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Gravitational lens

From Wikipedia, the free encyclopedia
(Redirected fromGravitational lensing)
Light bending by mass between source and observer
Part of a series of articles about
Gravitational lensing
Einstein ring
Formalism
Strong lensing
Microlensing
Weak lensing
General relativity
Spacetime curvature schematic
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 (seeEinstein ring).

Agravitational lens is matter, such as acluster of galaxies or apoint particle, that bendslight from a distant source as it travels toward an observer. The amount ofgravitational lensing is described byAlbert Einstein'sgeneral theory of relativity.[1][2] If light is treated ascorpuscles travelling at thespeed of light,Newtonian physics also predicts the bending of light, but only half of that predicted by general relativity.[3][4][5][6]

Orest Khvolson (1924)[7] andFrantisek Link (1936)[8] 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with being the first to discuss the effect in print, but it is more commonly associated with Einstein, who made unpublished calculations on it in 1912[9] and published an article on the subject in 1936.[10]

In 1937,Fritz Zwicky posited that galaxy clusters could act as gravitational lenses, a claim confirmed in 1979 by observation of theTwin QSO SBS 0957+561.

Description

[edit]
Gravitational lensing – intervening galaxy modifies appearance of a galaxy far behind it (video; artist's concept).
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.
An analysis of the distortion of SDP.81 caused by this effect has revealed star-forming clumps of matter.

Unlike anoptical 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 singlefocal 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".[11] 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 theSt. Petersburg physicistOrest Khvolson,[12] and quantified byAlbert Einstein in 1936. It is usually referred to in the literature as anEinstein 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 agalaxy group orcluster) 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:[13]: 399–401 [14]

Strong lensing
Where there are easily visible distortions such as the formation ofEinstein rings, 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 timesthat of the Sun will produce multiple images separated by only a fewarcseconds.Galaxy clusters can produce separations of several arcminutes. In both cases the galaxies and sources are quite distant, many hundreds ofmegaparsecs away from our Galaxy.
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 theshear 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 ofdark 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 ofsystematic error: the intrinsic shape of galaxies, the tendency of a camera'spoint spread function to distort the shape of a galaxy and the tendency ofatmospheric 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 theLambda-CDM model, and to provide a consistency check on other cosmological observations. They may also provide an important future constraint ondark energy.
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 theMilky Way in one typical case, with the background source being stars in a remote galaxy, or, in another case, an even more distantquasar. 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 starMACS 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 ofelectromagnetic radiation, not just visible light, and also in non-electromagnetic radiation, like gravitational waves. Weak lensing effects are being studied for thecosmic microwave background as well asgalaxy surveys. Strong lenses have been observed inradio andx-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

[edit]
One ofEddington's photographs of the 1919solar eclipse experiment, presented in his 1920 paper announcing its success

Henry Cavendish in 1784 (in an unpublished manuscript) andJohann Georg von Soldner in 1801 (published in 1804) had pointed out that Newtonian gravity predicts that starlight will bend around a massive object[15] as had already been supposed byIsaac Newton in 1704 in hisQueries No.1 in his bookOpticks.[16] The same value as Soldner's was calculated by Einstein in 1911 based on theequivalence principle alone.[13]: 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.[17]

The first observation of light deflection was performed by noting the change in position ofstars as they passed near the Sun on thecelestial sphere. The observations were performed in 1919 byArthur Eddington,Frank Watson Dyson, and their collaborators during thetotal solar eclipse on May 29.[18] Thesolar eclipse allowed the stars near the Sun to be observed. Observations were made simultaneously in the cities ofSobral, Ceará, Brazil and inSão Tomé and Príncipe on the west coast of Africa.[19] The observations demonstrated that the light fromstars passing close to theSun was slightly bent, so that stars appeared slightly out of position.[20]

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.
In the formation known asEinstein's Cross, four images of the same distantquasar 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."[21] 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.[10]

Although Einstein made unpublished calculations on the subject,[9] 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,[7] now referred to as theEinstein 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 journalScience.[10]

In 1937,Fritz Zwicky first considered the case where the newly discoveredgalaxies (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.[22]

In 1963 Yu. G. Klimov, S. Liebes, andSjur Refsdal recognized independently that quasars are an ideal light source for the gravitational lens effect.[23]

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 namedSBS 0957+561.) This gravitational lens was discovered byDennis Walsh, Bob Carswell, andRay Weymann using theKitt Peak National Observatory 2.1 metertelescope.[24]

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 asOptical Gravitational Lensing Experiment, or OGLE, that have characterized hundreds of such events, including those ofOGLE-2016-BLG-1190Lb andOGLE-2016-BLG-1195Lb.

Approximate Newtonian description

[edit]

Newton wondered whether 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.[25]

For a gravitational point-mass lens of massM{\displaystyle M}, a corpuscle of massm{\displaystyle m} feels aforce

F→=−GMmr2r^,{\displaystyle {\vec {F}}=-{\frac {GMm}{r^{2}}}{\hat {r}},}

wherer{\displaystyle r} is the lens-corpuscle separation. If we equate this force withNewton's second law, we can solve for the acceleration that the light undergoes:

a→=−GMr2r^.{\displaystyle {\vec {a}}=-{\frac {GM}{r^{2}}}{\hat {r}}.}

The light interacts with the lens from initial timet=0{\displaystyle t=0} tot{\displaystyle t}, and the velocity boost the corpuscle receives is

Δv→=−∫0tdt′GMr(t′)2r^(t′).{\displaystyle \Delta {\vec {v}}=-\int _{0}^{t}dt'\,{\frac {GM}{r(t')^{2}}}{\hat {r}}(t').}

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 isb (theimpact parameter), and the parallel distance isr∥{\displaystyle r_{\parallel }}, such thatr2=b2+r∥2{\displaystyle r^{2}=b^{2}+r_{\parallel }^{2}}. We additionally assume a constant speed of light along the parallel direction,dr∥≈cdt{\displaystyle dr_{\parallel }\approx c\,dt}, 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)[25]

θ=2GMc2r.{\displaystyle \theta ={\frac {2GM}{c^{2}r}}.}

Although this result appears to be half the prediction from general relativity, classical physics predicts that the speed of lightc{\displaystyle c} is observer-dependent (see, e.g., L. Susskind and A. Friedman 2018)[26] which was superseded by a universal speed of light inspecial relativity.

Explanation in terms of spacetime curvature

[edit]
See also:Kepler problem in general relativity
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 path of light depends on the shape of space (i.e. the metric). The gravitational attraction can be viewed as the motion of undisturbed objects in a background curvedgeometry or alternatively as the response of objects to aforce in a flat geometry. The angle of deflection is

θ=4GMc2r{\displaystyle \theta ={\frac {4GM}{c^{2}r}}}

toward the massM at a distancer from the affected radiation, whereG is theuniversal constant of gravitation, andc is the speed of light in vacuum.

Since theSchwarzschild radiusrs{\displaystyle r_{\text{s}}} is defined asrs=2Gm/c2{\displaystyle r_{\text{s}}=2Gm/c^{2}}, andescape velocityve{\displaystyle v_{\text{e}}} is defined asve=2Gm/r=βec{\textstyle v_{\text{e}}={\sqrt {2Gm/r}}=\beta _{\text{e}}c}, this can also be expressed in simple form as

θ=2rsr=2(vec)2=2βe2.{\displaystyle \theta =2{\frac {r_{\text{s}}}{r}}=2\left({\frac {v_{\text{e}}}{c}}\right)^{2}=2\beta _{\text{e}}^{2}.}

Search for gravitational lenses

[edit]
This image from the NASA/ESA Hubble Space Telescope shows the galaxy clusterMACS 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 instruments and data, a result similar to the northern survey can be expected. The use of the Australia Telescope 20 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 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.

Galaxy cluster SDSS J0915+3826 helps astronomers to study star formation in galaxies.[27]

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 theMilky Way galaxy hosted at least one orbiting planet within 0.5 to 10 AU.[28]

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.[29]

As of 2013[update] the most distant gravitational lens galaxy,J1000+0221, had been found usingNASA'sHubble Space Telescope.[30][31] 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 ofHubble Space Telescope andKeck telescope imaging and spectroscopy. The discovery and analysis of theIRC 0218 lens was published in theAstrophysical Journal Letters on June 23, 2014.[32]

Research published Sep 30, 2013 in the online edition ofPhysical Review Letters, led byMcGill University inMontreal,Québec, Canada, has discovered theB-modes, that are formed due to gravitational lensing effect, usingNational Science Foundation'sSouth 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.[33][34]

Abell 2744galaxy cluster - extremely distantgalaxies revealed bygravitational lensing (16 October 2014).[35][36]

Solar gravitational lens

[edit]
Main article:Solar gravitational lens

Albert Einstein predicted in 1936 that rays of light from the same direction that skirt the edges of theSun would converge to a focal point approximately 542AU from the Sun.[37] 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.[38] 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 asVoyager 1, and beyond the known planets and dwarf planets, though over thousands of years90377 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-cmhydrogen line, led to the suggestion byFrank Drake in the early days ofSETI that a probe could be sent to this distance. A multipurpose probe SETISAIL and laterFOCAL was proposed to the ESA in 1993, but is expected to be a difficult task.[39] 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.[40] A critique of the concept was given by Landis,[41] 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 inherentspherical aberration of the lens.

In 2020, NASA physicistSlava Turyshev presented his idea of Direct Multipixel Imaging and Spectroscopy of an Exoplanet with aSolar 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.[42]

Measuring weak lensing

[edit]
Galaxy cluster MACS J2129-0741 and lensed galaxy MACS2129-1.[43]

Kaiser, Squires and Broadhurst (1995),[44] Luppino & Kaiser (1997)[45] and Hoekstra et al. (1998) prescribed a method to invert the effects of thepoint 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.[46][47]

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.[48]

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.LSST) may need much better accuracy than KSB can provide.

Gallery

[edit]
  • Sunburst Arc galaxy.[49]
    Sunburst Arc galaxy.[49]
  • Gravitationally lensed Quasar.[50]
    Gravitationally lensed Quasar.[50]
  • In SDSS J0952+3434, the lower arc-shaped galaxy has the characteristic shape of a galaxy that has been gravitationally lensed.[51]
    In SDSS J0952+3434, the lower arc-shaped galaxy has the characteristic shape of a galaxy that has been gravitationally lensed.[51]
  • Warped and distorted around SDSS J1050+0017.[52]
    Warped and distorted around SDSS J1050+0017.[52]
  • Galaxy SPT0615-JD existed when the Universe was just 500 million years old.[53]
    GalaxySPT0615-JD existed when the Universe was just 500 million years old.[53]
  • Gravitational lenses found in the DESI Legacy Survey data[54]
    Gravitational lenses found in the DESI Legacy Survey data[54]
  • Gravitational lenses found in the DESI Legacy Survey data[55]
    Gravitational lenses found in the DESI Legacy Survey data[55]
  • The lensing phenomenon allows for features as small as about 100 light-years or less.[56]
    The lensing phenomenon allows for features as small as about 100 light-years or less.[56]
  • Detailed look at a gravitationally lensed type Ia supernova iPTF16geu.[57]
    Detailed look at a gravitationally lensedtype Ia supernovaiPTF16geu.[57]
  • "Smiley" image of galaxy cluster (SDSS J1038+4849) & gravitational lensing (an Einstein ring) (HST).[58]
    "Smiley" image ofgalaxy cluster (SDSS J1038+4849) & gravitational lensing (anEinstein ring) (HST).[58]
  • Abell 1689 - actual gravitational lensing effects (Hubble Space Telescope).
    Abell 1689 - actual gravitational lensing effects (Hubble Space Telescope).
  • Dark matter distribution - weak gravitational lensing (Hubble Space Telescope).
    Dark matter distribution - weak gravitational lensing (Hubble Space Telescope).
  • Gravitational lens discovered at redshift z = 1.53.[59]
    Gravitational lens discovered atredshift z = 1.53.[59]
  • Gravitational Lensing Graphic (January 8, 2020)
    Gravitational Lensing Graphic (January 8, 2020)
  • Hubble Showcases Hamilton's Object
    Hubble Showcases Hamilton's Object
  • Illustration of gravitational lensing[60]
    Illustration of gravitational lensing[60]
  • Gravitationally-lensed distant star-forming galaxies.[61]
    Gravitationally-lensed distant star-forming galaxies.[61]

See also

[edit]

References

[edit]

Notes

  1. ^Drakeford, Jason; Corum, Jonathan; Overbye, Dennis (March 5, 2015)."Einstein's Telescope - video (02:32)".New York Times. RetrievedDecember 27, 2015.
  2. ^Overbye, Dennis (March 5, 2015)."Astronomers Observe Supernova and Find They're Watching Reruns".New York Times. RetrievedMarch 5, 2015.
  3. ^Bernard F. Schutz (1985).A First Course in General Relativity (illustrated, herdruk ed.). Cambridge University Press. p. 295.ISBN 978-0-521-27703-7.
  4. ^Wolfgang Rindler (2006).Relativity: Special, General, and Cosmological (2nd ed.). OUP Oxford. p. 21.ISBN 978-0-19-152433-2.Extract of page 21
  5. ^Gabor Kunstatter; Jeffrey G Williams; D E Vincent (1992).General Relativity And Relativistic Astrophysics - Proceedings Of The 4th Canadian Conference. World Scientific. p. 100.ISBN 978-981-4554-87-9.Extract of page 100
  6. ^Pekka Teerikorpi; Mauri Valtonen; K. Lehto; Harry Lehto; Gene Byrd; Arthur Chernin (2008).The Evolving Universe and the Origin of Life: The Search for Our Cosmic Roots (illustrated ed.). Springer Science & Business Media. p. 165.ISBN 978-0-387-09534-9.Extract of page 165
  7. ^abTurner, Christina (February 14, 2006)."The Early History of Gravitational Lensing"(PDF). Archived fromthe original(PDF) on July 25, 2008.
  8. ^Bičák, Jiří; Ledvinka, Tomáš (2014).General Relativity, Cosmology and Astrophysics: Perspectives 100 years after Einstein's stay in Prague (illustrated ed.). Springer. pp. 49–50.ISBN 9783319063492.
  9. ^abTilman Sauer (2008). "Nova Geminorum 1912 and the Origin of the Idea of Gravitational Lensing".Archive for History of Exact Sciences.62 (1):1–22.arXiv:0704.0963.Bibcode:2008AHES...62....1S.doi:10.1007/s00407-007-0008-4.S2CID 17384823.
  10. ^abc"A brief history of gravitational lensing".Einstein Online.Max Planck Institute for Gravitational Physics. Archived fromthe original on 2016-07-01. Retrieved2016-06-29.
  11. ^Lodge, Oliver J. (December 1919)."Gravitation and Light".Nature.104 (2614): 354.Bibcode:1919Natur.104..354L.doi:10.1038/104354a0.ISSN 0028-0836.S2CID 4157815.
  12. ^"Gravity Lens – Part 2 (Great Moments in Science, ABS Science)".Australian Broadcasting Corporation. 5 November 2001.
  13. ^abSchneider, Peter; Ehlers, Jürgen; Falco, Emilio E. (1992).Gravitational Lenses. Springer-Verlag Berlin Heidelberg New York Press.ISBN 978-3-540-97070-5.
  14. ^Melia, Fulvio (2007).The Galactic Supermassive Black Hole. Princeton University Press. pp. 255–256.ISBN 978-0-691-13129-0.
  15. ^Soldner, J. G. V. (1804)."On the deflection of a light ray from its rectilinear motion, by the attraction of a celestial body at which it nearly passes by" .Berliner Astronomisches Jahrbuch:161–172.
  16. ^Newton, Isaac (1998).Opticks: or, a treatise of the reflexions, refractions, inflexions and colours of light. Also two treatises of the species and magnitude of curvilinear figures. Commentary by Nicholas Humez (Octavo ed.). Palo Alto, Calif.: Octavo.ISBN 978-1-891788-04-8. (Opticks was originally published in 1704).
  17. ^Will, C.M. (2006)."The Confrontation between General Relativity and Experiment".Living Reviews in Relativity.9 (1): 39.arXiv:gr-qc/0510072.Bibcode:2006LRR.....9....3W.doi:10.12942/lrr-2006-3.PMC 5256066.PMID 28179873.
  18. ^Dyson, F. W.; Eddington, A. S.; Davidson C. (1920)."A determination of the deflection of light by the Sun's gravitational field, from observations made at the total eclipse of 29 May 1919".Philosophical Transactions of the Royal Society.220A (571–581):291–333.Bibcode:1920RSPTA.220..291D.doi:10.1098/rsta.1920.0009.
  19. ^Stanley, Matthew (2003). "'An Expedition to Heal the Wounds of War': The 1919 Eclipse and Eddington as Quaker Adventurer".Isis.94 (1):57–89.Bibcode:2003Isis...94...57S.doi:10.1086/376099.PMID 12725104.S2CID 25615643.
  20. ^Dyson, F. W.; Eddington, A. S.; Davidson, C. (1 January 1920)."A Determination of the Deflection of Light by the Sun's Gravitational Field, from Observations Made at the Total Eclipse of May 29, 1919".Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences.220 (571–581):291–333.Bibcode:1920RSPTA.220..291D.doi:10.1098/rsta.1920.0009.
  21. ^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.)
  22. ^F. Zwicky (1937)."Nebulae as Gravitational lenses"(PDF).Physical Review.51 (4): 290.Bibcode:1937PhRv...51..290Z.doi:10.1103/PhysRev.51.290.Archived(PDF) from the original on 2013-12-26.
  23. ^Schneider Peter; Kochanek, Christopher; Wambsganss, Joachim (2006).Gravitational Lensing: Strong, Weak and Micro. Springer Verlag Berlin Heidelberg New York Press. p. 4.ISBN 978-3-540-30309-1.
  24. ^Walsh, D.; Carswell, R. F.; Weymann, R. J. (31 May 1979). "0957 + 561 A, B: twin quasistellar objects or gravitational lens?".Nature.279 (5712):381–384.Bibcode:1979Natur.279..381W.doi:10.1038/279381a0.PMID 16068158.S2CID 2142707.
  25. ^abMeneghetti, Massimo (2021).Introduction to Gravitational Lensing With Python Examples. Lecture Notes in Physics. Vol. 956. Springer.doi:10.1007/978-3-030-73582-1.ISBN 978-3-030-73582-1.S2CID 243826707.
  26. ^Susskind, Leonard; Friedman, Art (2018).Special Relativity and Classical Field Theory. Penguin Books.ISBN 9780141985015.
  27. ^"Helping Hubble".www.spacetelescope.org. Retrieved29 October 2018.
  28. ^Cassan, A.; Kubas, D.; Beaulieu, J.-P.; Dominik, M.; Horne, K.; Greenhill, J.; Wambsganss, J.; Menzies, J.; Williams, A. (2012). "One or more bound planets per Milky Way star from microlensing observations".Nature.481 (7380):167–169.arXiv:1202.0903.Bibcode:2012Natur.481..167C.doi:10.1038/nature10684.PMID 22237108.S2CID 2614136.
  29. ^DOE/Lawrence Berkeley National Laboratory (January 21, 2010)."Cosmology: Weak gravitational lensing improves measurements of distant galaxies".ScienceDaily.
  30. ^Sci-News.com (21 Oct 2013)."Most Distant Gravitational Lens Discovered".Sci-News.com. Archived fromthe original on 23 October 2013. Retrieved22 October 2013.
  31. ^van der Wel, A.; et al. (2013). "Discovery of a Quadruple Lens in CANDELS with a Record Lens Redshift".Astrophysical Journal Letters.777 (1): L17.arXiv:1309.2826.Bibcode:2013ApJ...777L..17V.doi:10.1088/2041-8205/777/1/L17.S2CID 55728208.
  32. ^Wong, K.; et al. (2014). "Discovery of a Strong Lensing Galaxy Embedded in a Cluster at z = 1.62".Astrophysical Journal Letters.789 (2): L31.arXiv:1405.3661.Bibcode:2014ApJ...789L..31W.doi:10.1088/2041-8205/789/2/L31.S2CID 56376674.
  33. ^NASA/Jet Propulsion Laboratory (October 22, 2013)."Long-sought pattern of ancient light detected".ScienceDaily. RetrievedOctober 23, 2013.
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