Witsenhausen's counterexample

2013-11-18T01:15:24Z

135.0.226.162: /* Attempts at obtaining a solution */

{{Multiple issues|orphan = January 2011|cleanup = January 2011}}

'''Relativistic images''' are [[image]]s of [[gravitational lens]]ing which result due to [[light]] deflections by angles <math>\hat{\alpha} > 3 \pi/2</math>.

This term was coined by Virbhadra and Ellis<ref name="Virbhadra and Ellis, 2000">K. S. Virbhadra and G. F. R. Ellis, Phys. Rev. D62, 084003 (2000).
</ref> in the year 2000 and is used by many [[researcher]]s working in this field<ref><pre>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].</pre></ref>
(see also in [http://search.arxiv.org:8081/?query=%22relativistic+images%22&qid=null [a<nowiki>]</nowiki>], [http://www.google.com/search?client=firefox-a&rls=org.mozilla%3Aen-US%3Aofficial&channel=s&hl=en&source=hp&q=%22relativistic+images%22&btnG=Google+Search [b<nowiki>]</nowiki>], and [http://search.yahoo.com/search;_ylt=A0geuz.5ynlMd2MBXD9XNyoA?p=%22relativistic+images%22&fr=sfp&xargs=0&pstart=1&b=1&xa=OsEZ9USraQ7tMnfDVpvwPg--,1283136569 [c<nowiki>]</nowiki>].)
They further defined a term ''relativistic Einstein rings''.<ref name="Virbhadra and Ellis, 2000"/>

==Relativistic Einstein Rings==
Relativistic [[Einstein ring]]s are ringed shaped images occurring due to light deflection <math>\hat{\alpha}>2 \pi</math> when the [[light source]], the lens (the deflector), and the [[Observation|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 [[magnification]]s as well as some additional problems.

==Effective Deflection Angle==
Virbhadra and Ellis<ref name="Virbhadra and Ellis, 2000"/> also defined a related term, the ''effective deflection angle'' of a relativistic image, which is defined as follows:<ref>K. S. Virbhadra, Phys. Rev. D79, 083004 (2009).</ref>

<math>\hat{\alpha}^{eff} = \hat{\alpha} - 2 n \pi</math>

where <math>\hat{\alpha}</math> is the Einstein [[deflection angle]]. The positive integer <math> n </math> stands for the number of turns a [[light ray]] makes around the lens. For primary and secondary images, <math> n = 0 </math>.

==References==
{{Reflist}}

{{DEFAULTSORT:Relativistic Images}}
[[Category:Gravitational lensing]]

formulasearchengine - User contributions [en]

Witsenhausen's counterexample