Wedge dislocations and three-dimensional gravity

Research Articles


The expression for the free energy of arbitrary static distribution of wedge dislocations in elastic media is proposed. In the framework of geometric theory of defects, the free energy is given by the Euclidean action for (1+2)-dimensional gravity interacting with N point particles. Relative movement of particles in gravity corresponds to bending of dislocations. The equations of equilibrium are obtained and analyzed. For two dislocations, the solution is found explicitly through hypergeometric functions.

Key words

dislocation gravity differential geometry 


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  1. 1.
    M. O. Katanaev and I. V. Volovich, “Theory of defects in solids and three-dimensional gravity,” Ann. Phys. 216, 1–28 (1992).MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    M. O. Katanaev, “Wedge dislocation in the geometric theory of defects,” Theor. Math. Phys. 135, 733–744 (2003).MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    M. O. Katanaev, “One-dimensional topologically nontrivial solutions in the Skyrme model,” Theor. Math. Phys. 138, 163–176 (2004).MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    M. O. Katanaev, “Geometric theory of defects,” Physics — Uspekhi, 48, 675–701 (2005).CrossRefGoogle Scholar
  5. 5.
    A. Bellini, M. Ciafaloni and P. Valtancoli, “Non-perturbative particle dynamics in (2+1)-gravity,” Phys. Lett. B 357, 532–538 (1995).MathSciNetCrossRefGoogle Scholar
  6. 6.
    A. Bellini, M. Ciafaloni and P. Valtancoli, “(2 + 1)-Gravity withmoving particles in an instantaneous gauge,” Nucl. Phys. B 454, 449–463 (1995).MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    P. Menotti and D. Seminara, “ADM approach to 2 +1-dimensional gravity coupled to particles,” Ann. Phys. 279, 282–310 (2000).MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    M. Welling, “Gravity in 2+1 dimensions as aRiemann-Hilbert problem,” Class. Quantum Grav. 13, 653–679 (1996).MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    R. Arnowitt, S. Deser and S. W. Misner, “The dynamics of general general relativity,” in Gravitation: an Introduction to Current Research (John Wiley & Sons, Inc., New York — London, 1962); arXiv:grqc/0405109.Google Scholar
  10. 10.
    P. A. M. Dirac, “The theory of gravitation in Hamiltonian form,” Proc. Roy. Soc. London A 246, 333–343 (1958).MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    V. S. Vladimirov, Equations of Mathematical Physics (Nauka, Moscow, 1988) [in Russian].MATHGoogle Scholar
  12. 12.
    L. Liouville, “Sur l’équation aux différences partielles,” J. Math. Pures Appl. 18, 71 (1853).Google Scholar
  13. 13.
    H. Poincaré, “Les Functions Fuchsiennes et l’équation Δu = eu,” J. Math. PureAppl. Ser. 5 4, 157 (1898).Google Scholar
  14. 14.
    V. V. Golubev, Lectures on Analytic Theory of Differential Equations (Gos. izd. tekhniko-teoreticheskoi lit. Moscow-Leningrad, 1950) [in Russian].Google Scholar
  15. 15.
    A. A. Bolibrukh, Fucshian Differential Equations and Holomorphic Fiber Bundles (MTsNMO, 2000) [in Russian].Google Scholar
  16. 16.
    M. O. Katanaev, “Effective action for scalar fields in two-dimensional gravity,” Ann. Phys. 296, 1–50 (2002).MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2012

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia

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