View source for Astronomy:Gravitational lens

{{Short description|Light bending by mass between source and observer}}
{{Gravitational lensing}}
{{General relativity sidebar |phenomena}}
[[File:Imageedit 6 2578820362.gif|thumb|231x231px|A light source passes behind a gravitational lens (invisible point mass placed in the center of the image). The aqua circle is the light source as it would be seen if there were no lens, while white spots are the multiple images of the source (see Einstein ring).]]

A '''gravitational lens''' is matter, such as a [[Astronomy:Galaxy cluster|cluster of galaxies]] or a [[Physics:Point particle|point particle]], that bends light from a distant source as it travels toward an observer. The amount of '''gravitational lensing''' is described by [[Biography:Albert Einstein|Albert Einstein]]'s [[Physics:General relativity|general theory of relativity]]<ref name="NYT-20150305-video">{{cite news |last1=Drakeford |first1=Jason |last2=Corum |first2=Jonathan |last3=Overbye |first3=Dennis |title=Einstein's Telescope - video (02:32) |url=https://www.nytimes.com/video/science/100000003552687/out-there-einsteins-telescope.html |date=March 5, 2015 |work={{wipe|New York Times}} |access-date=December 27, 2015 }}</ref><ref name="NYT-20150305">{{cite news |last=Overbye |first=Dennis |title=Astronomers Observe Supernova and Find They're Watching Reruns |url=https://www.nytimes.com/2015/03/06/science/astronomers-observe-supernova-and-find-theyre-watching-reruns.html |date=March 5, 2015 |work={{wipe|New York Times}} |access-date=March 5, 2015 }}</ref> with much greater accuracy than  Newtonian physics, which treats light as [[Physics:Corpuscular theory of light|corpuscles]] travelling at the [[Physics:Speed of light|speed of light]].<ref>{{cite book |title=A First Course in General Relativity |author1=Bernard F. Schutz |edition=illustrated, herdruk |publisher=Cambridge University Press |year=1985 |isbn=978-0-521-27703-7 |page=295 |url=https://books.google.com/books?id=qhDFuWbLlgQC}}</ref><ref>{{cite book |title=Relativity: Special, General, and Cosmological |author1=Wolfgang Rindler |edition=2nd |publisher=OUP Oxford |year=2006 |isbn=978-0-19-152433-2 |page=21 |url=https://books.google.com/books?id=UQpREAAAQBAJ}} [https://books.google.com/books?id=UQpREAAAQBAJ&pg=PA21 Extract of page 21]</ref><ref>{{cite book |title=General Relativity And Relativistic Astrophysics - Proceedings Of The 4th Canadian Conference |author1=Gabor Kunstatter |author2=Jeffrey G Williams |author3=D E Vincent |edition= |publisher=World Scientific |year=1992 |isbn=978-981-4554-87-9 |page=100 |url=https://books.google.com/books?id=ukJBDwAAQBAJ}} [https://books.google.com/books?id=ukJBDwAAQBAJ&pg=PA100 Extract of page 100]</ref><ref>{{cite book |title=The Evolving Universe and the Origin of Life: The Search for Our Cosmic Roots |author1=Pekka Teerikorpi |author2=Mauri Valtonen |author3=K. Lehto |author4=Harry Lehto |author5=Gene Byrd |author6=Arthur Chernin |edition=illustrated |publisher=Springer Science & Business Media |year=2008 |isbn=978-0-387-09534-9 |page=165 |url=https://books.google.com/books?id=sNmtKZnUg-sC}} [https://books.google.com/books?id=sNmtKZnUg-sC&pg=PA165 Extract of page 165]</ref>

[[Biography:Orest Khvolson|Orest Khvolson]] (1924)<ref name="Early History of Gravitational Lensing">{{cite web|url=http://www.nd.edu/~hps/turner.pdf|title=The Early History of Gravitational Lensing|first=Christina|last=Turner|date=February 14, 2006|url-status=dead|archive-url=https://web.archive.org/web/20080725085534/http://www.nd.edu/%7Ehps/turner.pdf|archive-date=July 25, 2008}}</ref> and Frantisek Link (1936)<ref>{{cite book |title=General Relativity, Cosmology and Astrophysics: Perspectives 100 years after Einstein's stay in Prague |edition=illustrated |first1=Jiří |last1=Bičák |first2=Tomáš |last2=Ledvinka |publisher=Springer |year=2014 |isbn=9783319063492 |pages=49–50 |url=https://books.google.com/books?id=Ax0qBAAAQBAJ&pg=PA49}}</ref> are generally credited with being the first to discuss the effect in print, but it is more commonly associated with Einstein, who made unpublished calculations on it in 1912<ref name="origin of the idea">{{Cite journal |author1=Tilman Sauer |year=2008 |title=Nova Geminorum 1912 and the Origin of the Idea of Gravitational Lensing |journal=Archive for History of Exact Sciences |volume=62 |issue=1 |pages=1–22 |arxiv=0704.0963 |bibcode=2008AHES...62....1S |doi=10.1007/s00407-007-0008-4 |s2cid=17384823}}</ref> and published an article on the subject in 1936.<ref name=":0" />

In 1937, [[Biography:Fritz Zwicky|Fritz Zwicky]] posited that galaxy clusters could act as gravitational lenses, a claim confirmed in 1979 by observation of the Twin QSO SBS 0957+561.

==Description==
[[File:Artist's impression of gravitational lensing of a distant merger.ogv|thumb|Gravitational lensing – intervening galaxy modifies appearance of a galaxy far behind it (video; artist's concept).]]
[[File:Gravitational lensing of distant star-forming galaxies (schematic) -vid-.webm|thumb|left|This schematic image shows how light from a distant galaxy is distorted by the gravitational effects of a foreground galaxy, which acts like a lens and makes the distant source appear distorted, but magnified, forming characteristic rings of light, known as Einstein rings.]]
[[File:Gravitational lensing of distant star-forming galaxies (schematic) 2.webm|thumb|An analysis of the distortion of SDP.81 caused by this effect has revealed star-forming clumps of matter.]]

Unlike an [[Physics:Lens (optics)|optical lens]], a point-like gravitational lens produces a maximum deflection of light that passes closest to its center, and a minimum deflection of light that travels furthest from its center. Consequently, a gravitational lens has no single [[Physics:Focus (optics)|focal point]], but a focal line.  The term "lens" in the context of gravitational light deflection was first used by O.J. Lodge, who remarked that it is "not permissible to say that the solar gravitational field acts like a lens, for it has no focal length".<ref>{{Cite journal |last=Lodge |first=Oliver J. |date=December 1919 |title=Gravitation and Light |journal=Nature |language=en |volume=104 |issue=2614 |pages=354 |doi=10.1038/104354a0 |bibcode=1919Natur.104..354L |s2cid=4157815 |issn=0028-0836|doi-access=free }}</ref> If the (light) source, the massive lensing object, and the observer lie in a straight line, the original light source will appear as a ring around the massive lensing object (provided the lens has circular symmetry). If there is any misalignment, the observer will see an arc segment instead. 

This phenomenon was first mentioned in 1924 by the St. Petersburg physicist [[Biography:Orest Khvolson|Orest Khvolson]],<ref>{{cite web| url = http://www.abc.net.au/science/k2/moments/gmis9737.htm| title = Gravity Lens – Part 2 (Great Moments in Science, ABS Science)| website = Australian Broadcasting Corporation| date = 5 November 2001}}</ref> and quantified by [[Biography:Albert Einstein|Albert Einstein]] in 1936. It is usually referred to in the literature as an [[Physics:Einstein ring|Einstein ring]], since Khvolson did not concern himself with the flux or radius of the ring image. More commonly, where the lensing mass is complex (such as a [[Astronomy:Galaxy group|galaxy group]] or [[Astronomy:Galaxy cluster|cluster]]) and does not cause a spherical distortion of spacetime, the source will resemble partial arcs scattered around the lens. The observer may then see multiple distorted images of the same source; the number and shape of these depending upon the relative positions of the source, lens, and observer, and the shape of the gravitational well of the lensing object.

There are three classes of gravitational lensing:<ref name="Schneider, Peter; Ehlers, Jürgen; Falco, Emilio E. 1992">{{cite book |author1=Schneider, Peter |title=Gravitational Lenses |author2=Ehlers, Jürgen |author3=Falco, Emilio E. |date=1992 |publisher=Springer-Verlag Berlin Heidelberg New York Press |isbn=978-3-540-97070-5}}</ref>{{rp|399-401}}<ref>{{cite book | title = The Galactic Supermassive Black Hole | author = Melia, Fulvio |  publisher = Princeton University Press | date = 2007 | isbn = 978-0-691-13129-0 | pages = 255–256}}</ref>

; [[Physics:Strong gravitational lensing|Strong lensing]]: Where there are easily visible distortions such as the formation of [[Physics:Einstein ring|Einstein ring]]s, arcs, and multiple images. Despite being considered "strong", the effect is in general relatively small, such that even a galaxy with a mass more than 100 billion times [[Astronomy:Solar mass|that of the Sun]] will produce multiple images separated by only a few [[Arcsecond|arcsecond]]s. [[Astronomy:Galaxy cluster|Galaxy cluster]]s can produce separations of several arcminutes. In both cases the galaxies and sources are quite distant, many hundreds of megaparsecs away from our Galaxy.
; [[Physics:Weak gravitational lensing|Weak lensing]]: Where the distortions of background sources are much smaller and can only be detected by analyzing large numbers of sources in a statistical way to find coherent distortions of only a few percent. The lensing shows up statistically as a preferred stretching of the background objects perpendicular to the direction to the centre of the lens. By measuring the shapes and orientations of large numbers of distant galaxies, their orientations can be averaged to measure the [[Shear matrix|shear]] of the lensing field in any region. This, in turn, can be used to reconstruct the mass distribution in the area: in particular, the background distribution of [[Astronomy:Dark matter|dark matter]] can be reconstructed. Since galaxies are intrinsically elliptical and the weak gravitational lensing signal is small, a very large number of galaxies must be used in these surveys. These weak lensing surveys must carefully avoid a number of important sources of systematic error: the intrinsic shape of galaxies, the tendency of a camera's [[Physics:Point spread function|point spread function]] to distort the shape of a galaxy and the tendency of atmospheric seeing to distort images must be understood and carefully accounted for. The results of these surveys are important for cosmological parameter estimation, to better understand and improve upon the [[Astronomy:Lambda-CDM model|Lambda-CDM model]], and to provide a consistency check on other cosmological observations. They may also provide an important future constraint on [[Astronomy:Dark energy|dark energy]].
; [[Physics:Gravitational microlensing|Microlensing]]: Where no distortion in shape can be seen but the amount of light received from a background object changes in time. The lensing object may be stars in the Milky Way in one typical case, with the background source being stars in a remote galaxy, or, in another case, an even more distant [[Astronomy:Quasar|quasar]]. In extreme cases, a star in a distant galaxy can act as a microlens and magnify another star much farther away. The first example of this was the star [[Astronomy:MACS J1149 Lensed Star 1|MACS J1149 Lensed Star 1]] (also known as Icarus), thanks to the boost in flux due to the microlensing effect. 

Gravitational lenses act equally on all kinds of [[Physics:Electromagnetic radiation|electromagnetic radiation]], not just visible light, and also in non-electromagnetic radiation, like gravitational waves. Weak lensing effects are being studied for the [[Astronomy:Cosmic microwave background|cosmic microwave background]] as well as galaxy surveys. Strong lenses have been observed in [[Social:Radio|radio]] and [[Physics:X-ray|x-ray]] regimes as well. If a strong lens produces multiple images, there will be a relative time delay between two paths: that is, in one image the lensed object will be observed before the other image.

==History==
[[File:1919 eclipse negative.jpg|right|thumb|150px|One of Eddington's photographs of the 1919 solar eclipse experiment, presented in his 1920 paper announcing its success]]

[[Biography:Henry Cavendish|Henry Cavendish]] in 1784 (in an unpublished manuscript) and [[Biography:Johann Georg von Soldner|Johann Georg von Soldner]] in 1801 (published in 1804) had pointed out that Newtonian gravity predicts that starlight will bend around a massive object<ref>{{Cite journal |author=Soldner, J. G. V. |date=1804 |title=On the deflection of a light ray from its rectilinear motion, by the attraction of a celestial body at which it nearly passes by |journal=Berliner Astronomisches Jahrbuch |pages =161–172|title-link=s:Translation:On the Deflection of a Light Ray from its Rectilinear Motion }}</ref> as had already been supposed by [[Biography:Isaac Newton|Isaac Newton]] in 1704 in his ''Queries'' No.1 in his book ''[[Physics:Opticks|Opticks]]''.<ref name=Opticks>{{cite book|last=Newton|first=Isaac|title=Opticks: or, a treatise of the reflexions, refractions, inflexions and colours of light. Also two treatises of the species and magnitude of curvilinear figures.|date=1998|publisher=Octavo|location=Palo Alto, Calif.|isbn=978-1-891788-04-8|edition=Octavo |others=Commentary by Nicholas Humez}} (''Opticks'' was originally published in 1704).</ref> The same value as Soldner's was calculated by Einstein in 1911 based on the [[Astronomy:Equivalence principle|equivalence principle]] alone.<ref name="Schneider, Peter; Ehlers, Jürgen; Falco, Emilio E. 1992"/>{{rp|3}} However, Einstein noted in 1915, in the process of completing general relativity, that his (and thus Soldner's) 1911-result is only half of the correct value. Einstein became the first to calculate the correct value for light bending.<ref>{{Cite journal |author=Will, C.M.|date=2006 |title=The Confrontation between General Relativity and Experiment |journal=Living Reviews in Relativity |volume =9 |issue=1 |page=39 |doi=10.12942/lrr-2006-3|arxiv = gr-qc/0510072 |bibcode = 2006LRR.....9....3W |pmid=28179873 |pmc=5256066 }}</ref>

The first observation of light deflection was performed by noting the change in position of [[Astronomy:Star|star]]s as they passed near the Sun on the [[Celestial sphere|celestial sphere]]. The observations were performed in 1919 by Arthur Eddington, Frank Watson Dyson, and their collaborators during the total [[Earth:Solar eclipse|solar eclipse]] on [[Astronomy:Solar eclipse of May 29, 1919|May 29]].<ref>{{cite journal| last = Dyson| first = F. W.| title = A determination of the deflection of light by the Sun's gravitational field, from observations made at the total eclipse of 29 May 1919| journal = Philosophical Transactions of the Royal Society | volume = 220A | issue = 571–581| pages = 291–333| date = 1920| author2 = Eddington, A. S. | author3 = Davidson C. | doi=10.1098/rsta.1920.0009 | bibcode=1920RSPTA.220..291D| url = https://zenodo.org/record/1432106| doi-access = free}}</ref> The solar eclipse allowed the stars near the Sun to be observed. Observations were made simultaneously in the cities of Sobral, Ceará, Brazil and in São Tomé and Príncipe on the west coast of Africa.<ref>{{cite journal| last = Stanley| first = Matthew| title = 'An Expedition to Heal the Wounds of War': The 1919 Eclipse and Eddington as Quaker Adventurer| journal = Isis | volume = 94| issue = 1| pages = 57–89| date = 2003| doi = 10.1086/376099| pmid = 12725104| bibcode = 2003Isis...94...57S| s2cid = 25615643}}</ref> The observations demonstrated that the light from [[Astronomy:Star|star]]s passing close to the [[Astronomy:Sun|Sun]] was slightly bent, so that stars appeared slightly out of position.<ref>{{cite journal|last=Dyson|first=F. W.|date=1 January 1920|title=A Determination of the Deflection of Light by the Sun's Gravitational Field, from Observations Made at the Total Eclipse of May 29, 1919|journal=Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences|volume=220|issue=571–581|pages=291–333|bibcode=1920RSPTA.220..291D|doi=10.1098/rsta.1920.0009|author2=Eddington, A. S.|author3=Davidson, C.|url=https://zenodo.org/record/1432106|doi-access=free}}</ref>

[[Image:Gravitational lens-full.jpg|thumb|left|Bending light around a massive object from a distant source. The orange arrows show the apparent position of the background source. The white arrows show the path of the light from the true position of the source.]]
[[Image:Einstein cross.jpg|thumb|In the formation known as Einstein's Cross, four images of the same distant quasar appear around a foreground galaxy due to strong gravitational lensing.]]
The result was considered spectacular news and made the front page of most major newspapers. It made Einstein and his theory of general relativity world-famous. When asked by his assistant what his reaction would have been if general relativity had not been confirmed by Eddington and Dyson in 1919, Einstein said "Then I would feel sorry for the dear Lord. The theory is correct anyway."<ref>Rosenthal-Schneider, Ilse: Reality and Scientific Truth. Detroit: Wayne State University Press, 1980. p 74. (See also Calaprice, Alice: ''The New Quotable Einstein''. Princeton: Princeton University Press, 2005. p 227.)</ref> In 1912, Einstein had speculated that an observer could see multiple images of a single light source, if the light were deflected around a mass.  This effect would make the mass act as a kind of gravitational lens. However, as he only considered the effect of deflection around a single star, he seemed to conclude that the phenomenon was unlikely to be observed for the foreseeable future since the necessary alignments between stars and observer would be highly improbable. Several other physicists speculated about gravitational lensing as well, but all reached the same conclusion that it would be nearly impossible to observe.<ref name=":0">{{Cite web|url=http://www.einstein-online.info/spotlights/grav_lensing_history|title=A brief history of gravitational lensing|website=Einstein Online|publisher=[[Organization:Max Planck Institute for Gravitational Physics|Max Planck Institute for Gravitational Physics]]|access-date=2016-06-29|archive-url=https://web.archive.org/web/20160701154224/http://www.einstein-online.info/spotlights/grav_lensing_history|archive-date=2016-07-01|url-status=dead}}</ref>

Although Einstein made unpublished calculations on the subject,<ref name="origin of the idea"/> the first discussion of the gravitational lens in print was by Khvolson, in a short article discussing the "halo effect" of gravitation when the source, lens, and observer are in near-perfect alignment,<ref name="Early History of Gravitational Lensing" /> now referred to as the [[Physics:Einstein ring|Einstein ring]].

In 1936, after some urging by Rudi W. Mandl, Einstein reluctantly published the short article "Lens-Like Action of a Star By the Deviation of Light In the Gravitational Field" in the journal ''Science''.<ref name=":0" />

In 1937, [[Biography:Fritz Zwicky|Fritz Zwicky]] first considered the case where the newly discovered [[Astronomy:Galaxy|galaxies]] (which were called 'nebulae' at the time) could act as both source and lens, and that, because of the mass and sizes involved, the effect was much more likely to be observed.<ref>{{cite journal|title=Nebulae as Gravitational lenses|author=F. Zwicky|journal=[[Physics:Physical Review|Physical Review]]|year=1937|issue=4|volume=51|page=290|doi=10.1103/PhysRev.51.290|url=http://authors.library.caltech.edu/5611/1/ZWIpr37a.pdf |archive-url=https://web.archive.org/web/20131226004748/http://authors.library.caltech.edu/5611/1/ZWIpr37a.pdf |archive-date=2013-12-26 |url-status=live|bibcode = 1937PhRv...51..290Z }}</ref>

In 1963 Yu. G. Klimov, S. Liebes, and Sjur Refsdal recognized independently that quasars are an ideal light source for the gravitational lens effect.<ref name="Schneider, Peter; Kochanek, Christopher; Wambsganss, Joachim">{{cite book|title=Gravitational Lensing: Strong, Weak and Micro|pages=4|author1=Schneider Peter|author2=Kochanek, Christopher|author3=Wambsganss, Joachim|publisher=Springer Verlag Berlin Heidelberg New York Press|date=2006|isbn=978-3-540-30309-1}}</ref>

It was not until 1979 that the first gravitational lens would be discovered. It became known as the "Twin QSO" since it initially looked like two identical quasistellar objects. (It is officially named SBS 0957+561.) This gravitational lens was discovered by Dennis Walsh, Bob Carswell, and [[Biography:Ray Weymann|Ray Weymann]] using the [[Astronomy:Kitt Peak National Observatory|Kitt Peak National Observatory]] 2.1 meter [[Astronomy:Telescope|telescope]].<ref>{{cite journal|last=Walsh|first=D.|author2=Carswell, R. F. |author3=Weymann, R. J. |title=0957 + 561 A, B: twin quasistellar objects or gravitational lens?|journal=Nature|date=31 May 1979|volume=279|issue=5712|pages=381–384|doi=10.1038/279381a0|bibcode = 1979Natur.279..381W|pmid=16068158|s2cid=2142707}}</ref>

In the 1980s, astronomers realized that the combination of CCD imagers and computers would allow the brightness of millions of stars to be measured each night. In a dense field, such as the galactic center or the Magellanic clouds, many microlensing events per year could potentially be found. This led to efforts such as [[Astronomy:Optical Gravitational Lensing Experiment|Optical Gravitational Lensing Experiment]], or OGLE, that have characterized hundreds of such events, including those of [[Astronomy:OGLE-2016-BLG-1190Lb|OGLE-2016-BLG-1190Lb]] and [[Astronomy:OGLE-2016-BLG-1195Lb|OGLE-2016-BLG-1195Lb]].

==Approximate Newtonian description==
Newton wondered if light, in the form of corpuscles, would be bent due to gravity. The Newtonian prediction for light deflection refers to the amount of deflection a corpuscle would feel under the effect of gravity, and therefore one should read "Newtonian" in this context as the referring to the following calculations and not a belief that Newton held in the validity of these calculations.<ref name="Meneghetti">{{cite book |last1=Meneghetti |first1=Massimo |title=Introduction to Gravitational Lensing With Python Examples |series=Lecture Notes in Physics |date=2021 |volume=956 |publisher=Springer |doi=10.1007/978-3-030-73582-1 |isbn=978-3-030-73582-1 |s2cid=243826707 |url=https://doi.org/10.1007/978-3-030-73582-1}}</ref> 

For a gravitational point-mass lens of mass <math>M</math>, a corpuscle of mass <math>m</math> feels a [[Physics:Newton's law of universal gravitation|force]] 

:<math>\vec F = -\frac{GMm}{r^2} \hat r ~,</math>

where <math>r</math> is the lens-corpuscle separation. If we equate this force with Newton's second law, we can solve for the acceleration that the light undergoes

:<math>\vec a = -\frac{GM}{r^2}\hat r</math>.

The light interacts with the lens from initial time <math>t=0</math> to <math>t</math>, and the velocity boost the corpuscle receives is

:<math>\Delta \vec v = -\int_0^{t} dt' ~ \frac{GM}{r(t')^2} \hat r(t') ~.</math>

If one assumes that initially the light is far enough from the lens to neglect gravity, the perpendicular distance between the light's initial trajectory and the lens is b (the [[Physics:Impact parameter|impact parameter]]), and the parallel distance is <math>r_{\parallel}</math> such that <math>r^2 = b^2 + r_{\parallel}^2</math>. We additionally assume a constant speed of light along the parallel direction, <math>dr_{\parallel} \approx c dt</math>, and that the light is only being deflected a small amount. After plugging these assumptions into the above equation and further simplifying, one can solve for the velocity boost in the perpendicular direction. The angle of deflection between the corpuscle’s initial and final trajectories is therefore (see, e.g., M. Meneghetti 2021)<ref name="Meneghetti"></ref>

:<math>\theta = \frac{2GM}{c^2 r} ~.</math>
  
Although this result appears to be half the prediction from general relativity, classical physics predicts that the speed of light <math>c</math> is observer-dependent (see, e.g., L. Susskind and A. Friedman 2018)<ref>{{cite book |last1=Susskind |first1=Leonard |last2=Friedman |first2=Art |title=Special Relativity and Classical Field Theory |date=2018 |publisher=Penguin Books |isbn=9780141985015 |url=https://www.penguin.co.uk/books/303699/special-relativity-and-classical-field-theory-by-leonard-susskind-and-art-friedman/9780141985015}}</ref> which was superseded by a universal speed of light in [[Physics:Special relativity|special relativity]].

==Explanation in terms of spacetime curvature==
[[Image:Black hole lensing web.gif|thumb|left|Simulated gravitational lensing (black hole passing in front of a background galaxy).]]
In general relativity, light follows the curvature of spacetime, hence when light passes around a massive object, it is bent. This means that the light from an object on the other side will be bent towards an observer's eye, just like an ordinary lens. In general relativity the speed of light depends on the gravitational potential (i.e. the metric) and this bending can be viewed as a consequence of the light traveling along a gradient in light speed. Light rays are the boundary between the future, the spacelike, and the past regions. The gravitational attraction can be viewed as the motion of undisturbed objects in a background curved ''[[Geometry|geometry]]'' or alternatively as the response of objects to a ''force'' in a flat geometry. The angle of deflection is:

:<math>\theta = \frac{4GM}{c^2 r}</math>

toward the mass ''M'' at a distance ''r'' from the affected radiation, where ''G'' is the [[Physics:Gravitational constant|universal constant of gravitation]] and ''c'' is the speed of light in vacuum.

Since the [[Astronomy:Schwarzschild radius|Schwarzschild radius]] <math>r_\text{s}</math> is defined as <math>r_\text{s} = {2Gm}/{c^2}</math> and [[Astronomy:Escape velocity|escape velocity]] <math>v_\text{e}</math> is defined as <math display="inline">v_\text{e} = \sqrt{2Gm/r} = \beta_\text{e} c</math>, this can also be expressed in simple form as

:<math>\theta = 2 \frac{r_\text{s}}{r} = 2 \left(\frac{v_\text{e}}{c}\right)^2 = 2\beta_\text{e}^2</math>

==Search for gravitational lenses==
<!---[[File:Heic1401a-Abell2744-20140107.jpg|thumb|Abell 2744 galaxy cluster - Hubble Frontier Fields view (7 January 2014).<ref name="NASA-20140107">{{cite news |last1=Clavin |first1=Whitney |last2=Jenkins |first2=Ann |last3=Villard |first3=Ray |title=NASA's Hubble and Spitzer Team up to Probe Faraway Galaxies |url=http://www.jpl.nasa.gov/news/news.php?release=2014-007 |date=7 January 2014 |work=NASA |access-date=8 January 2014 }}</ref>]]--->
[[File:MACS J1206.jpg|thumb|This image from the NASA/ESA Hubble Space Telescope shows the galaxy cluster MACS J1206.]]

Most of the gravitational lenses in the past have been discovered accidentally. A search for gravitational lenses in the northern hemisphere (Cosmic Lens All Sky Survey, CLASS), done in radio frequencies using the Very Large Array (VLA) in New Mexico, led to the discovery of 22 new lensing systems, a major milestone. This has opened a whole new avenue for research ranging from finding very distant objects to finding values for cosmological parameters so we can understand the universe better.

A similar search in the southern hemisphere would be a very good step towards complementing the northern hemisphere search as well as obtaining other objectives for study. If such a search is done using well-calibrated and well-parameterized instrument and data, a result similar to the northern survey can be expected. The use of the Australia Telescope 20&nbsp;GHz (AT20G) Survey data collected using the Australia Telescope Compact Array (ATCA) stands to be such a collection of data. As the data were collected using the same instrument maintaining a very stringent quality of data we should expect to obtain good results from the search. The AT20G survey is a blind survey at 20&nbsp;GHz frequency in the radio domain of the electromagnetic spectrum. Due to the high frequency used, the chances of finding gravitational lenses increases as the relative number of compact core objects (e.g. quasars) are higher (Sadler et al. 2006). This is important as the lensing is easier to detect and identify in simple objects compared to objects with complexity in them. This search involves the use of interferometric methods to identify candidates and follow them up at higher resolution to identify them. Full detail of the project is currently under works for publication.

[[File:Helping Hubble SDSS J0915+3826.tif|thumb|left|Galaxy cluster SDSS J0915+3826 helps astronomers to study star formation in galaxies.<ref>{{cite web |title=Helping Hubble |url=https://www.spacetelescope.org/images/potw1844a/ |website=www.spacetelescope.org |access-date=29 October 2018}}</ref>]]

Microlensing techniques have been used to search for planets outside our solar system.  A statistical analysis of specific cases of observed microlensing over the time period of 2002 to 2007 found that most stars in the [[Astronomy:Milky Way|Milky Way]] galaxy hosted at least one orbiting planet within 0.5 to 10 AU.<ref>{{Cite journal|last1=Cassan|first1=A.|last2=Kubas|first2=D.|last3=Beaulieu|first3=J.-P.|last4=Dominik|first4=M.|last5=Horne|first5=K.|last6=Greenhill|first6=J.|last7=Wambsganss|first7=J.|last8=Menzies|first8=J.|last9=Williams|first9=A.|title=One or more bound planets per Milky Way star from microlensing observations|journal=Nature|volume=481|issue=7380|pages=167–169|doi=10.1038/nature10684|arxiv = 1202.0903 |bibcode = 2012Natur.481..167C |year=2012|pmid=22237108|s2cid=2614136}}</ref>

In 2009, weak gravitational lensing was used to extend the mass-X-ray-luminosity relation to older and smaller structures than was previously possible to improve measurements of distant galaxies.<ref>{{cite web| url = https://www.sciencedaily.com/releases/2010/01/100119172846.htm| title = Cosmology: Weak gravitational lensing improves measurements of distant galaxies |date=January 21, 2010 |work=ScienceDaily |author=DOE/Lawrence Berkeley National Laboratory}}</ref>

{{As of|2013}} the most distant gravitational lens galaxy, [[Astronomy:J1000+0221|J1000+0221]], had been found using [[Organization:NASA|NASA]]'s [[Astronomy:Hubble Space Telescope|Hubble Space Telescope]].<ref>{{cite web|last=Sci-News.com|title=Most Distant Gravitational Lens Discovered|url=http://www.sci-news.com/astronomy/science-gravitational-lens-01478.html|website=Sci-News.com|access-date=22 October 2013|date=21 Oct 2013 |archive-url=https://web.archive.org/web/20131023092313/http://www.sci-news.com/astronomy/science-gravitational-lens-01478.html|archive-date=23 October 2013|url-status=dead}}</ref><ref>{{cite journal|last=van der Wel|first=A.|display-authors=0|title=Discovery of a Quadruple Lens in CANDELS with a Record Lens Redshift|journal=Astrophysical Journal Letters|date=2013|arxiv=1309.2826|bibcode = 2013ApJ...777L..17V |doi = 10.1088/2041-8205/777/1/L17|volume=777|issue=1|pages=L17|s2cid=55728208}}</ref>  While it remains the most distant quad-image lensing galaxy known, an even more distant two-image lensing galaxy was subsequently discovered by an international team of astronomers using a combination of [[Astronomy:Hubble Space Telescope|Hubble Space Telescope]] and Keck telescope imaging and spectroscopy.  The discovery and analysis of the IRC 0218 lens was published in the ''Astrophysical Journal Letters'' on June 23, 2014.<ref>{{cite journal|last=Wong|first=K.|display-authors=0|title=Discovery of a Strong Lensing Galaxy Embedded in a Cluster at z = 1.62|journal=Astrophysical Journal Letters|date=2014|volume=789|issue=2|pages=L31|doi=10.1088/2041-8205/789/2/L31|bibcode=2014ApJ...789L..31W|arxiv = 1405.3661 |s2cid=56376674}}</ref>

Research published Sep 30, 2013 in the online edition of ''[[Physics:Physical Review Letters|Physical Review Letters]]'', led by McGill University in {{wipe|Montreal}}, Québec, Canada, has discovered the B-modes, that are formed due to gravitational lensing effect, using [[Organization:National Science Foundation|National Science Foundation]]'s [[Astronomy:South Pole Telescope|South Pole Telescope]] and with help from the Herschel space observatory. This discovery would open the possibilities of testing the theories of how our universe originated.<ref>{{cite journal|last=NASA/Jet Propulsion Laboratory|date=October 22, 2013|title=Long-sought pattern of ancient light detected|journal=ScienceDaily|access-date=October 23, 2013|url=https://www.sciencedaily.com/releases/2013/10/131022101009.htm}}</ref><ref>{{cite journal|author=Hanson, D. <!---|authors=D. Hanson, S. Hoover, A. Crites, P. A. R. Ade, K. A. Aird, J. E. Austermann, J. A. Beall, A. N. Bender, B. A. Benson, L. E. Bleem, J. J. Bock, J. E. Carlstrom, C. L. Chang, H. C. Chiang, H-M. Cho, A. Conley, T. M. Crawford, T. de Haan, M. A. Dobbs, W. Everett, J. Gallicchio, J. Gao, E. M. George, N. W. Halverson, N. Harrington, J. W. Henning, G. C. Hilton, G. P. Holder, W. L. Holzapfel, J. D. Hrubes, N. Huang, J. Hubmayr, K. D. Irwin, R. Keisler, L. Knox, A. T. Lee, E. Leitch, D. Li, C. Liang, D. Luong-Van, G. Marsden, J. J. McMahon, J. Mehl, S. S. Meyer, L. Mocanu, T. E. Montroy, T. Natoli, J. P. Nibarger, V. Novosad, S. Padin, C. Pryke, C. L. Reichardt, J. E. Ruhl, B. R. Saliwanchik, J. T. Sayre, K. K. Schaffer, B. Schulz, G. Smecher, A. A. Stark, K. T. Story, C. Tucker, K. Vanderlinde, J. D. Vieira, M. P. Viero, G. Wang, V. Yefremenko, O. Zahn, M. Zemcov--->
|title=Detection of B-Mode Polarization in the Cosmic Microwave Background with Data from the South Pole Telescope|journal=Physical Review Letters|date=Sep 30, 2013|volume=111|issue=14|pages=141301|series=14|doi=10.1103/PhysRevLett.111.141301|pmid=24138230|arxiv = 1307.5830 |bibcode = 2013PhRvL.111n1301H |s2cid=9437637|display-authors=0}}</ref>
{{wide image|14-283-Abell2744-DistantGalaxies-20141016.jpg|800px|align-cap=center|[[Astronomy:Abell 2744|Abell 2744]] [[Astronomy:Galaxy cluster|galaxy cluster]] - extremely distant galaxies revealed by [[Physics:Gravitational lensing|gravitational lensing]] (16 October 2014).<ref name="NASA-20140107">{{cite news |last1=Clavin |first1=Whitney |last2=Jenkins |first2=Ann |last3=Villard |first3=Ray |title=NASA's Hubble and Spitzer Team up to Probe Faraway Galaxies |url=http://www.jpl.nasa.gov/news/news.php?release=2014-007 |date=7 January 2014 |work=[[Organization:NASA|NASA]] |access-date=8 January 2014 |archive-date=9 July 2018 |archive-url=https://web.archive.org/web/20180709164545/https://www.jpl.nasa.gov/news/news.php?release=2014-007 |url-status=dead }}</ref><ref name="NASA-20141016">{{cite web |last1=Chou |first1=Felecia |last2=Weaver |first2=Donna |title=RELEASE 14-283 - NASA's Hubble Finds Extremely Distant Galaxy through Cosmic Magnifying Glass |url=http://www.nasa.gov/press/2014/october/nasa-s-hubble-finds-extremely-distant-galaxy-through-cosmic-magnifying-glass/ |date=16 October 2014 |work=[[Organization:NASA|NASA]] |access-date=17 October 2014 }}</ref>}}

==Solar gravitational lens==
{{Main|Astronomy:Solar gravitational lens}}
Albert Einstein predicted in 1936 that rays of light from the same direction that skirt the edges of the [[Astronomy:Sun|Sun]] would converge to a focal point approximately 542 [[Astronomy:Astronomical unit|AU]] from the Sun.<ref>{{cite journal|author= Einstein, Albert|s2cid=38450435|title=Lens-Like Action of a Star by the Deviation of Light in the Gravitational Field|journal=Science|volume=84|date=1936|pages=506–507|bibcode = 1936Sci....84..506E |doi = 10.1126/science.84.2188.506|pmid=17769014|issue=2188}}</ref> Thus, a probe positioned at this distance (or greater) from the Sun could use the Sun as a gravitational lens for magnifying distant objects on the opposite side of the Sun.<ref>{{cite journal | last1 = Eshleman | first1 = Von R | year = 1979 | title = Gravitational lens of the sun: its potential for observations and communications over interstellar distances | journal = Science | volume = 205 | issue = 4411| pages = 1133–1135 | doi = 10.1126/science.205.4411.1133 | pmid = 17735051 | bibcode = 1979Sci...205.1133E | s2cid = 27692082 }}</ref> A probe's location could shift around as needed to select different targets relative to the Sun.

This distance is far beyond the progress and equipment capabilities of space probes such as ''[[Astronomy:Voyager 1|Voyager 1]]'', and beyond the known planets and dwarf planets, though over thousands of years [[Astronomy:90377 Sedna|90377 Sedna]] will move farther away on its highly elliptical orbit.  The high gain for potentially detecting signals through this lens, such as microwaves at the 21-cm [[Physics:Hydrogen line|hydrogen line]], led to the suggestion by Frank Drake in the early days of SETI that a probe could be sent to this distance.  A multipurpose probe SETISAIL and later [[Engineering:FOCAL (spacecraft)|FOCAL]] was proposed to the ESA in 1993, but is expected to be a difficult task.<ref name="Landis">Geoffrey A. Landis, [https://arxiv.org/abs/1604.06351 "Mission to the Gravitational Focus of the Sun: A Critical Analysis,"] ArXiv, paper 1604.06351, Cornell University, 21 Apr 2016 (downloaded 30 April 2016)</ref>  If a probe does pass 542 AU, magnification capabilities of the lens will continue to act at farther distances, as the rays that come to a focus at larger distances pass further away from the distortions of the Sun's corona.<ref>{{cite book|url=https://books.google.com/books?id=nGBhKDE1SFEC&pg=PA6|title=Deep Space Flight and Communications: Exploiting the Sun as a Gravitational Lens|author=Claudio Maccone|date=2009|publisher=Springer|isbn=9783540729433}}</ref> A critique of the concept was given by Landis,<ref>Landis, Geoffrey A., “Mission to the Gravitational Focus of the Sun: A Critical Analysis,” paper AIAA-2017-1679, AIAA Science and Technology Forum and Exposition 2017, Grapevine TX, January 9–13, 2017. [https://arxiv.org/abs/1604.06351 Preprint] at arXiv.org (accessed 24 December 2016).</ref> who discussed issues including interference of the solar corona, the high magnification of the target, which will make the design of the mission focal plane difficult, and an analysis of the inherent [[Physics:Spherical aberration|spherical aberration]] of the lens.

In 2020, NASA physicist [[Biography:Slava Turyshev|Slava Turyshev]] presented his idea of Direct Multipixel Imaging and Spectroscopy of an Exoplanet with a [[Astronomy:Solar gravitational lens|Solar Gravitational Lens]] Mission. The lens could reconstruct the exoplanet image with ~25 km-scale surface resolution, enough to see surface features and signs of habitability.<ref>{{Cite web|last=Hall|first=Loura|date=2020-04-06|title=Direct Multipixel Imaging and Spectroscopy of an Exoplanet|url=http://www.nasa.gov/directorates/spacetech/niac/2020_Phase_I_Phase_II/Direct_Multipixel_Imaging_and_Spectroscopy_of_an_Exoplanet|access-date=2020-08-05|website=NASA}}</ref>

==Measuring weak lensing==
[[File:Galaxy cluster MACS J2129-0741 and lensed galaxy MACS2129-1.jpg|thumb|Galaxy cluster MACS J2129-0741 and lensed galaxy MACS2129-1.<ref>{{cite web|title=Galaxy cluster MACS J2129-0741 and lensed galaxy MACS2129-1|url=https://www.spacetelescope.org/images/opo1726a/|website=www.spacetelescope.org|access-date=23 June 2017}}</ref>]]
Kaiser, Squires and Broadhurst (1995),<ref name=KSB-1995>{{cite journal|last1=Kaiser|first1=Nick|last2=Squires|first2=Gordon|last3=Broadhurst|first3=Tom|title=A Method for Weak Lensing Observations|journal=The Astrophysical Journal|date=August 1995|volume=449|pages=460–475|doi=10.1086/176071|arxiv = astro-ph/9411005 |bibcode = 1995ApJ...449..460K |s2cid=119382511}}</ref> Luppino & Kaiser (1997)<ref name=LK-1997>{{cite journal|last1=Luppino|first1=G. A.|last2=Kaiser|first2=Nick|title=Detection of Weak Lensing by a Cluster of Galaxies at z = 0.83|journal=The Astrophysical Journal|date=20 January 1997|volume=475|issue=1|pages=20–28|doi=10.1086/303508|arxiv = astro-ph/9601194 |bibcode = 1997ApJ...475...20L |s2cid=6294167}}</ref> and Hoekstra et al. (1998) prescribed a method to invert the effects of the [[Physics:Point spread function|point spread function]] (PSF) smearing and shearing, recovering a shear estimator uncontaminated by the systematic distortion of the PSF. This method (KSB+) is the most widely used method in weak lensing shear measurements.<ref>{{cite book |title=Statistical Challenges in Modern Astronomy IV: Proceedings of a Conference Held at Pennsylvania State University, University Park, Pennsylvania, USA, 12–15 June 2006, Volume 371 |edition=illustrated |first1=Gutti Jogesh |last1=Babu |first2=Eric D. |last2=Feigelson |publisher=Astronomical Society of the Pacific |year=2007 |isbn=978-1-58381-240-2 |page=66 |url=https://books.google.com/books?id=dHPvAAAAMAAJ}}</ref><ref>{{cite book |title=A Pan-Chromatic View of Clusters of Galaxies and the Large-Scale Structure |edition=illustrated |first1=Manolis |last1=Plionis |first2=O. |last2=López-Cruz |first3=D. |last3=Hughes |publisher=Springer Science & Business Media |year=2008 |isbn=978-1-4020-6940-6 |page=233 |url=https://books.google.com/books?id=TnovKp-nJuMC&pg=PA233}}</ref>
 
Galaxies have random rotations and inclinations. As a result, the shear effects in weak lensing need to be determined by statistically preferred orientations. The primary source of error in lensing measurement is due to the convolution of the PSF with the lensed image. The KSB method measures the ellipticity of a galaxy image. The shear is proportional to the ellipticity. The objects in lensed images are parameterized according to their weighted quadrupole moments. For a perfect ellipse, the weighted quadrupole moments are related to the weighted ellipticity. KSB calculate how a weighted ellipticity measure is related to the shear and use the same formalism to remove the effects of the PSF.<ref>{{cite book |title=Gravitational Lensing: An Astrophysical Tool |edition=illustrated |first1=Frederic Courbin, Dante Minniti |last1=Frederic Courbin, Dante Minniti |publisher=Springer |year=2008 |isbn=978-3-540-45857-9 |page=69 |url=https://books.google.com/books?id=TsxrCQAAQBAJ&pg=PA69}}</ref>

KSB's primary advantages are its mathematical ease and relatively simple implementation. However, KSB is based on a key assumption that the PSF is circular with an anisotropic distortion. This is a reasonable assumption for cosmic shear surveys, but the next generation of surveys (e.g. [[Astronomy:Large Synoptic Survey Telescope|LSST]]) may need much better accuracy than KSB can provide.

==Gallery==
<gallery>
Image:The Sunburst Arc PSZ1 G311.65-18.48.jpg|Sunburst Arc galaxy.<ref>{{cite web|title=Hubble Captures a Dozen Sunburst Arc Doppelgangers|url=https://spacetelescope.org/news/heic1920/|website=www.spacetelescope.org|access-date=11 November 2019|language=en}}</ref>
Image:Hubble’s view on distant quasar J043947.08+163415.7.tif|Gravitationally lensed Quasar.<ref>{{cite web|title=Hubble sees the brightest quasar in the early Universe|url=https://www.spacetelescope.org/news/heic1902/|website=www.spacetelescope.org|access-date=10 January 2019|language=en}}</ref>
Image:On the hunt for newborn stars.tif|In SDSS J0952+3434, the lower arc-shaped galaxy has the characteristic shape of a galaxy that has been gravitationally lensed.<ref>{{cite web|title=On the hunt for newborn stars|url=https://www.spacetelescope.org/images/potw1842a/|website=www.spacetelescope.org|access-date=15 October 2018}}</ref>
Image:Warped and distorted SDSS J1050+0017.jpg|Warped and distorted around SDSS J1050+0017.<ref>{{cite web|title=Warped and distorted|url=https://www.spacetelescope.org/images/potw1839a/|website=www.spacetelescope.org|access-date=24 September 2018}}</ref>
Image:Stretched out image of distant galaxy.jpg|Galaxy [[Astronomy:SPT0615-JD|SPT0615-JD]] existed when the Universe was just 500 million years old.<ref>{{cite web|title=Stretched out image of distant galaxy|url=https://www.spacetelescope.org/images/opo1802a/|website=www.spacetelescope.org|access-date=16 January 2018}}</ref>
Image:Cosmic snake pregnant with stars.tif|Gravitational lenses found in the DESI Legacy Survey data<ref>{{cite web|title=Cosmic snake pregnant with stars|url=http://www.spacetelescope.org/images/potw1747a/|website=www.spacetelescope.org|access-date=20 November 2017}}</ref>
Image:Gravitational_lenses_found_in_the_DESI_Legacy_Survey_data.jpg|Gravitational lenses found in the DESI Legacy Survey data<ref>{{cite web|title=Doubling the Number of Known Gravitational Lenses|url=https://noirlab.edu/public/news/noirlab2104/|website=noirlab.edu|access-date=3 February 2021}}</ref>
Image:Hubble captures gallery of ultra-bright galaxies.jpg|The lensing phenomenon allows for features as small as about 100 light-years or less.<ref>{{cite web|title=Hubble captures gallery of ultra-bright galaxies|url=https://www.spacetelescope.org/images/opo1724a/|website=www.spacetelescope.org|access-date=8 June 2017}}</ref>
Image:Detailed look at a gravitationally lensed supernova.jpg|Detailed look at a gravitationally lensed [[Astronomy:Type Ia supernova|type Ia supernova]] iPTF16geu.<ref>{{cite web|title=Detailed look at a gravitationally lensed supernova|url=https://www.spacetelescope.org/images/heic1710a/|website=www.spacetelescope.org|access-date=21 April 2017}}</ref>
Image:HST-Smiling-GalaxyClusterSDSS-J1038+4849-20150210.jpg|"Smiley" image of [[Astronomy:Galaxy cluster|galaxy cluster]] (SDSS J1038+4849) & gravitational lensing (an [[Physics:Einstein ring|Einstein ring]]) ([[Astronomy:Hubble Space Telescope|HST]]).<ref name="NASA-20150210">{{cite web|last1=Loff|first1=Sarah|last2=Dunbar|first2=Brian|title=Hubble Sees A Smiling Lens|url=http://www.nasa.gov/content/hubble-sees-a-smiling-lens/|date=February 10, 2015|work=[[Organization:NASA|NASA]]|access-date=February 10, 2015 }}</ref>
Image:Gravitationell-lins-4.jpg|[[Astronomy:Abell 1689|Abell 1689]] - actual gravitational lensing effects ([[Astronomy:Hubble Space Telescope|Hubble Space Telescope]]).
Image:COSMOS 3D dark matter map.jpg|[[Astronomy:Dark matter|Dark matter]] distribution - weak gravitational lensing ([[Astronomy:Hubble Space Telescope|Hubble Space Telescope]]).
Image:The most distant gravitational lens yet discovered.jpg|Gravitational lens discovered at [[Astronomy:Redshift|redshift]] z = 1.53.<ref>{{cite news|title=Most distant gravitational lens helps weigh galaxies|url=http://www.spacetelescope.org/news/heic1319/|access-date=18 October 2013|newspaper=ESA/Hubble Press Release}}</ref>
Image:PIA23641-GravitationalLensing-20200108.jpg|Gravitational Lensing Graphic (January 8, 2020)
File:Hubble Showcases Hamilton’s Object.jpg|Hubble Showcases Hamilton's Object
File:Gravitational lensing by a black hole.jpg|Illustration of gravitational lensing<ref>{{cite web |title=Hubble Determines Mass of Isolated Black Hole Roaming Our Milky Way Galaxy |url=https://hubblesite.org/contents/news-releases/2022/news-2022-001?news=true |website=HubbleSite.org |access-date=16 June 2022 |language=en}}</ref>
Gravitationally-lensed distant star-forming galaxy.jpg|Gravitationally-lensed distant star-forming galaxies.<ref>{{cite news|title=ALMA Rewrites History of Universe's Stellar Baby Boom|url=http://www.eso.org/public/news/eso1313/|access-date=2 April 2013|newspaper=ESO |date=13 March 2013}}</ref>
</gallery>

==See also==
* {{Annotated link|Terrestrial atmospheric lens}}
* {{Annotated link|Galactic lens}}
* {{Annotated link|Gravitational lensing formalism}}
* {{Annotated link|Strong gravitational lensing}}
** {{Annotated link|Einstein cross}}
** {{Annotated link|Einstein ring}}
* {{Annotated link|Weak gravitational lensing}}
* {{Annotated link|Gravitational microlensing}}
* {{Annotated link|SN Refsdal}}
* {{Annotated link|Euclid (spacecraft)}}

==Historical papers and references==
* {{cite journal | first = O | last = Khvolson | title = Über eine mögliche Form fiktiver Doppelsterne | journal = Astronomische Nachrichten | volume = 221 | issue = 20 | pages = 329–330 | date = 1924 | bibcode = 1924AN....221..329C | doi = 10.1002/asna.19242212003}}
* {{cite journal | first = Albert | last = Einstein | s2cid = 38450435 |  title = Lens-like Action of a Star by the Deviation of Light in the Gravitational Field | journal = Science | volume = 84 | pages = 506–7 | date = 1936 | doi = 10.1126/science.84.2188.506 | pmid = 17769014 | issue = 2188 | jstor=1663250|bibcode = 1936Sci....84..506E }}
* {{cite journal | first = Jürgen | last = Renn |author2 = Tilman Sauer |author-link2=Tilman Sauer |author3 = John Stachel |author-link3=John Stachel |title = The Origin of Gravitational Lensing: A Postscript to Einstein's 1936 Science paper | journal = Science | volume = 275 | pages = 184–6 | date = 1997 | doi = 10.1126/science.275.5297.184 | pmid =8985006 | issue = 5297|bibcode = 1997Sci...275..184R | s2cid = 43449111 }}

==References==
;Notes
{{Reflist|30em}}

;Bibliography
* "''[http://www.sciencenews.org/view/generic/id/33082/title/Math_Trek__Accidental_astrophysicists Accidental Astrophysicists] {{Webarchive|url=https://web.archive.org/web/20120216201236/http://www.sciencenews.org/view/generic/id/33082/title/Math_Trek__Accidental_astrophysicists |date=2012-02-16 }}''". Science News, June 13, 2008.
* "''[https://web.archive.org/web/20101210135526/http://www.cinespa.ucr.ac.cr/software/xfgl/index.html XFGLenses]''". A Computer Program to visualize Gravitational Lenses, Francisco Frutos-Alfaro
* "''[https://web.archive.org/web/20070929124804/http://www.btinternet.com/~Boughen/G-LenS G-LenS]''". A Point Mass Gravitational Lens Simulation, Mark Boughen.
* Newbury, Pete, "''[https://web.archive.org/web/20110523100723/http://www.iam.ubc.ca/old_pages/newbury/lenses/lenses.html Gravitational Lensing]''". Institute of Applied Mathematics, The University of British Columbia.
* Cohen, N., "Gravity's Lens: Views of the New Cosmology", Wiley and Sons, 1988.
* "''[http://cfa-www.harvard.edu/~rschild/qgl.html Q0957+561 Gravitational Lens]''". Harvard.edu.
* Bridges, Andrew, "''[http://www.signonsandiego.com/news/science/20040215-1033-ca-farthestgalaxy.html Most distant known object in universe discovered]''". Associated Press. February 15, 2004. (Farthest galaxy found by gravitational lensing, using Abell 2218 and Hubble Space Telescope.)
* [https://web.archive.org/web/20070310222247/http://sciencecareers.sciencemag.org/career_development/previous_issues/articles/2006_05_12/analysing_corporations_and_the_cosmos/(parent)/12100 Analyzing Corporations ... and the Cosmos] An unusual career path in gravitational lensing.
* "''[http://cfa-www.harvard.edu/castles HST images of strong gravitational lenses]''". Harvard-Smithsonian Center for Astrophysics.
* "''[https://arxiv.org/abs/astro-ph/0404309 A planetary microlensing event]''" and "''[https://arxiv.org/abs/astro-ph/0505451 A Jovian-mass Planet in Microlensing Event OGLE-2005-BLG-071]''", the first extra-solar planet detections using microlensing.
* [http://xstructure.inr.ac.ru/x-bin/theme3.py?level=2&index1=-188353 Gravitational lensing on arxiv.org]
* [http://www.aoc.nrao.edu/~smyers/class.html NRAO CLASS home page]
* [http://www.atnf.csiro.au/research/AT20G AT20G survey]
* [http://cdsads.u-strasbg.fr/full/1981Ap%26SS..78..199B ''A diffraction limit on the gravitational lens effect'' (Bontz, R. J. and Haugan, M. P. "Astrophysics and Space Science" vol. 78, no. 1, p. 199-210. August 1981)]

;Further reading
* {{cite journal
 | author = Blandford & Narayan
 | date = 1992
 | title = Cosmological applications of gravitational lensing
 | journal = Annual Review of Astronomy and Astrophysics
 | pages = 311–358
 | issue = 1
 | volume = 30
 | bibcode = 1992ARA&A..30..311B
 | doi = 10.1146/annurev.aa.30.090192.001523
 | last2 = Narayan
 | first2 = R
}}
* {{cite journal|title=Weak Gravitational Lensing|author1=Matthias Bartelmann|author2=Peter Schneider|date=2000-08-17|url=http://www.mpa-garching.mpg.de/Lenses/WLRevEls.pdf|url-status=dead|archive-url=https://web.archive.org/web/20070226035150/http://www.mpa-garching.mpg.de/Lenses/WLRevEls.pdf|archive-date=2007-02-26}}
* {{Cite journal | first1=Dmitry | last1=Khavinson | first2=Genevra | last2=Neumann | url=https://www.ams.org/notices/200806/tx080600666p.pdf |archive-url=https://web.archive.org/web/20080907121928/http://www.ams.org/notices/200806/tx080600666p.pdf |archive-date=2008-09-07 |url-status=live | title=From Fundamental Theorem of Algebra to Astrophysics: A "Harmonious" Path | journal=Notices of the AMS |date=June–July 2008 | volume=55 | issue=6 | pages=666–675 }}.
* {{cite book | first1=Arlie O. | last1=Petters |  first2=Harold | last2=Levine | first3=Joachim | last3=Wambsganss | url=https://www.springer.com/birkhauser/physics/book/978-0-8176-3668-5 | title=Singularity Theory and Gravitational Lensing | series=Progress in Mathematical Physics | volume=21 | publisher=Birkhäuser | date=2001 }}
* [http://www.matnat.uio.no/proveforelesninger/g-i/haugan-stein-vidar-hagfors.xml ''Tools for the evaluation of the possibilities of using parallax measurements of gravitationally lensed sources'' (Stein Vidar Hagfors Haugan. June 2008)]

==External links==
* [http://www.pdxjustice.org/node/54 Video: Evalyn Gates – ''Einstein's Telescope: The Search for Dark Matter and Dark Energy in the Universe''] {{Webarchive|url=https://web.archive.org/web/20180902235413/http://pdxjustice.org/node/54 |date=2018-09-02 }}, presentation in Portland, Oregon, on April 19, 2009, from the author's recent book tour.
* [https://web.archive.org/web/20100129190931/http://www.astronomycast.com/stars/episode-37-gravitational-lensing/ Audio: Fraser Cain and Dr. Pamela Gay – Astronomy Cast: Gravitational Lensing, May 2007]

{{Relativity}}

{{DEFAULTSORT:Gravitational Lens}}
[[Category:Astrophysics]]
[[Category:Effects of gravitation]]
[[Category:Gravitational lensing| ]]
[[Category:Large-scale structure of the cosmos]]

{{Sourceattribution|Gravitational lens}}