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

Aristophanes Dimakis ; Nils Kanning ; Folkert Müller-Hoissen

Symmetry, integrability and geometry: methods and applications, Tome 7 (2011)

Résumé

The non-autonomous chiral model equation for an $m\times m$ 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-Demiański–Newman metrics.

Détail

Keywords: bidifferential calculus, chiral model, Ernst equation, Sylvester equation.

Bibliographie

[1] Dimakis A., Müller-Hoissen F., “Bidifferential graded algebras and integrable systems”, Discrete Contin. Dyn. Syst., 2009, suppl. (2009), 208–219 ; arXiv: 0805.4553 | DOI | MR | Zbl

[2] Kinnersley W., “Symmetries of the stationary Einstein–Maxwell field equations. I”, J. Math. Phys., 18 (1977), 1529–1537 | DOI

[3] Maison D., “Are the stationary, axially symmetric Einstein equations completely integrable?”, Phys. Rev. Lett., 41 (1978), 521–522 | DOI | MR

[4] Maison D., “On the complete integrability of the stationary, axially symmetric Einstein equations”, J. Math. Phys., 20 (1979), 871–877 | DOI | MR

[5] Belinskiǐ 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 | MR

[6] Belinskiǐ V., Sakharov V., “Stationary gravitational solitons with axial symmetry”, Soviet Phys. JETP, 50 (1979), 1–9 | MR

[7] Belinski V., Verdaguer E., Gravitational solitons, Cambridge Monographs on Mathematical Physics, Cambridge University Press, Cambridge, 2001 | DOI | MR | Zbl

[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 | DOI | MR

[9] Lee S.-C., “Generalized Neugebauer–Kramer transformations for nonlinear sigma models”, Phys. Lett. B, 164 (1985), 75–79 | DOI | MR

[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 | DOI | MR | Zbl

[11] Wei Y.-H., “Extended Ernst equation and five-dimensional axisymmetric Kaluza–Klein solutions”, Classical Quantum Gravity, 18 (2001), 2163–2170 | DOI | MR | Zbl

[12] Harmark T., “Stationary and axisymmetric solutions of higher-dimensional general relativity”, Phys. Rev. D, 70 (2004), 124002, 25 pp. ; arXiv: hep-th/0408141 | DOI | MR

[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 pp. ; arXiv: hep-th/0507250 | DOI | MR | Zbl

[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 ; arXiv: hep-th/0512350 | DOI | MR | Zbl

[15] Iguchi H., Mishima T., “Solitonic generation of vacuum solutions in five-dimensional general relativity”, Phys. Rev. D, 74 (2006), 024029, 17 pp. ; arXiv: hep-th/0605090 | DOI | MR

[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 pp. ; arXiv: hep-th/0512252 | DOI | MR

[17] Yazadjiev S.S., “5D Einstein–Maxwell solitons and concentric rotating dipole black rings”, Phys. Rev. D, 78 (2008), 064032, 11 pp. ; arXiv: 0805.1600 | DOI | MR

[18] Emparan R., Reall H.S., “Black holes in higher dimensions”, Living Rev. Relativity, 11 (2008), 6, 87 pp. ; arXiv: 0801.3471 | Zbl

[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 pp. ; arXiv: 0912.3199 | DOI | MR | Zbl

[20] Mikhailov A.V., Yaremchuk A.I., “Cylindrically symmetric solutions of the non-linear chiral field model ($\sigma$ model)”, Nuclear Phys. B, 202 (1982), 508–522 | DOI | MR

[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 | DOI | MR

[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 | MR | Zbl

[24] Neugebauer G., “A general integral of the axially symmetric Einstein equations”, J. Phys. A: Math. Gen., 13 (1980), L19–L21 | DOI | MR

[25] Neugebauer G., Kramer D., “Einstein–Maxwell solitons”, J. Phys. A: Math. Gen., 16 (1983), 1927–1936 | DOI | MR

[26] Kramer D., Neugebauer G., Matos T., “Bäcklund transforms of chiral fields”, J. Math. Phys., 32 (1991), 2727–2730 | DOI | MR | Zbl

[27] Korotkin D., “Finite-gap solutions of the stationary axisymmetric Einstein equation in vacuum”, Theoret. and Math. Phys., 77 (1988), 25–41 | DOI | MR

[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 | DOI | MR

[29] Ruiz E., Manko V., Martín J., “Extended $N$-soliton solution of the Einstein–Maxwell equations”, Phys. Rev. D, 51 (1995), 4192–4197 | DOI | MR

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[32] Dimakis A., Müller-Hoissen F., “Solutions of matrix NLS systems and their discretizations: a unified treatment”, Inverse Problems, 26 (2010), 095007, 55 pp. | DOI | MR | Zbl

[33] Dimakis A., Müller-Hoissen F., “Bidifferential calculus approach to AKNS hierarchies and their solutions”, SIGMA, 6 (2010), 055, 27 pp. ; arXiv: 1004.1627 | DOI | MR | Zbl

[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 | MR

[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 | DOI

[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 | DOI | MR

[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 | DOI | MR | Zbl

[38] Ward R.S., “Stationary axisymmetric space-times: a new approach”, Gen. Relativity Gravitation, 15 (1983), 105–109 | DOI | MR | Zbl

[39] Mason L.J., Woodhouse N.M.J., Integrability, self-duality, and twistor theory, London Mathematical Society Monographs. New Series, 15, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1996 | MR

[40] Klein C., Richter O., Ernst equation and Riemann surfaces. Analytical and numerical methods, Lecture Notes in Physics, 685, Springer-Verlag, Berlin, 2005 | MR | Zbl

[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 | DOI | MR | Zbl

[43] Bernstein D.S., Matrix mathematics. Theory, facts, and formulas, 2nd ed., Princeton University Press, Princeton, NJ, 2009 | MR | Zbl

[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 | MR

[45] Eichenherr H., Forger M., “More about non-linear sigma models on symmetric spaces”, Nuclear Phys. B, 164 (1980), 528–535 | DOI | MR

[46] Mazur P.O., “Proof of uniqueness of the Kerr–Newman black hole solution”, J. Phys. A: Math. Gen., 15 (1982), 3173–3180 | DOI | MR | Zbl

[47] Gürses M., “Inverse scattering, differential geometry, Einstein–Maxwell solitons and one soliton Bäcklund transformations”, Solutions of Einstein's Equations: Techniques and Results (Retzbach, 1983), Lecture Notes in Phys., 205, eds. C. Hoenselaers and W. Dietz, Springer, Berlin, 1984, 199–234 | DOI | MR

[48] Harrison B.K., “New solutions of the Einstein–Maxwell equations from old”, J. Math. Phys., 9 (1968), 1744–1752 | DOI | Zbl

[49] Kinnersley W., “Generation of stationary Einstein–Maxwell fields”, J. Math. Phys., 14 (1973), 651–653 | DOI | MR

[50] Aguilar-Sánchez J., García A.A., Manko V.S., “Demiański–Newman solution revisited”, Gravit. Cosmol., 7 (2001), 149–152 ; arXiv: gr-qc/0106011 | MR | Zbl

[51] Kramer D., Neugebauer G., “The superposition of two Kerr solutions”, Phys. Lett. A, 75 (1980), 259–261 | DOI | MR

[52] Neugebauer G., Hennig J., “Non-existence of stationary two-black-hole configurations”, Gen. Relativity Gravitation, 41 (2009), 2113–2130 ; arXiv: 0905.4179 | DOI | MR | Zbl

[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 | DOI | MR

[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 | DOI | MR | Zbl

[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 | DOI | MR | Zbl

[57] Marchenko V., Nonlinear equations and operator algebras, Mathematics and its Applications, 17, D. Reidel Publishing Co., Dordrecht, 1988 | MR | Zbl