On the homogeneous Hilbert problem for effecting Kinnersley-Chitre transformations

  • Isidore Hauser
Conference paper
Part of the Lecture Notes in Physics book series (LNP, volume 205)


For each real number zo, the multiplicative group of all 2 x 2 unimodular matrix functions v(τ) of a complex variable τ which are holomorphic at zo and are real for real τ is a realization of the Geroch group; the union of these realizations is denoted by K(R). The HHP (homogeneous Hilbert problem) of Hauser and Ernst is given an advantageous reformulation which employs K(R) and which dispenses with the use of contours. The use of and the potentials which occur in the HHP are discussed. A complex extension K(C) of K(R) is introduced by dropping the reality conditions, n-parameter families of members of K(C) are discussed and examples of analytic continuation in the complex parameter space are given.Some basic solutions of the HHP are reviewed; in particular an alternative form and derivation of Cosgrove's solution for the double Harrison family are given.


Fundamental Solution Branch Point Analytic Continuation Complex Extension Translational Mapping 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    W. Kinnersley and D. Chitre, J. Math. Phys. 18, 1538 (1977); J. Math. Phys. 19, 1926 (1978); J. Math.Phys. 19, 2037 (1978).Google Scholar
  2. 2.
    R. Geroch. J. Math. Phys. 12, 918 (1971); J. Math. Phys. 13, 394 (1972).Google Scholar
  3. 3.
    I. Hauser and F. J. Ernst, Phys. Rev. D20, 362 (1979).Google Scholar
  4. 4.
    I. Hauser and F. J. Ernst, J. Math. Phys. 20, 1783 (1979); J. Math. Phys. 21, 1418 (1980). (Electrovacs).Google Scholar
  5. 5.
    I. Hauser and F. J. Ernst, J. Math. Phys. 21, 1126 (1980).Google Scholar
  6. 6.
    I. Hauser and F. J. Ernst, J. Math. Phys. 22, 1051 (1981).Google Scholar
  7. 7.
    C. M. Cosgrove, J. Math. Phys. 22, 2624 (1981).(A preview of our form of the solution for the double Harrison was given by D. Guo and F. Ernst, J. Math. Phys., 23, 1359 (1982).)Google Scholar
  8. 8.
    C. M. Cosgrove, J. Math. Phys. 23, 615 (1982). The idea of dropping the contour in our HHP was proposed by Cosgrove in Sec. 1 of this paper.Google Scholar
  9. 9.
    B. K. Harrison, Phys. Rev. Lett. 41, 1197 (1978); Phys. Rev. D21, 1695 (1980).Google Scholar
  10. 10.
    V. A. Belinsky and V. E. Zakharov, Sov. Phys. JETP 48, 985 (1978); Sov. Phys. JETP 50, 1 (1979).Google Scholar
  11. 11.
    N. I. Muskhelishvili, Singular Integral Equations (Noordhoff, Grongingen, 1953) especially Chap. 18. Our X± is the transpose of his X±, and our G is his (GT)−l.To say that G satisfies a Hölder condition on L means there exist 0 < μ > l and A > 0 such that ‖ G(s′)-G(s)‖ < A|s′-s μ for all s,s′ on L where ‖ M‖ denotes the norm of M.Google Scholar
  12. 12.
    This means that there exists no other holomorphic function fa b(τ) of τ such that va b(τ) = fa b(τ) for all τ in at least one neighborhood of zo and such that dom va b is a proper subset of dom fa b.Google Scholar
  13. 13.
    The domain of v as defined here may not be connected, but it always has a connected component which covers zo.Google Scholar
  14. 14.
    The corresponding transformation of F is given by Eq. (53) in the reference in footnote 6.Google Scholar
  15. 15.
    This implies that Λ must be chosen so that it lies in that connected component of dom v which covers the origin.Google Scholar
  16. 16.
    A special case of this relation was derived by B. Xanthopoulos (preprint) prior to the reference in footnote 6.Google Scholar
  17. 17.
    A complex-analytic atlas A on a set M is here defined as a collection of one-by-one functions (regarded as sets of ordered pairs) which map subsets of M onto open subsets of cn such that (l) the union of the domains of these functions is M (2) if σl, σ2 are members of A with respective domains Ul, U2, then σl, (Ul ⋂ U2) is open in cn (3) and σl · (σ2)−l is a complex-analytic mapping of σ2 (Ul ⋂ U2) onto σl (Ul ⋂ U2). A topology is introduced into M, by using the set of all domains of the charts in a maximal (saturated) A as a basis.Google Scholar
  18. 18.
    A different derivation and form of this solution is given by Cosgrove in the reference in footnote 8. See his Eq. (3.7).Google Scholar
  19. 19.
    C. M. Cosgrove, J. Phys. A: Math. Gen. 10, 1481 (1977); J. Phys. A: Math. Gen. 10, 2093 (1977); J. Phys. A: Math. Gen. 11, 2405 (1978).Google Scholar
  20. 20.
    M. Yamazaki, J. Math. Phys. 19, 1847 (1978).Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • Isidore Hauser
    • 1
  1. 1.Department of PhysicsIllinois Institute of TechnologyChicagoU.S.A.

Personalised recommendations