black hole shadows and spacetime instabilities. Phys. Rev. D96(2), 024039 (2017)ADSGoogle Scholar
  33. 33.
    Shipley, J., Dolan, S .R.: Binary black hole shadows, chaotic scattering and the Cantor set. Class. Quantum Gravity 33(17), 175001 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  34. 34.
    Liebling, S.L., Palenzuela, C.: Dynamical boson stars. Living Rev. Relativ. 15, 6 (2012)ADSzbMATHCrossRefGoogle Scholar
  35. 35.
    Kleihaus, B., Kunz, J., List, M.: Rotating boson stars and Q-balls. Phys. Rev. D 72, 064002 (2005)ADSMathSciNetCrossRefGoogle Scholar
  36. 36.
    Schunck, F.E., Mielke, E.W.: Rotating boson star as an effective mass torus in general relativity. Phys. Lett. A 249, 389–394 (1998)ADSCrossRefGoogle Scholar
  37. 37.
    Bick, E., Steffen, F.D. (eds.): Topology and Geometry in Physics. Springer, Berlin (2005)zbMATHGoogle Scholar
  38. 38.
    Naber, G.L.: Topology, Geometry, and Gauge Fields. Springer, New York (2000)zbMATHCrossRefGoogle Scholar
  39. 39.
    Hod, S.: On the number of light rings in curved spacetimes of ultra-compact objects. Phys. Lett. B 776, 1–4 (2018)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Geroch, R.P.: Topology in general relativity. J. Math. Phys. 8, 782–786 (1967)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  41. 41.
    Rubakov, V.A.: The null energy condition and its violation. Phys. Usp. 57, 128–142 (2014). [Usp. Fiz. Nauk 184, no. 2, 137 (2014)]Google Scholar
  42. 42.
    Dolan, S.R., Shipley, J.O.: Stable photon orbits in stationary axisymmetric electrovacuum spacetimes. Phys. Rev. D94(4), 044038 (2016)ADSMathSciNetGoogle Scholar
  43. 43.
    Kerr, R.P.: Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11, 237–238 (1963)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  44. 44.
    Robinson, D.: Four decades of black holes uniqueness theorems. In: Wiltshire, D., Visser, M., Scott, S.M. (eds.) The Kerr Spacetime: Rotating Black Holes in General Relativity. Cambridge University Press, Cambridge (2009)Google Scholar
  45. 45.
    Chrusciel, P.T., Costa, J.L., Heusler, M.: Stationary black holes: uniqueness and beyond. Living Rev. Relativ. 15, 7 (2012)ADSzbMATHCrossRefGoogle Scholar
  46. 46.
    Heusler, M.: Stationary black holes: uniqueness and beyond. Living Rev. Relativ. 1, 6 (1998)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  47. 47.
    Wilkins, D.C.: Bound geodesics in the Kerr metric. Phys. Rev. D 5, 814–822 (1972)ADSCrossRefGoogle Scholar
  48. 48.
    Nickalls, R.W.D.: Viète, descartes and the cubic equation. Math. Gaz. 90(518), 203–208 (2006)CrossRefGoogle Scholar
  49. 49.
    Zakharov, A.F., De Paolis, F., Ingrosso, G., Nucita, A.A.: Measuring the black hole parameters in the Galactic center with RADIOASTRON. New Astron. 10, 479–489 (2005)ADSCrossRefGoogle Scholar
  50. 50.
    Herdeiro, C., Radu, E.: Construction and physical properties of Kerr black holes with scalar hair. Class. Quantum Gravity 32(14), 144001 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  51. 51.
    Herdeiro, C.A.R., Radu, E.: Kerr black holes with scalar hair. Phys. Rev. Lett. 112, 221101 (2014)ADSCrossRefGoogle Scholar
  52. 52.
    Muhleman, D.O., Ekers, R.D., Fomalont, E.B.: Radio interferometric test of the general relativistic light bending near the Sun. Phys. Rev. Lett. 24, 1377–1380 (1970)ADSCrossRefGoogle Scholar
  53. 53.
    Perlick, V.: Ray Optics, Fermat’s Principle, and Applications to General Relativity. Lecture notes in physics monographs. Springer, Berlin (2000)zbMATHGoogle Scholar
  54. 54.
    Tsupko, OYu., Bisnovatyi-Kogan, G.S.: Gravitational lensing in plasma: relativistic images at homogeneous plasma. Phys. Rev. D87(12), 124009 (2013)ADSGoogle Scholar
  55. 55.
    Bisnovatyi-Kogan, G.S., Tsupko, O.Y.: Gravitational lensing in plasmic medium. Plasma Phys. Rep. 41, 562 (2015)ADSCrossRefGoogle Scholar
  56. 56.
    Abdujabbarov, A., Juraev, B., Ahmedov, B., Stuchlík, Z.: Shadow of rotating wormhole in plasma environment. Astrophys. Space Sci. 361(7), 226 (2016)ADSMathSciNetCrossRefGoogle Scholar
  57. 57.
    Abdujabbarov, A., Toshmatov, B., Stuchlík, Z., Ahmedov, B.: Shadow of the rotating black hole with quintessential energy in the presence of plasma. Int. J. Mod. Phys. D26(06), 1750051 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  58. 58.
    Perlick, V., Tsupko, OYu.: Light propagation in a plasma on Kerr spacetime: separation of the Hamilton–Jacobi equation and calculation of the shadow. Phys. Rev. D95(10), 104003 (2017)ADSMathSciNetGoogle Scholar
  59. 59.
    Herdeiro, C., Radu, E., Runarsson, H.: Kerr black holes with Proca hair. Class. Quantum Gravity 33(15), 154001 (2016)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  60. 60.
    East, W.E., Pretorius, F.: Superradiant instability and backreaction of massive vector fields around Kerr black holes. Phys. Rev. Lett. 119(4), 041101 (2017)ADSCrossRefGoogle Scholar
  61. 61.
    Herdeiro, C.A.R., Radu, E.: Kerr black holes with synchronised hair: an analytic model and dynamical formation. Phys. Rev. Lett. 119(26) (2017).  https://doi.org/10.1103/PhysRevLett.119.261101
  62. 62.
    Hod, S.: Stationary scalar clouds around rotating black holes. Phys. Rev. D 86, 104026 (2012)ADSCrossRefGoogle Scholar
  63. 63.
    Hod, S.: Stationary resonances of rapidly-rotating Kerr black holes. Eur. Phys. J. C 73, 2378 (2013)ADSCrossRefGoogle Scholar
  64. 64.
    Perlick, V.: Gravitational lensing from a spacetime perspective. Living Rev. Relativ. 7, 9 (2004)ADSzbMATHCrossRefGoogle Scholar
  65. 65.
    Bohn, A., Throwe, W., Hébert, F., Henriksson, K., Bunandar, D., et al.: What does a binary black hole merger look like? Class. Quantum Gravity 32(6), 065002 (2015)ADSCrossRefGoogle Scholar
  66. 66.
    Cunha, P.V.P., Herdeiro, C.A.R., Radu, E., Runarsson, H.F.: Shadows of Kerr black holes with scalar hair. Phys. Rev. Lett. 115(21), 211102 (2015)ADSzbMATHCrossRefGoogle Scholar
  67. 67.
    Wang, M., Chen, S., Jing, J.: Shadow casted by a Konoplya–Zhidenko rotating non-Kerr black hole. JCAP 1710(10), 051 (2017)ADSMathSciNetCrossRefGoogle Scholar
  68. 68.
    José, J., Saletan, E.: Classical Dynamics: A Contemporary Approach. Cambridge University Press, Cambridge (1998)zbMATHCrossRefGoogle Scholar
  69. 69.
    Brito, R., Cardoso, V., Herdeiro, C.A.R., Radu, E.: Proca stars: gravitating Bose–Einstein condensates of massive spin 1 particles. Phys. Lett. B 752, 291–295 (2016)ADSCrossRefGoogle Scholar
  70. 70.
    Pikel’Ner, S.B.: Book review: Ya. B. Zel’dovich and I. D. Novikov. The theory of gravitation and stellar evolution. Sov. Astron. 17, 562 (1974)ADSGoogle Scholar
  71. 71.
    Chakraborty, S., SenGupta, S.: Strong gravitational lensing—a probe for extra dimensions and Kalb–Ramond field. JCAP 1707(07), 045 (2017)ADSMathSciNetCrossRefGoogle Scholar
  72. 72.
    Cunha, P.V.P., Herdeiro, C.A.R., Kleihaus, B., Kunz, J., Radu, E.: Shadows of Einstein-dilaton–Gauss–Bonnet black holes. Phys. Lett. B 768, 373–379 (2017)ADSzbMATHCrossRefGoogle Scholar
  73. 73.
    Ostrogradsky, M.: Mémoires sur les équations différentielles, relatives au problème des isopérimètres. Mem. Acad. St. Petersbourg 6(4), 385–517 (1850)Google Scholar
  74. 74.
    Lovelock, D.: The Einstein tensor and its generalizations. J. Math. Phys. 12, 498–501 (1971)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  75. 75.
    Zwiebach, B.: Curvature squared terms and string theories. Phys. Lett. 156B, 315–317 (1985)ADSCrossRefGoogle Scholar
  76. 76.
    Kanti, P., Mavromatos, N.E., Rizos, J., Tamvakis, K., Winstanley, E.: Dilatonic black holes in higher curvature string gravity. Phys. Rev. D 54, 5049–5058 (1996)ADSMathSciNetCrossRefGoogle Scholar
  77. 77.
    Kanti, P., Mavromatos, N.E., Rizos, J., Tamvakis, K., Winstanley, E.: Dilatonic black holes in higher curvature string gravity. 2: linear stability. Phys. Rev. D 57, 6255–6264 (1998)ADSMathSciNetCrossRefGoogle Scholar
  78. 78.
    Torii, T., Yajima, H., Maeda, K.-I.: Dilatonic black holes with Gauss–Bonnet term. Phys. Rev. D 55, 739–753 (1997)ADSMathSciNetCrossRefGoogle Scholar
  79. 79.
    Alexeev, S.O., Pomazanov, M.V.: Black hole solutions with dilatonic hair in higher curvature gravity. Phys. Rev. D 55, 2110–2118 (1997)ADSCrossRefGoogle Scholar
  80. 80.
    Melis, M., Mignemi, S.: Global properties of charged dilatonic Gauss–Bonnet black holes. Phys. Rev. D 73, 083010 (2006)ADSMathSciNetCrossRefGoogle Scholar
  81. 81.
    Chen, C.-M., Gal’tsov, D.V., Orlov, D.G.: Extremal black holes in D = 4 Gauss–Bonnet gravity. Phys. Rev. D 75, 084030 (2007)ADSMathSciNetCrossRefGoogle Scholar
  82. 82.
    Chen, C.-M., Gal’tsov, D.V., Orlov, D.G.: Extremal dyonic black holes in D = 4 Gauss–Bonnet gravity. Phys. Rev. D 78, 104013 (2008)ADSMathSciNetCrossRefGoogle Scholar
  83. 83.
    Kleihaus, B., Kunz, J., Radu, E.: Rotating black holes in dilatonic Einstein–Gauss–Bonnet theory. Phys. Rev. Lett. 106, 151104 (2011)ADSCrossRefGoogle Scholar
  84. 84.
    Kleihaus, B., Kunz, J., Mojica, S., Radu, E.: Spinning black holes in Einstein–Gauss–Bonnet-dilaton theory: nonperturbative solutions. Phys. Rev. D93(4), 044047 (2016)ADSMathSciNetGoogle Scholar
  85. 85.
    Pani, P., Cardoso, V.: Are black holes in alternative theories serious astrophysical candidates? The case for Einstein-dilaton–Gauss–Bonnet black holes. Phys. Rev. D 79, 084031 (2009)ADSCrossRefGoogle Scholar
  86. 86.
    Pani, P., Macedo, C.F.B., Crispino, L.C.B., Cardoso, V.: Slowly rotating black holes in alternative theories of gravity. Phys. Rev. D 84, 087501 (2011)ADSCrossRefGoogle Scholar
  87. 87.
    Ayzenberg, D., Yunes, N.: Slowly-rotating black holes in Einstein-dilaton–Gauss–Bonnet gravity: quadratic order in spin solutions. Phys. Rev. D 90, 044066 (2014). [Erratum: Phys. Rev.D91,no.6,069905 (2015)] ADSCrossRefGoogle Scholar
  88. 88.
    Maselli, A., Pani, P., Gualtieri, L., Ferrari, V.: Rotating black holes in Einstein-dilaton–Gauss–Bonnet gravity with finite coupling. Phys. Rev. D92(8), 083014 (2015)ADSMathSciNetGoogle Scholar
  89. 89.
    Herdeiro, C.A.R., Radu, E.: Asymptotically flat black holes with scalar hair: a review. Int. J. Mod. Phys. D24(09), 1542014 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  90. 90.
    Amarilla, L., Eiroa, E.F.: Shadow of a rotating braneworld black hole. Phys. Rev. D 85, 064019 (2012)ADSCrossRefGoogle Scholar
  91. 91.
    Yumoto, A., Nitta, D., Chiba, T., Sugiyama, N.: Shadows of multi-black holes: analytic exploration. Phys. Rev. D 86, 103001 (2012)ADSCrossRefGoogle Scholar
  92. 92.
    Abdujabbarov, A., Atamurotov, F., Kucukakca, Y., Ahmedov, B., Camci, U.: Shadow of Kerr–Taub–NUT black hole. Astrophys. Space Sci. 344, 429–435 (2013)ADSCrossRefGoogle Scholar
  93. 93.
    Amarilla, L., Eiroa, E.F.: Shadow of a Kaluza–Klein rotating dilaton black hole. Phys. Rev. D87(4), 044057 (2013)ADSGoogle Scholar
  94. 94.
    Nedkova, P.G., Tinchev, V.K., Yazadjiev, S.S.: Shadow of a rotating traversable wormhole. Phys. Rev. D88(12), 124019 (2013)ADSGoogle Scholar
  95. 95.
    Atamurotov, F., Abdujabbarov, A., Ahmedov, B.: Shadow of rotating Hor̃ava–Lifshitz black hole. Astrophys. Space Sci. 348, 179–188 (2013)ADSCrossRefGoogle Scholar
  96. 96.
    Atamurotov, F., Abdujabbarov, A., Ahmedov, B.: Shadow of rotating non-Kerr black hole. Phys. Rev. D88(6), 064004 (2013)ADSGoogle Scholar
  97. 97.
    Li, Z., Bambi, C.: Measuring the Kerr spin parameter of regular black holes from their shadow. JCAP 1401, 041 (2014)ADSCrossRefGoogle Scholar
  98. 98.
    Tinchev, V.K., Yazadjiev, S.S.: Possible imprints of cosmic strings in the shadows of galactic black holes. Int. J. Mod. Phys. D 23, 1450060 (2014)ADSzbMATHCrossRefGoogle Scholar
  99. 99.
    Wei, S.-W., Liu, Y.-X.: Observing the shadow of Einstein–Maxwell-dilaton-axion black hole. JCAP 1311, 063 (2013)ADSCrossRefGoogle Scholar
  100. 100.
    Tsukamoto, N., Li, Z., Bambi, C.: Constraining the spin and the deformation parameters from the black hole shadow. JCAP 1406, 043 (2014)ADSCrossRefGoogle Scholar
  101. 101.
    Grenzebach, A., Perlick, V., Lammerzahl, C.: Photon regions and shadows of Kerr–Newman–NUT black holes with a cosmological constant. Phys. Rev. D89(12), 124004 (2014)ADSGoogle Scholar
  102. 102.
    Lu, R.-S., Broderick, A.E., Baron, F., Monnier, J.D., Fish, V.L., Doeleman, S.S., Pankratius, V.: Imaging the supermassive black hole shadow and jet base of M87 with the event horizon telescope. Astrophys. J. 788, 120 (2014)ADSCrossRefGoogle Scholar
  103. 103.
    Papnoi, U., Atamurotov, F., Ghosh, S.G., Ahmedov, B.: Shadow of five-dimensional rotating Myers–Perry black hole. Phys. Rev. D90(2), 024073 (2014)ADSGoogle Scholar
  104. 104.
    Sakai, N., Saida, H., Tamaki, T.: Gravastar shadows. Phys. Rev. D90(10), 104013 (2014)ADSGoogle Scholar
  105. 105.
    Psaltis, D., Ozel, F., Chan, C.-K., Marrone, D.P.: A general relativistic null hypothesis test with event horizon telescope observations of the black-hole shadow in Sgr A*. Astrophys. J. 814(2), 115 (2015)ADSCrossRefGoogle Scholar
  106. 106.
    Wei, S.-W., Cheng, P., Zhong, Y., Zhou, X.-N.: Shadow of noncommutative geometry inspired black hole. JCAP 1508(08), 004 (2015)ADSMathSciNetCrossRefGoogle Scholar
  107. 107.
    Abdolrahimi, S., Mann, R.B., Tzounis, C.: Distorted local shadows. Phys. Rev. D91(8), 084052 (2015)ADSMathSciNetGoogle Scholar
  108. 108.
    Moffat, J.W.: Modified gravity black holes and their observable shadows. Eur. Phys. J. C75(3), 130 (2015)ADSCrossRefGoogle Scholar
  109. 109.
    Grenzebach, A.: Aberrational effects for shadows of black holes. Fund. Theor. Phys. 179, 823–832 (2015)MathSciNetzbMATHGoogle Scholar
  110. 110.
    Vincent, F .H., Meliani, Z., Grandclement, P., Gourgoulhon, E., Straub, O.: Imaging a boson star at the Galactic center. Class. Quantum Gravity 33(10), 105015 (2016)ADSCrossRefGoogle Scholar
  111. 111.
    Grenzebach, A., Perlick, V., Lammerzahl, C.: Photon regions and shadows of accelerated black holes. Int. J. Mod. Phys. D 24, 1542024 (2015)ADSMathSciNetzbMATHCrossRefGoogle Scholar
  112. 112.
    Abdujabbarov, A.A., Rezzolla, L., Ahmedov, B.J.: A coordinate-independent characterization of a black hole shadow. Mon. Not. R. Astron. Soc. 454(3), 2423–2435 (2015)ADSCrossRefGoogle Scholar
  113. 113.
    Ortiz, N., Sarbach, O., Zannias, T.: Shadow of a naked singularity. Phys. Rev. D92(4), 044035 (2015)ADSGoogle Scholar
  114. 114.
    Ghasemi-Nodehi, M., Li, Z., Bambi, C.: Shadows of CPR black holes and tests of the Kerr metric. Eur. Phys. J. C 75, 315 (2015)ADSCrossRefGoogle Scholar
  115. 115.
    Ohgami, T., Sakai, N.: Wormhole shadows. Phys. Rev. D91(12), 124020 (2015)ADSMathSciNetGoogle Scholar
  116. 116.
    Atamurotov, F., Ghosh, S.G., Ahmedov, B.: Horizon structure of rotating Einstein–Born–Infeld black holes and shadow. Eur. Phys. J. C76(5), 273 (2016)ADSCrossRefGoogle Scholar
  117. 117.
    Perlick, V., Tsupko, OYu., Bisnovatyi-Kogan, G.S.: Influence of a plasma on the shadow of a spherically symmetric black hole. Phys. Rev. D92(10), 104031 (2015)ADSMathSciNetGoogle Scholar
  118. 118.
    Bambi, C.: Testing the Kerr paradigm with the black hole shadow. In: 14th Marcel Grossmann Meeting on General Relativity on Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories (MG14) Rome, Italy, July 12–18 (2015)Google Scholar
  119. 119.
    Atamurotov, F., Ahmedov, B.: Optical properties of black hole in the presence of plasma: shadow. Phys. Rev. D 92, 084005 (2015)ADSCrossRefGoogle Scholar
  120. 120.
    Yang, L., Li, Z.: Shadow of a dressed black hole and determination of spin and viewing angle. Int. J. Mod. Phys. D25(02), 1650026 (2015)ADSCrossRefGoogle Scholar
  121. 121.
    Tinchev, V.K.: The shadow of generalized Kerr black holes with exotic matter. Chin. J. Phys. 53, 110113 (2015)MathSciNetGoogle Scholar
  122. 122.
    Amir, M., Ghosh, S.G.: Shapes of rotating nonsingular black hole shadows. Phys. Rev. D94(2), 024054 (2016)ADSMathSciNetGoogle Scholar
  123. 123.
    Johannsen, T., Broderick, A.E., Plewa, P.M., Chatzopoulos, S., Doeleman, S.S., Eisenhauer, F., Fish, V.L., Genzel, R., Gerhard, O., Johnson, M.D.: Testing general relativity with the shadow size of Sgr A*. Phys. Rev. Lett. 116(3), 031101 (2016)ADSCrossRefGoogle Scholar
  124. 124.
    Abdujabbarov, A., Amir, M., Ahmedov, B., Ghosh, S.G.: Shadow of rotating regular black holes. Phys. Rev. D93(10), 104004 (2016)ADSMathSciNetGoogle Scholar
  125. 125.
    Cunha, P.V.P., Herdeiro, C.A.R., Radu, E., Runarsson, H.F.: Shadows of Kerr black holes with and without scalar hair. Int. J. Mod. Phys. D 25(9) (2016).  https://doi.org/10.1142/S0218271816410212
  126. 126.
    Huang, Y., Chen, S., Jing, J.: Double shadow of a regular phantom black hole as photons couple to Weyl tensor. Eur. Phys. J. C 76, 594 (2016)ADSCrossRefGoogle Scholar
  127. 127.
    Dastan, S., Saffari, R., Soroushfar, S.: Shadow of a charged rotating black hole in \(f(R)\) gravity (2016). arXiv:1606.06994
  128. 128.
    Younsi, Z., Zhidenko, A., Rezzolla, L., Konoplya, R., Mizuno, Y.: A new method for shadow calculations: application to parameterised axisymmetric black holes. Phys. Rev. D 94, 084025 (2016)ADSMathSciNetCrossRefGoogle Scholar
  129. 129.
    Ohgami, T., Sakai, N.: Wormhole shadows in rotating dust. Phys. Rev. D94(6), 064071 (2016)ADSMathSciNetGoogle Scholar
  130. 130.
    Mureika, J.R., Varieschi, G.U.: Black hole shadows in fourth-order conformal Weyl gravity. Can. J. Phys. 95, 1299 (2016)ADSCrossRefGoogle Scholar
  131. 131.
    Sharif, M., Iftikhar, S.: Shadow of a charged rotating non-commutative black hole. Eur. Phys. J. C76(11), 630 (2016)ADSCrossRefGoogle Scholar
  132. 132.
    Tsupko, OYu.: Analytical calculation of black hole spin using deformation of the shadow. Phys. Rev. D95(10), 104058 (2017)ADSMathSciNetGoogle Scholar
  133. 133.
    Bisnovatyi-Kogan, G., Tsupko, O.: Gravitational lensing in presence of plasma: strong lens systems, black hole lensing and shadow. Universe 3(3), 57 (2017)ADSCrossRefGoogle Scholar
  134. 134.
    Amir, M., Singh, B.P., Ghosh, S.G.: Shadows of rotating five-dimensional EMCS black holes (2017). arXiv:1707.09521
  135. 135.
    Alhamzawi, A.: Observing the shadow of modified gravity black hole. Int. J. Mod. Phys. D26(14), 1750156 (2017)ADSMathSciNetCrossRefGoogle Scholar
  136. 136.
    Tsukamoto, N.: Black hole shadow in an asymptotically-flat, stationary, and axisymmetric spacetime: the Kerr–Newman and rotating regular black holes (2017). arXiv:1708.07427
  137. 137.
    Mars, M., Paganini, C .F., Oancea, M .A.: The fingerprints of black holes-shadows and their degeneracies. Class. Quantum Gravity 35(2), 025005 (2018)ADSzbMATHMathSciNetCrossRefGoogle Scholar
  138. 138.
    Wang, M., Chen, S., Jing, J.: Shadows of a compact object with magnetic dipole by chaotic lensing (2017). arXiv:1710.07172
  139. 139.
    Singh, B.P.: Rotating charge black holes shadow in quintessence (2017). arXiv:1711.02898
  140. 140.
    Eiroa, E.F., Sendra, C.M.: Shadow cast by rotating braneworld black holes with a cosmological constant. Eur. Phys. J. C78(2), 91 (2018)ADSCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Departamento de FísicaUniversidade de Aveiro and CIDMAAveiroPortugal
  2. 2.Centro de Astrofísica e Gravitação - CENTRA, Departamento de Física, Instituto Superior Técnico - ISTUniversidade de Lisboa - ULLisboaPortugal

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