Witsenhausen's counterexample

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Relativistic images are images of gravitational lensing which result due to light deflections by angles $\hat{α} > 3 π / 2$ .

This term was coined by Virbhadra and Ellis^[1] in the year 2000 and is used by many researchers working in this field^[2] (see also in [a], [b], and [c].) They further defined a term relativistic Einstein rings.^[1]

Relativistic Einstein Rings

Relativistic Einstein rings are ringed shaped images occurring due to light deflection $\hat{α} > 2 π$ when the light source, the lens (the deflector), and the observer are perfectly aligned.

Obviously, relativistic Einstein rings are relativistic images for the case when the source, lens, and observer are aligned. Relativistic images are very much demagnified and are extremely difficult to observe due to their poor magnifications as well as some additional problems.

Effective Deflection Angle

Virbhadra and Ellis^[1] also defined a related term, the effective deflection angle of a relativistic image, which is defined as follows:^[3]

${\hat{α}}^{e f f} = \hat{α} - 2 n π$

where $\hat{α}$ is the Einstein deflection angle. The positive integer $n$ stands for the number of turns a light ray makes around the lens. For primary and secondary images, $n = 0$ .

References

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↑ ^1.0 ^1.1 ^1.2 K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D62, 084003 (2000).

↑

A. Y. Bin-Nun,  Phys. Rev. D 81, 123011 (2010);
O. Yu. Tsupko and G. S. Bisnovatyi-Kogan, Gravitation and Cosmology, 15,  184 (2009);
E. F. Eiroa, G. E. Romero, and D. F. Torres, Phys. Rev. D66, 024010 (2002);
T. Kanti Dey, S. Sen, Mod. Phys. Lett. A23, 953 (2008);
E. F. Eiroa, Phys. Rev. D71,  083010 (2005);
V. Bozza, Phys. Rev.  D67, 103006 (2003);
A. Bhadra, Phys.Rev. D67 (2003) 103009;
K. Sarkar, A. Bhadra, Class. Quant. Grav. 23 (2006) 6101;
Y. Liu, S. Chen, Jiliang Jing, Phys. Rev. D81:124017, 2010;
G. N. Gyulchev, S. S. Yazadjiev, Phys. Rev. D75, 023006,(2007);
S. Chen, J.Jing  Phys. Rev. D80, 024036, (2009);
G. N. Gyulchev, S. S. Yazadjiev, Phys. Rev. D78,  083004, (2008);
A.O. Petters, Mon. Not .Roy. Astron. Soc. 338 (2003) 457;
A. S. Majumdar, N. Mukherjee,     Int. J. Mod. Phys. D14 (2005) 1095;
M. Sereno, Phys.Rev. D69 (2004) 023002;
I. Zh. Stefanov, S. S. Yazadjiev, G. G. Gyulchev, arXiv:1003.1609v2 [gr-qc];
N. Mukherjee, A. S. Majumdar, Gen. Rel. Grav. 39, 583, (2007);
 Clarissa-Marie Claudel, K.S. Virbhadra, and G. F. R. Ellis,  J. Math. Phys. 42, 818-838 (2001)
K. S. Virbhadra and  G. F. R. Ellis,  Phys. Rev. D65, 103004 (2002); K. S. Virbhadra,  Phys. Rev. D79, 083004 (2009;)
T. Ghosh, S. Sengupta, Phys.Rev. D81, 044013,2010;
R. Whisker, Phys.Rev. D71 (2005) 064004;
S. Rahvar, A. Mehrabi, M. Dominik, arXiv:1008.1033v1 [astro-ph.SR];
C.  Leiva, J. Saavedra, J. Villanueva, Mod. Phys. Lett. A24, 1443, (2009);
R. Sosnovskiy, arXiv:0806.4379v4 [physics.gen-ph];
C. R. Keeton, A. O. Petters, Phys. Rev. D73 (2006) 044024;
K. K. Nandi, Yuan-Zhong Zhang, Alexander V. Zakharov, Phys. Rev. D74 (2006) 024020;
M. Sereno, F. De Luca, Phys. Rev. D78, 023008, (2008);
F. De Paolis, A. Geralico, G. Ingrosso, A.A. Nucita, A. Qadir, Astron.Astrophys. 415 (2004) 1;
C. Stornaiolo, Gen.Rel.Grav. 34 (2002) 2089;
C.R. Keeton, A. O. Petters, Phys. Rev. D72 (2005) 104006;
T. Harko, Z.  Kovács, Francisco S. N. Lobo,  arXiv:0908.2874v1 [gr-qc];
M. Sereno, Phys. Rev. D78, 083003,(2008);
M. Sereno, Phys. Rev. D77, 043004,(2008);
S. V. Iyer and A. O. Petters, Gen. Rel. Grav. 39, 1563,(2007);
T. Harko, Z. Kovács, F. S. N. Lobo, Phys. Rev. D80, 044021, (2009);
M. Sereno, Mon. Not. Roy. Astron. Soc. 344, 942 2003;
C. G. Boehmer, T. Harko,  Class. Quant. Grav. 24, 3191,(2007);
A. N. Aliev, P. Talazan, Phys. Rev. D80, 044023 (2009);
V. Perlick, arXiv:0708.0178v1 [gr-qc];
V. Bozza, arXiv:0911.2187v2 [gr-qc].

↑ K. S. Virbhadra, Phys. Rev. D79, 083004 (2009).

[Virbhadra_and_Ellis,_2000-1] 1.0 ^1.1 ^1.2 K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D62, 084003 (2000).

[2] 
A. Y. Bin-Nun, Phys. Rev. D 81, 123011 (2010); O. Yu. Tsupko and G. S. Bisnovatyi-Kogan, Gravitation and Cosmology, 15, 184 (2009); E. F. Eiroa, G. E. Romero, and D. F. Torres, Phys. Rev. D66, 024010 (2002); T. Kanti Dey, S. Sen, Mod. Phys. Lett. A23, 953 (2008); E. F. Eiroa, Phys. Rev. D71, 083010 (2005); V. Bozza, Phys. Rev. D67, 103006 (2003); A. Bhadra, Phys.Rev. D67 (2003) 103009; K. Sarkar, A. Bhadra, Class. Quant. Grav. 23 (2006) 6101; Y. Liu, S. Chen, Jiliang Jing, Phys. Rev. D81:124017, 2010; G. N. Gyulchev, S. S. Yazadjiev, Phys. Rev. D75, 023006,(2007); S. Chen, J.Jing Phys. Rev. D80, 024036, (2009); G. N. Gyulchev, S. S. Yazadjiev, Phys. Rev. D78, 083004, (2008); A.O. Petters, Mon. Not .Roy. Astron. Soc. 338 (2003) 457; A. S. Majumdar, N. Mukherjee, Int. J. Mod. Phys. D14 (2005) 1095; M. Sereno, Phys.Rev. D69 (2004) 023002; I. Zh. Stefanov, S. S. Yazadjiev, G. G. Gyulchev, arXiv:1003.1609v2 [gr-qc]; N. Mukherjee, A. S. Majumdar, Gen. Rel. Grav. 39, 583, (2007); Clarissa-Marie Claudel, K.S. Virbhadra, and G. F. R. Ellis, J. Math. Phys. 42, 818-838 (2001) K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D65, 103004 (2002); K. S. Virbhadra, Phys. Rev. D79, 083004 (2009;) T. Ghosh, S. Sengupta, Phys.Rev. D81, 044013,2010; R. Whisker, Phys.Rev. D71 (2005) 064004; S. Rahvar, A. Mehrabi, M. Dominik, arXiv:1008.1033v1 [astro-ph.SR]; C. Leiva, J. Saavedra, J. Villanueva, Mod. Phys. Lett. A24, 1443, (2009); R. Sosnovskiy, arXiv:0806.4379v4 [physics.gen-ph]; C. R. Keeton, A. O. Petters, Phys. Rev. D73 (2006) 044024; K. K. Nandi, Yuan-Zhong Zhang, Alexander V. Zakharov, Phys. Rev. D74 (2006) 024020; M. Sereno, F. De Luca, Phys. Rev. D78, 023008, (2008); F. De Paolis, A. Geralico, G. Ingrosso, A.A. Nucita, A. Qadir, Astron.Astrophys. 415 (2004) 1; C. Stornaiolo, Gen.Rel.Grav. 34 (2002) 2089; C.R. Keeton, A. O. Petters, Phys. Rev. D72 (2005) 104006; T. Harko, Z. Kovács, Francisco S. N. Lobo, arXiv:0908.2874v1 [gr-qc]; M. Sereno, Phys. Rev. D78, 083003,(2008); M. Sereno, Phys. Rev. D77, 043004,(2008); S. V. Iyer and A. O. Petters, Gen. Rel. Grav. 39, 1563,(2007); T. Harko, Z. Kovács, F. S. N. Lobo, Phys. Rev. D80, 044021, (2009); M. Sereno, Mon. Not. Roy. Astron. Soc. 344, 942 2003; C. G. Boehmer, T. Harko, Class. Quant. Grav. 24, 3191,(2007); A. N. Aliev, P. Talazan, Phys. Rev. D80, 044023 (2009); V. Perlick, arXiv:0708.0178v1 [gr-qc]; V. Bozza, arXiv:0911.2187v2 [gr-qc].

[3] K. S. Virbhadra, Phys. Rev. D79, 083004 (2009).

[1]

[2]

[3]

Witsenhausen's counterexample

Relativistic Einstein Rings

Effective Deflection Angle

References

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