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1997 | 41 | 1 | 221-232
Tytuł artykułu

Integrability and Einstein's equations

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
1. Introduction. In recent years, there has been considerable interest in Oxford and elsewhere in the connections between Einstein's equations, the (anti-) self-dual Yang-Mills (SDYM) equations, and the theory of integrable systems. The common theme running through this work is that, to a greater or lesser extent, all three areas involve questions that can be addressed by twistor methods. In this paper, I shall review progress, with particular emphasis on the known and potential applications in relativity. Some of the results are well-established, others are more recent, and a few appear here for the first time.
Słowa kluczowe
Rocznik
Tom
41
Numer
1
Strony
221-232
Opis fizyczny
Daty
wydano
1997
Twórcy
  • Wadham College, Oxford OX1 3PN, U.K.
Bibliografia
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  • Atiyah, M. F., Hitchin, N. J. and Singer, I. M. (1978), Self-duality in four-dimensional Riemannian geometry. Proc. Roy. Soc. Lond., A 362, 425-61.
  • Atiyah, M. F. and Ward, R. S. (1977), Instantons and algebraic geometry. Commun. Math. Phys., 55, 111-24.
  • Berger, B. K., Chruściel, P. T. and Moncrief, V. (1995), On 'asymptotically flat' spacetimes with $G_2$-invariant Cauchy surfaces. Ann. Phys., 237, 322-54.
  • Calvert, G. and Woodhouse, N. M. J. (1996), Painlevé transcendents and Einstein's equation. Class. Quantum Grav., 13, L33-9.
  • Chakravarty, S., Mason, L. J. and Newman, E. T. (1991), Canonical structures on anti-self-dual four-manifolds and the diffeomorphism group. J. Math. Phys., 32, 1458-64.
  • Chandrasekhar, S. (1986), Cylindrical gravitational waves. Proc. Roy. Soc. Lond., A408, 209-32.
  • Cosgrove, C. M. (1977), New family of exact stationary axisymmetric gravitational fields generalising the Tomimatsu-Sato solutions. J. Phys., A10, 1481-524.
  • Dunajski, M., Mason, L. J. and Woodhouse, N. M. J. (1996), From 2D integrable systems to self-dual gravity. In preparation.
  • Fletcher, J. (1990), Non-Hausdorff twistor spaces and the global geometry of space-time. D. Phil. thesis, University of Oxford.
  • Fletcher, J. and Woodhouse, N. M. J. (1990), Twistor characterization of stationary axisymmetric solutions of Einstein's equations. In Twistors in mathematics and physics. Eds. T. N. Bailey and R. J. Baston. London Mathematical Society Lecture Notes in Mathematics, 156. Cambridge University Press, Cambridge.
  • Geroch, R. (1971), A method for generating new solutions of Einstein's equations. J. Math. Phys., 12, 918-24.
  • Geroch, R. (1972), A method for generating new solutions of Einstein's equations, II. J. Math. Phys., 13, 394-404.
  • Gohberg, I. C. and Krein, M. G. (1958), Systems of integral equations on the half line with kernels depending on the difference of the arguments. Uspekhi Mat. Nauk, 13, 3-72. (Russian)
  • Hitchin, N. J. (1995), Twistor spaces, Einstein metrics and isomonodromic deformations. J. Diff. Geom., 42, 30-112.
  • Hoenselaers, C. and Dietz, W. (eds.) (1984), Solutions of Einstein's equations: techniques and results. Lecture Notes in Physics, 205. Springer, Berlin.
  • Ince, E. L. (1956), Ordinary differential equations. Dover, New York.
  • Kinnersley, W. (1977), Symmetries of the stationary Einstein-Maxwell field equations, I. J. Math. Phys., 18, 1529-37.
  • Kinnersley, W. and Chitre, D. M. (1977-8), Symmetries of the stationary Einstein-Maxwell field equations, II-IV. J. Math. Phys., 18, 1538-42, 19, 1926-31, 2037-42.
  • Léauté, B. and Marcilhacy, G. (1979), Sur certaines particulières transcendantes des équations d'Einstein. Ann. Inst. H. Poincaré, 31, 363-75.
  • Mason, L. J., Chakravarty, S. and Newman, E. T. (1988), Bäcklund transformations for the anti-self-dual Yang-Mills equations. J. Math. Phys., 29, 4, 1005-13.
  • Mason, L. J. and Newman, E. T. (1989). A connection between the Einstein and Yang-Mills equations. Commun. Math. Phys., 121, 659-68.
  • Mason, L. J. and Sparling, G. A. J. (1989), Nonlinear Schrödinger and Korteweg de Vries are reductions of self-dual Yang-Mills. Phys. Lett., A137, 29-33.
  • Mason, L. J. and Woodhouse, N. M. J. (1993), Self-duality and the Painlevé transcendents. Nonlinearity, 6, 569-81.
  • Mason, L. J. and Woodhouse, N. M. J. (1996), Integrability, self-duality, and twistor theory. London Mathematical Society Monographs. Oxford University Press, Oxford.
  • Maszczyk, R. (1995), The symmetry transformation--self-dual Yang-Mills fields and self-dual metrics. Ph. D. Thesis, Warsaw University.
  • Maszczyk, R., Mason, L. J. and Woodhouse, N. M. J. (1994), Self-dual Bianchi metrics and the Painlevé transcendents. Class. Quantum Grav., 11, 65-71.
  • Newman, E. T. (1978), Source-free Yang-Mills theories. Phys. Rev., D18, 2901-2908.
  • Penrose, R. (1976), Nonlinear gravitons and curved twistor theory. Gen. Rel. Grav., 7, 31-52.
  • Penrose, R. (1992), Twistors as spin 3/2 charges. In Gravitation and modern cosmology. Eds. A. Zichichi and N. Sánchez. Plenum Press, New York.
  • Persides, S. and Xanthopoulos, B. C. (1988), Some new stationary axisymmetric asymptotically flat space-times obtained from Painlevé transcendents. J. Math. Phys., 29, 674-80.
  • Tod, K. P. (1994), Self-dual Einstein metrics from the Painlevé VI equation. Phys. Lett., A190, 221-4.
  • Ward, R. S. (1977), On self-dual gauge fields. Phys. Lett., 61A, 81-2.
  • Ward, R. S. (1983), Stationary axisymmetric space-times: a new approach. Gen. Rel. Grav., 15, 105-9.
  • Ward, R. S. (1985), Integrable and solvable systems and relations among them. Phil. Trans. R. Soc., A315, 451-7.
  • Ward, R. S. (1990 a), Integrable systems in twistor theory. In Twistors in mathematics and physics. Eds. T. N. Bailey and R. J. Baston. London Mathematical Society Lecture Notes in Mathematics, 156. Cambridge University Press, Cambridge.
  • Ward, R. S. (1990 b), The SU(∞) chiral model and self-dual vacuum spaces. Class. Quantum Grav., 7, L217-22.
  • Ward, R. S. (1992), Infinite-dimensional gauge groups and special nonlinear gravitons. J. Geom. Phys., 8, 317-25.
  • Witten, L. (1979), Static axially symmetric solutions of self-dual SU(2) gauge fields in Euclidean four-dimensional space. Phys. Rev., D19, 718-20.
  • Woodhouse, N. M. J. (1989), Cylindrical gravitational waves. Class. Quantum Grav., 6, 933-43.
  • Woodhouse, N. M. J. (1990 a), Ward's splitting construction for stationary axisymmetric solutions of the Einstein-Maxwell equations. Class. Quantum Grav., 7, 257-60.
  • Woodhouse, N. M. J. (1990 b), Spinors, twistors, and complex methods. In: General relativity and Gravitation 1989, pp. 93-8 (eds Ashby, Bartlett, and Wyss), CUP.
  • Woodhouse, N. M. J. and Mason, L. J. (1988), The Geroch group and non-Hausdorff twistor spaces. Nonlinearity, 1, 73-114.
  • Yang, C. N. (1977), Condition of self-duality for SU(2) gauge fields on Euclidean four-dimensional space. Phys. Rev. Lett., 38, 1377-9.
Typ dokumentu
Bibliografia
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