General Relativity and Gravitation

, Volume 42, Issue 9, pp 2011–2046 | Cite as

Mathematics of gravitational lensing: multiple imaging and magnification

  • A. O. Petters
  • M. C. WernerEmail author
Review Article


The mathematical theory of gravitational lensing has revealed many generic and global properties. Beginning with multiple imaging, we review Morse-theoretic image counting formulas and lower bound results, and complex-algebraic upper bounds in the case of single and multiple lens planes. We discuss recent advances in the mathematics of stochastic lensing, discussing a general formula for the global expected number of minimum lensed images as well as asymptotic formulas for the probability densities of the microlensing random time delay functions, random lensing maps, and random shear, and an asymptotic expression for the global expected number of micro-minima. Multiple imaging in optical geometry and a spacetime setting are treated. We review global magnification relation results for model-dependent scenarios and cover recent developments on universal local magnification relations for higher order caustics.


Gravitational lensing Singularities 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Aazami A.B., Petters A.O.: A universal magnification theorem for higher-order caustic singularities. J. Math. Phys. 50, 032501 (2009)CrossRefMathSciNetADSGoogle Scholar
  2. 2.
    Aazami A.B., Petters A.O.: A universal magnification theorem II. Generic caustics up to codimension five. J. Math. Phys. 50, 082501 (2009)CrossRefMathSciNetADSGoogle Scholar
  3. 3.
    Aazami, A.B., Petters, A.O.: A universal magnification theorem III. Caustics beyond codimension five. J. Math. Phys. (2009), math-ph/0909.5235 (to appear)Google Scholar
  4. 4.
    Abramowicz M.A., Carter B., Lasota J.P.: Optical reference geometry for stationary and static dynamics. Gen. Relativ. Gravit. 20, 1173 (1988)zbMATHCrossRefMathSciNetADSGoogle Scholar
  5. 5.
    Adler R., Taylor J.: Random Fields and Geometry. Wiley, London (1981)Google Scholar
  6. 6.
    Arnold V.I.: Normal forms for functions near degenerate critical points, the Weyl groups of A k, D k, E k and Lagrangian singularities. Func. Anal. Appl. 6, 254 (1973)CrossRefGoogle Scholar
  7. 7.
    Arnold V.I.: Evolution of singularities of potential flows in collision-free media and the metamorphoses of caustics in three-dimensional space. J. Sov. Math. 32, 229 (1986)CrossRefGoogle Scholar
  8. 8.
    Arnold V.I., Gusein-Zade S.M., Varchenko A.N.: Singularities of Differentiable Maps, vol. 1. Birkhäuser, Boston (1985)Google Scholar
  9. 9.
    Arnold V.I., Gusein-Zade S.M., Varchenko A.N.: Singularities of Differentiable Maps, vol. 2. Birkhäuser, Boston (1985)Google Scholar
  10. 10.
    Atiyah M.F., Bott R.: A Lefschetz fixed point formula for elliptic complexes: I. Appl. Ann. Math. 86, 374 (1967)MathSciNetGoogle Scholar
  11. 11.
    Atiyah M.F., Bott R.: A Lefschetz fixed point formula for elliptic complexes: II. Appl. Ann. Math. 88, 451 (1968)MathSciNetGoogle Scholar
  12. 12.
    Azais J.M., Wschebor M.: On the distribution of the maximum of a Gaussian field with d parameters. Ann. Appl. Probab. 15(1A), 254 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Bayer J., Dyer C.C.: Maximal lensing: mass constraints on point lens configurations. Gen. Relativ. Gravit. 39, 1413 (2007)zbMATHCrossRefMathSciNetADSGoogle Scholar
  14. 14.
    Blandford R.D.: Gravitational lenses. Q. J. R. Astron. Soc. 31, 305 (1990)ADSGoogle Scholar
  15. 15.
    Blandford R., Narayan R.: Fermat’s principle, caustics, and the classification of gravitational lens images. Astrophys. J. 310, 568 (1986)CrossRefADSGoogle Scholar
  16. 16.
    Burke W.: Multiple gravitational imaging by distributed masses. Astrophys. J. Lett. 244, L1 (1981)CrossRefADSGoogle Scholar
  17. 17.
    Castrigiano D., Hayes S.: Catastrophe Theory. Addison-Wesley, Reading (2004)zbMATHGoogle Scholar
  18. 18.
    Chiba M.: Probing dark matter substructure in lens galaxies. Astrophys. J. 565, 17 (2002)CrossRefADSGoogle Scholar
  19. 19.
    Dalal N.: The magnification invariant of simple galaxy lens models. Astrophys. J. 509, 13 (1998)CrossRefADSGoogle Scholar
  20. 20.
    Dalal N., Rabin J.M.: Magnification relations in gravitational lensing via multidimensional residue integrals. J. Math. Phys. 42, 1818 (2001)zbMATHCrossRefMathSciNetADSGoogle Scholar
  21. 21.
    Ehlers J., Newman E.T.: The theory of caustics and wave front singularities with physical applications. J. Math. Phys. 41, 3344 (2000)zbMATHCrossRefMathSciNetADSGoogle Scholar
  22. 22.
    Evans N.W., Hunter C.: Lensing properties of cored galaxy models. Astrophys. J. 575, 68 (2002)CrossRefADSGoogle Scholar
  23. 23.
    Evans N.W., Witt H.J.: Are there sextuplet and octuplet image systems?. Mon. Not. R. Astron. Soc. 327, 1260 (2001)CrossRefADSGoogle Scholar
  24. 24.
    Frankel T.: Gravitational Curvature: An Introduction to Einstein’s Theory. W. H. Freeman, San Francisco (1979)zbMATHGoogle Scholar
  25. 25.
    Friedrich H., Stewart M.J.: Characteristic initial data and wavefront singularities in general relativity. Proc. R. Soc. Lond. A 385, 345 (1983)zbMATHCrossRefMathSciNetADSGoogle Scholar
  26. 26.
    Forrester P.J., Honner G.: Exact statistical properties of the zeros of complex random polynomials. J. Phys. A Math. Gen. 32, 2961 (1999)zbMATHCrossRefMathSciNetADSGoogle Scholar
  27. 27.
    Giannoni F., Lombardi M.: Gravitational lenses: odd or even images?. Class. Quantum Grav. 16, 1 (1999)MathSciNetADSGoogle Scholar
  28. 28.
    Giannoni F., Masiello A., Piccione P.: A Morse theory for light rays on stably causal Lorentzian manifolds. Ann. Inst. H. Poincaré Phys. Theor. 69, 359 (1998)zbMATHMathSciNetGoogle Scholar
  29. 29.
    Gibbons G.W.: No glory in cosmic string theory. Phys. Lett. B 308, 237 (1993)CrossRefMathSciNetADSGoogle Scholar
  30. 30.
    Gibbons G.W., Herdeiro C.A.R., Warnick C., Werner M.C.: Stationary metrics and optical Zermelo–Randers–Finsler geometry. Phys. Rev. D 79, 044022 (2009)CrossRefMathSciNetADSGoogle Scholar
  31. 31.
    Gibbons G.W., Warnick C.M.: Universal properties of the near-horizon optical geometry. Phys. Rev. D 79, 064031 (2009)CrossRefMathSciNetADSGoogle Scholar
  32. 32.
    Gibbons G.W., Werner M.C.: Applications of the Gauss–Bonnet theorem to gravitational lensing. Class. Quantum Grav. 25, 235009 (2008)CrossRefMathSciNetADSGoogle Scholar
  33. 33.
    Gilmore R.: Catastrophe Theory for Scientists and Engineers. Dover, New York (1981)zbMATHGoogle Scholar
  34. 34.
    Golubitsky M., Guillemin V.: Stable Mappings and Their Singularities. Springer, Berlin (1973)zbMATHGoogle Scholar
  35. 35.
    Gottlieb D.H.: A gravitational lens need not produce an odd number of images. J. Math. Phys. 35, 5507 (1994)zbMATHCrossRefMathSciNetADSGoogle Scholar
  36. 36.
    Granot J., Schechter P.L., Wambsganss J.: The mean number of extra microimage pairs for macrolensed quasars. Astrophys. J. 583, 575 (2003)CrossRefADSGoogle Scholar
  37. 37.
    Griffiths P., Harris J.: Principles of Algebraic Geometry. Wiley, New York (1994)zbMATHGoogle Scholar
  38. 38.
    Hunter C., Evans N.W.: Lensing properties of scale-free galaxies. Astrophys. J. 554, 1227 (2001)CrossRefADSGoogle Scholar
  39. 39.
    Katz N., Balbus S., Paczyński B.: Random scattering approach to gravitational microlensing. Astrophys. J. 306, 2 (1986)CrossRefADSGoogle Scholar
  40. 40.
    Keeton, C.R.: Gravitational lensing with stochastic substructure: Effects of the clump mass function and spatial distribution. (2009)
  41. 41.
    Keeton C., Gaudi S., Petters A.O.: Identifying lenses with small-scale structure. I. Cusp lenses. Astrophys. J. 598, 138 (2003)CrossRefADSGoogle Scholar
  42. 42.
    Keeton C., Gaudi S., Petters A.O.: Identifying lenses with small-scale structure. II. Fold lenses. Astrophys. J. 635, 35 (2005)CrossRefADSGoogle Scholar
  43. 43.
    Khavinson D., Neumann G.: On the number of zeros of certain rational harmonic functions. Proc. Am. Math. Soc. 134, 1077 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  44. 44.
    Kovner I.: Fermat principle in arbitrary gravitational fields. Astrophys. J. 351, 114 (1990)CrossRefADSGoogle Scholar
  45. 45.
    Li W.V., Wei A.: On the expected number of zeros of random harmonic polynomials. Proc. AMS 137, 195 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Low R.: Stable singularities of wave-fronts in general relativity. J. Math. Phys. 39, 3332 (1998)zbMATHCrossRefMathSciNetADSGoogle Scholar
  47. 47.
    Majthay A.: Foundations of Catastrophe Theory. Pitman, Boston (1985)zbMATHGoogle Scholar
  48. 48.
    Mao S., Petters A.O., Witt H.: Properties of point masses on a regular polygon and the problem of maximum number of images. In: Piran, T. (eds) Proceedings of the Eighth Marcel Grossman Meeting on General Relativity, World Scientific, Singapore (1997)Google Scholar
  49. 49.
    Mao S., Schneider P.: Evidence for substructure in lens galaxies?. Mon. Not. R. Astron. Soc. 295, 587 (1998)CrossRefADSGoogle Scholar
  50. 50.
    McKenzie R.H.: A gravitational lens produces an odd number of images. J. Math. Phys. 26, 1592 (1985)zbMATHCrossRefMathSciNetADSGoogle Scholar
  51. 51.
    Metcalf R.B., Madau P.: Compound gravitational lensing as a probe of dark matter substructure within galaxy halos. Astrophys. J. 563, 9 (2001)CrossRefADSGoogle Scholar
  52. 52.
    Milnor J.: Dynamics in One Complex Variable. Princeton University Press, Princeton (2006)zbMATHGoogle Scholar
  53. 53.
    Narasimha D., Subramanian K.: ‘Missing image’ in gravitational lens systems?. Nature 310, 112 (1986)Google Scholar
  54. 54.
    Nityananda R., Ostriker J.P.: Gravitational lensing by stars in a galaxy halo—theory of combined weak and strong scattering. J. Astrophys. Astron. 5, 235 (1984)CrossRefADSGoogle Scholar
  55. 55.
    Orban de Xivry, G., Marshall, P.: An atlas of predicted exotic gravitational lenses. astro-ph/0904.1454 (2009)Google Scholar
  56. 56.
    Padmanabhan T., Subramanian K.: The focusing equation, caustics and the condition for multiple imaging by thick gravitational lenses. Mon. Not. R. Astron. Soc. 233, 265 (1988)ADSGoogle Scholar
  57. 57.
    Perlick V.: On Fermat’s principle in general relativity: I. The general case. Class. Quantum Grav. 7, 1319 (1990)zbMATHCrossRefMathSciNetADSGoogle Scholar
  58. 58.
    Perlick V.: On Fermat’s principle in general relativity: II. The conformally stationary case. Class. Quantum Grav. 7, 1849 (1990)zbMATHCrossRefMathSciNetADSGoogle Scholar
  59. 59.
    Perlick V.: Infinite dimensional Morse theory and Fermat’s principle in general relativity I. J. Math. Phys. 36, 6915 (1995)zbMATHCrossRefMathSciNetADSGoogle Scholar
  60. 60.
    Perlick V.: Criteria for multiple imaging in Lorentzian manifolds. Class. Quantum Grav. 13, 529 (1996)zbMATHCrossRefMathSciNetADSGoogle Scholar
  61. 61.
    Perlick V.: Global properties of gravitational lens maps in a Lorentzian manifold setting. Commun. Math. Phys. 220, 403 (2001)zbMATHCrossRefMathSciNetADSGoogle Scholar
  62. 62.
    Perlick V.: Ray Optics, Fermat’s Principle, and Applications to General Relativity. Springer, Berlin (2000)zbMATHGoogle Scholar
  63. 63.
    Petters, A.O.: Singularities in gravitational microlensing. Ph.D. Thesis, MIT, Department of Mathematics (1991)Google Scholar
  64. 64.
    Petters A.O.: Morse theory and gravitational microlensing. J. Math. Phys. 33, 1915 (1992)CrossRefMathSciNetADSGoogle Scholar
  65. 65.
    Petters A.O.: Multiplane gravitational lensing. I. Morse theory and image counting. J. Math. Phys. 36, 4263 (1995)zbMATHCrossRefMathSciNetADSGoogle Scholar
  66. 66.
    Petters A.O.: Arnold’s singularity theory and gravitational lensing. J. Math. Phys. 33, 3555 (1993)CrossRefMathSciNetADSGoogle Scholar
  67. 67.
    Petters A.O.: Multiplane gravitational lensing III: upper bound on number of images. J. Math. Phys. 38, 1605 (1997)zbMATHCrossRefMathSciNetADSGoogle Scholar
  68. 68.
    Petters A.O., Levine H., Wambsganss J.: Singularitiy Theory and Gravitational Lensing. Birkäuser, Boston (2001)Google Scholar
  69. 69.
    Petters A.O., Rider B., Teguia A.M.: A mathematical theory of stochastic microlensing I. Random time delay functions and lensing maps. J. Math. Phys. 50, 072503 (2009)CrossRefMathSciNetADSGoogle Scholar
  70. 70.
    Petters, A.O., Rider, B., Teguia, A.M.: A mathematical theory of stochastic microlensing II. Random images, shear, and the Kac-Rice formula, to appear in J. Math. Phys. (2009), astro-ph/0807.4984v2Google Scholar
  71. 71.
    Petters A.O., Wicklin F.W.: Fixed points due to gravitational lenses. J. Math. Phys. 39, 1011 (1998)zbMATHCrossRefMathSciNetADSGoogle Scholar
  72. 72.
    Poston T., Stewart I.: Catastrophe Theory and its Applications. Dover, New York (1978)zbMATHGoogle Scholar
  73. 73.
    Rhie S.H.: Infimum microlensing amplification of the maximum number of images of n-point lens systems. Astrophys. J. 484, 67 (1997)CrossRefADSGoogle Scholar
  74. 74.
    Rhie, S.H.: n-point gravitational lenses with 5(n−1) images. astro-ph/0305166 (2003)Google Scholar
  75. 75.
    Renn J., Sauer T., Stachel J.: The origin of gravitational lensing: a postscipt to Einstein’s 1936 Science Paper. Science 275, 184 (1997)CrossRefMathSciNetADSGoogle Scholar
  76. 76.
    Schechter P.L., Wambsganss J.: Quasar microlensing at high magnification and the role of dark matter: enhanced fluctuations and suppressed saddle points. Astrophys. J. 580, 685 (2002)CrossRefADSGoogle Scholar
  77. 77.
    Schneider P., Ehlers J., Falco E.E.: Gravitational Lenses. Springer, Berlin (1992)Google Scholar
  78. 78.
    Schneider P., Weiss A.: The two-point mass lens: detailed investigation of a special asymmetric gravitational lens. Astron. Astrophys. 164, 237 (1986)ADSGoogle Scholar
  79. 79.
    Schneider P., Weiss A.: The gravitational lens equation near cusps. Astron. Astrophys. 260, 1 (1992)MathSciNetADSGoogle Scholar
  80. 80.
    Shin E.M., Evans N.W.: The Milky Way Galaxy as a strong gravitational lens. Mon. Not. R. Astron. Soc. 374, 1427 (2007)CrossRefADSGoogle Scholar
  81. 81.
    Shub M., Smale S.: Complexity of Bezout’s theorem. II. Volumes and Probabilities, Computational Algebraic Geometry, Nice (1992), Progress in Mathematics, vol. 109. Birkhäuser, Boston (1993)Google Scholar
  82. 82.
    Sodin M., Tsirelson B.: Random complex zeroes, I. Asymptotic normality. Israel J. Math. 144, 125 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  83. 83.
    Sodin M., Tsirelson B.: Random complex zeroes, II. Perturbed lattice. Israel J. Math. 152, 105 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  84. 84.
    Sodin M., Tsirelson B.: Random complex zeroes, III. Decay of the hole probability. Israel J. Math. 147, 371 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  85. 85.
    Subramanian K., Cowling S.: On local conditions for multiple imaging by bounded, smooth gravitational lenses. Mon. Not. R. Astron. Soc. 219, 333 (1986)ADSGoogle Scholar
  86. 86.
    Wambsganss J., Witt H.J., Schneider P.: Gravitational microlensing - powerful combination of ray-shooting and parametric representation of caustics. Astron. Astrophys. 258, 591 (1992)ADSGoogle Scholar
  87. 87.
    Werner M.C.: A Lefschetz fixed point theorem in gravitational lensing. J. Math. Phys. 48, 052501 (2007)CrossRefMathSciNetADSGoogle Scholar
  88. 88.
    Werner M.C.: Geometry of universal magnification invariants. J. Math. Phys. 50, 082504 (2009)CrossRefMathSciNetADSGoogle Scholar
  89. 89.
    Witt H.: Investigation of high amplification events in light curves of gravitationally lensed quasars. Astron. Astrophys. 236, 311 (1990)ADSGoogle Scholar
  90. 90.
    Witt H.J., Mao S.: On the minimum magnification between caustic crossings for microlensing by binary and multiple Stars. Astrophys. J. Lett. 447, 105 (1995)CrossRefADSGoogle Scholar
  91. 91.
    Witt H.J., Mao S.: On the magnification relations in quadruple lenses: a moment approach. Mon. Not. R. Astron. Soc. 311, 689 (2000)CrossRefADSGoogle Scholar
  92. 92.
    Zakharov A.: On the magnification of gravitational lens images near cusps. Astron. Astrophys. 293, 1 (1995)ADSGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsDuke UniversityDurhamUSA
  2. 2.Department of PhysicsDuke UniversityDurhamUSA

Personalised recommendations