Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 7 (2011), 118, 27 pages      arXiv:1106.4122      https://doi.org/10.3842/SIGMA.2011.118

The Non-Autonomous Chiral Model and the Ernst Equation of General Relativity in the Bidifferential Calculus Framework

Aristophanes Dimakis a, Nils Kanning b and Folkert Müller-Hoissen c
a) Department of Financial and Management Engineering, University of the Aegean, 41, Kountourioti Str., 82100 Chios, Greece
b) Institute for Mathematics and Institute for Physics, Humboldt University, Rudower Chaussee 25, 12489 Berlin, Germany
c) Max-Planck-Institute for Dynamics and Self-Organization, Bunsenstrasse 10, 37073 Göttingen, Germany

Received August 31, 2011, in final form December 16, 2011; Published online December 23, 2011

Abstract
The non-autonomous chiral model equation for an m×m matrix function on a two-dimensional space appears in particular in general relativity, where for m=2 a certain reduction of it determines stationary, axially symmetric solutions of Einstein's vacuum equations, and for m=3 solutions of the Einstein-Maxwell equations. Using a very simple and general result of the bidifferential calculus approach to integrable partial differential and difference equations, we generate a large class of exact solutions of this chiral model. The solutions are parametrized by a set of matrices, the size of which can be arbitrarily large. The matrices are subject to a Sylvester equation that has to be solved and generically admits a unique solution. By imposing the aforementioned reductions on the matrix data, we recover the Ernst potentials of multi-Kerr-NUT and multi-Deminski-Newman metrics.

Key words: bidifferential calculus; chiral model; Ernst equation; Sylvester equation.

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References

  1. Dimakis A., Müller-Hoissen F., Bidifferential graded algebras and integrable systems, Discrete Contin. Dyn. Syst. 2009 (2009), suppl., 208-219, arXiv:0805.4553.
  2. Kinnersley W., Symmetries of the stationary Einstein-Maxwell field equations. I, J. Math. Phys. 18 (1977), 1529-1537.
  3. Maison D., Are the stationary, axially symmetric Einstein equations completely integrable?, Phys. Rev. Lett. 41 (1978), 521-522.
  4. Maison D., On the complete integrability of the stationary, axially symmetric Einstein equations, J. Math. Phys. 20 (1979), 871-877.
  5. Belinski V., Zakharov V., Integration of the Einstein equations by means of the inverse scattering problem technique and construction of exact soliton solutions, Soviet Phys. JETP 48 (1978), 985-994.
  6. Belinski V., Sakharov V., Stationary gravitational solitons with axial symmetry, Soviet Phys. JETP 50 (1979), 1-9.
  7. Belinski V., Verdaguer E., Gravitational solitons, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2001.
  8. Stephani H., Kramer D., MacCallum M., Hoenselaers C., Herlt E., Exact solutions of Einstein's field equations, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2003.
  9. Lee S.-C., Generalized Neugebauer-Kramer transformations for nonlinear sigma models, Phys. Lett. B 164 (1985), 75-79.
  10. Lee S.-C., Soliton transformations for axially symmetric higher-dimensional gravity. II. Belinkii-Zakharov N-soliton transformations, J. Math. Phys. 28 (1987), 901-904.
  11. Wei Y.-H., Extended Ernst equation and five-dimensional axisymmetric Kaluza-Klein solutions, Classical Quantum Gravity 18 (2001), 2163-2170.
  12. Harmark T., Stationary and axisymmetric solutions of higher-dimensional general relativity, Phys. Rev. D 70 (2004), 124002, 25 pages, hep-th/0408141.
  13. Pomeransky A.A., Complete integrability of higher-dimensional Einstein equations with additional symmetry and rotating black holes, Phys. Rev. D 73 (2006), 044004, 5 pages, hep-th/0507250.
  14. Azuma T., Koikawa T., An infinite number of stationary soliton solutions to the five-dimensional vacuum Einstein equation, Progr. Theoret. Phys. 116 (2006), 319-328, hep-th/0512350.
  15. Iguchi H., Mishima T., Solitonic generation of vacuum solutions in five-dimensional general relativity, Phys. Rev. D 74 (2006), 024029, 17 pages, hep-th/0605090.
  16. Tomizawa S., Morisawa Y., Yasui Y., Vacuum solutions of five dimensional Einstein equations generated by inverse scattering method, Phys. Rev. D 73 (2006), 064009, 8 pages, hep-th/0512252.
  17. Yazadjiev S.S., 5D Einstein-Maxwell solitons and concentric rotating dipole black rings, Phys. Rev. D 78 (2008), 064032, 11 pages, arXiv:0805.1600.
  18. Emparan R., Reall H.S., Black holes in higher dimensions, Living Rev. Relativity 11 (2008), 6, 87 pages, arXiv:0801.3471.
  19. Figueras P., Jamsin E., Rocha J.V., Virmani A., Integrability of five-dimensional minimal supergravity and charged rotating black holes, Classical Quantum Gravity 27 (2010), 135011, 37 pages, arXiv:0912.3199.
  20. Mikhailov A.V., Yaremchuk A.I., Cylindrically symmetric solutions of the non-linear chiral field model (σ model), Nuclear Phys. B 202 (1982), 508-522.
  21. Gutshabash E.Sh., Lipovskii V.D., Exact solutions of a nonlinear sigma-model in a curved space and the theory of magnetically ordered media with variable saturation magnetization, J. Math. Sci. 77 (1995), 3063-3068.
  22. Alekseev G.A., N-soliton solutions of Einstein-Maxwell equations, JETP Lett. 32 (1980), 277-279.
  23. Alekseev G.A., Exact solutions in the general theory of relativity, Proc. Steklov Inst. Math. 176 (1988), 215-262.
  24. Neugebauer G., A general integral of the axially symmetric Einstein equations, J. Phys. A: Math. Gen. 13 (1980), L19-L21.
  25. Neugebauer G., Kramer D., Einstein-Maxwell solitons, J. Phys. A: Math. Gen. 16 (1983), 1927-1936.
  26. Kramer D., Neugebauer G., Matos T., Bäcklund transforms of chiral fields, J. Math. Phys. 32 (1991), 2727-2730.
  27. Korotkin D., Finite-gap solutions of the stationary axisymmetric Einstein equation in vacuum, Theoret. and Math. Phys. 77 (1989), 1018-1031.
  28. Manko V., Martín J., Ruiz E., Extended family of the electrovac two-solitons for the Einstein-Maxwell equations, Phys. Rev. D 51 (1995), 4187-4191.
  29. Ruiz E., Manko V., Martín J., Extended N-soliton solution of the Einstein-Maxwell equations, Phys. Rev. D 51 (1995), 4192-4197.
  30. Masuda T., Sasa N., Fukuyama T., Neugebauer-Kramer solutions of the Ernst equation in Hirota's direct method, J. Phys. A: Math. Gen. 31 (1998), 5717-5731.
  31. Burtsev S.P., Zakharov V.E., Mikhailov A.V., Inverse scattering method with variable spectral parameter, Theoret. and Math. Phys. 70 (1987), 227-240.
  32. Dimakis A., Müller-Hoissen F., Solutions of matrix NLS systems and their discretizations: a unified treatment, Inverse Problems 26 (2010), 095007, 55 pages.
  33. Dimakis A., Müller-Hoissen F., Bidifferential calculus approach to AKNS hierarchies and their solutions, SIGMA 6 (2010), 055, 27 pages, arXiv:1004.1627.
  34. Dimakis A., Kanning N., Müller-Hoissen F., Bidifferential calculus, matrix SIT and sine-Gordon equations, Acta Polytechnica 51 (2011), 33-37, arXiv:1011.1737.
  35. Witten L., Static axially symmetric solutions of self-dual SU(2) gauge fields in Euclidean four-dimensional space, Phys. Rev. D 19 (1979), 718-720.
  36. Forgács P., Horváth Z., Palla L., Generating the Bogomolny-Prasad-Sommerfield one-monopole solution by a Bäcklund transformation, Phys. Rev. Lett. 45 (1980), 505-508.
  37. Bais F.A., Sasaki R., On the complete integrability of the static axially symmetric self-dual gauge field equations for an arbitrary group, Nuclear Phys. B 195 (1982), 522-540.
  38. Ward R.S., Stationary axisymmetric space-times: a new approach, Gen. Relativity Gravitation 15 (1983), 105-109.
  39. Mason L.J., Woodhouse N.M.J., Integrability, self-duality, and twistor theory, London Mathematical Society Monographs. New Series, Vol. 15, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996.
  40. Klein C., Richter O., Ernst equation and Riemann surfaces. Analytical and numerical methods, Lecture Notes in Physics, Vol. 685, Springer-Verlag, Berlin, 2005.
  41. Kanning N., Integrable Systeme in der Allgemeinen Relativitätstheorie: ein Bidifferentialkalkül-Zugang, Diploma Thesis, University of Göttingen, 2010.
  42. Higham N.J., Computing real square roots of a real matrix, Linear Algebra Appl. 88/89 (1987), 405-430.
  43. Bernstein D.S., Matrix mathematics. Theory, facts, and formulas, 2nd ed., Princeton University Press, Princeton, NJ, 2009.
  44. Zakharov V.E., Mikhailov A.V., Relativistically invariant two-dimensional models of field theory which are integrable by means of the inverse scattering problem method, Soviet Phys. JETP 47 (1978), 1017-1027.
  45. Eichenherr H., Forger M., More about non-linear sigma models on symmetric spaces, Nuclear Phys. B 164 (1980), 528-535.
  46. Mazur P.O., Proof of uniqueness of the Kerr-Newman black hole solution, J. Phys. A: Math. Gen. 15 (1982), 3173-3180.
  47. Gürses M., Inverse scattering, differential geometry, Einstein-Maxwell solitons and one soliton Bäcklund transformations, in Solutions of Einstein's Equations: Techniques and Results (Retzbach, 1983), Editors C. Hoenselaers and W. Dietz, Lecture Notes in Phys., Vol. 205, Springer, Berlin, 1984, 199-234.
  48. Harrison B.K., New solutions of the Einstein-Maxwell equations from old, J. Math. Phys. 9 (1968), 1744-1752.
  49. Kinnersley W., Generation of stationary Einstein-Maxwell fields, J. Math. Phys. 14 (1973), 651-653.
  50. Aguilar-Sánchez J., García A.A., Manko V.S., Demianski-Newman solution revisited, Gravit. Cosmol. 7 (2001), 149-152, gr-qc/0106011.
  51. Kramer D., Neugebauer G., The superposition of two Kerr solutions, Phys. Lett. A 75 (1980), 259-261.
  52. Neugebauer G., Hennig J., Non-existence of stationary two-black-hole configurations, Gen. Relativity Gravitation 41 (2009), 2113-2130, arXiv:0905.4179.
  53. Gürses M., Xanthopoulos B., Axially symmetric, static self-dual SU(3) gauge fields and stationary Einstein-Maxwell metrics, Phys. Rev. D 26 (1982), 1912-1915.
  54. Alekseev G., Thirty years of studies of integrable reductions of Einstein's field equations, arXiv:1011.3846.
  55. Bhatia R., Rosenthal P., How and why to solve the operator equation AX−XB=Y, Bull. London Math. Soc. 29 (1997), 1-21.
  56. Garimella R.V., Hrynkiv V., Sourour A.R., A solution of an operator equation related to the KdV equation, Linear Algebra Appl. 418 (2006), 788-792.
  57. Marchenko V., Nonlinear equations and operator algebras, Mathematics and its Applications, Vol. 17, D. Reidel Publishing Co., Dordrecht, 1988.

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