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Bäcklund transformations in general relativity

  • Francisco Javier Chinea
Workshop
Part of the Lecture Notes in Physics book series (LNP, volume 189)

Keywords

Gauge Transformation Real Constant Constant Negative Curvature Pseudospherical Surface Permutability Property 
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References

  1. [1]
    F. J. Ernst, New formulation of the axially symmetric gravitational field problem. Phys. Rev. 187, 1175–1178 (1968).CrossRefGoogle Scholar
  2. [2]
    D. Maison, Are the stationary, axially symmetric Einstein equations completely integrable? Phys. Rev. Lett. 41, 521–522 (1978); ib. On the complete integrability of the stationary, axially symmetric Einstein equations. J. Math. Phys. 20, 871–877 (1979).CrossRefGoogle Scholar
  3. [3]
    R. Geroch, A method for generating new solutions of Einstein's equation. II. J. Math. Phys. 13, 394–404 (1972).CrossRefGoogle Scholar
  4. [4]
    W. Kinnersley, Symmetries of the stationary Einstein-Maxwell field equations. I. J. Math. Phys. 18, 1529–1537 (1977).CrossRefGoogle Scholar
  5. [5]
    W. Kinnersley and D. M. Chitre, Group transformation that generates the Kerr and Tomimatsu-Sato metrics. Phys. Rev. Lett. 40, 1608–1610 (1978).CrossRefGoogle Scholar
  6. [6]
    C. Hoenselaers, W. Kinnerley, and B. C. Xanthopoulos, Generation of asymptotically flat, stationary space-times with any number of parameters. Phys. Rev. Lett. 42, 481–482 (1979); ib. Symmetries of the stationary Einstein-Maxwell equations. VI. Transformations which generate asymptotically flat spacetimes with arbitrary multipole moments. J. Math. Phys. 20, 2530-2536 (1979).CrossRefGoogle Scholar
  7. [7]
    V. A. Belinskiî and V. E. Zakharov, Integration of the Einstein equations by means of the inverse scattering problem technique and construction of exact soliton solutions. Zh. Eksp. Teor. Fiz. 75, 1955–1971 (1978) [Sov. Phys. JETP 48, 985-994 (1978)].Google Scholar
  8. [8]
    I. Hauser and F. J. Ernst, A homogeneous Hilbert problem for the Kinnersley-Chitre transformations. J. Math. Phys. 21, 1126–1140 (1980); see also Professor Ernst's lectures in these Proceedings.CrossRefGoogle Scholar
  9. [9]
    B. K. Harrison, Bäcklund transformation for the Ernst equation of general relativity. Phys. Rev. Lett. 41, 1197–1200 (1978); 1835 (E)(1978).CrossRefGoogle Scholar
  10. [10]
    G. Neugebauer, Bäcklund transformations of axially symmetric stationary gravitational fields. J. Phys. A12, L67–L70 (1979).Google Scholar
  11. [11]
    G. Neugebauer and D. Kramer, Generation of the Kerr-NUT solution from flat space-time by Bäcklund transformations. Exp. Tech. Phys. 28, 3–8 (1980); G. Neugebauer, A general integral of the axially symmetric stationary Einstein equations. J. Phys. A13, L19-L21 (1980); D. Kramer and G. Neugebauer, The superposition of two Kerr solutions. Phys. Lett. A75, 259–261 (1980).Google Scholar
  12. [12]
    B. K. Harrison, New large family of vacuum solutions of the equations of general relativity. Phys. Rev. D21, 1695–1697 (1980).Google Scholar
  13. [13]
    M. Omote and M. Wadati, Bäcklund transformations for the Ernst equation. J. Math. Phys. 22, 961–964 (1981).CrossRefGoogle Scholar
  14. [14]
    F. J. Chinea, Integrability formulation and Bäcklund transformations for gravitational fields with symmetries. Phys. Rev. D24, 1053–1055 (1981); D26, 2175 (E)(1982); ib. Bundle connections and Bäcklund transformations for gravitational fields with isometries. Physicsa A114, 151–153 (1982).Google Scholar
  15. [15]
    C. M. Cosgrove, Relationships between the group-theoretic and soliton-theoretic techniques for generating stationary axisymmetric gravitational solutions. J. Math. Phys. 21, 2417–2447 (1980).CrossRefGoogle Scholar
  16. [16]
    G. Darboux, Leçons sur la Théorie Générale des Surfaces, vol. 3, ch. XII–XIII. Gauthier-Villars, Paris, 1894; L. Bianchi, Lezioni di Geometria Differenziale, vol. 1, Part 2, ch. XV–XVI. N. Zanichelli, Bologna 1927; L. P. Eisenhart, A Treatise on the Differential Geometry of Curves and Surfaces, chapter VIII. Dover, New York 1960.Google Scholar
  17. [17]
    E. Goursat, Le Problème de Bäcklund. Gauthier-Villars, Paris, 1925.Google Scholar
  18. [18]
    A. Neveu and N. Papanicolaou, Integrability of the classical \(\left[ {\bar \psi _i \psi _i } \right]_2^2 \) and \(\left[ {\bar \psi _i \psi _i } \right]_2^2 - \left[ {\bar \psi _i \gamma _5 \psi _i } \right]_2^2 \) interactions. Commun. Math. Phys. 58, 31–64 (1978); M. Crampin, Solitons and SL(2,R). Phys. Lett. A68, 170-172 (1978); R. Sasaki, Soliton equations and pseudospherical surfaces. Nucl. Phys. B154, 343–357 (1979); F. J. Chinea, On the intrinsic geometry of certain nonlinear equations: The sine-Gordon equation. J. Math. Phys. 21, 1588–1592 (1980).Google Scholar
  19. [19]
    K. Pohlmeyer, Integrable hamiltonian systems and interactions through quadratic constraints. Commun. Math. Phys. 4B, 207–221 (1976).Google Scholar
  20. [20]
    V. E. Zakharov and A. B. Shabat, Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Zh. Eksp. Teor. Fiz. 61, 118–134 (1971) [Sov. Phys. JETP 34, 62–69 (1972)]; M. J. Ablowitz, D. J. Kaup, A. C. Newell, and H. Segur, The inverse scattering transform —Fourier analysis for nonlinear problems. Stud. Appl. Math. 53, 249–315 (1974).Google Scholar
  21. [21]
    M. Crampin, F. A. E. Pirani, and D. C. Robinson, The soliton connection. Lett. Math. Phys. 2, 15–19 (1977).Google Scholar
  22. [22]
    T. Lewis, Some special solutions of the equations of axially symmetric gravitational fields. Proc. Roy. Soc. (London) A13B, 176–192 (1932); A. Papapetrou, Eine rotationssymmetrische Lösung in der allgemeinen Relativitätstheorie. Ann. Phys. (Leipzig) 12, 309–315 (1953).Google Scholar
  23. [23]
    J. Ehlers, in Les Théories Relativistes de la Gravitation. CNRS, Paris 1959.Google Scholar
  24. [24]
    F. J. Chinea, New Bäcklund transformations and superposition principle for gravitational fields with symmetries. Phys. Rev. Lett. 50, 221–224 (1983).Google Scholar

Copyright information

© Springer-Verlag 1983

Authors and Affiliations

  • Francisco Javier Chinea
    • 1
  1. 1.Departamento de Métodos Matemáticos de la Física Facultad de Ciencias FísicasUniversidad de MadridSpain

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