Abstract
We consider the classes of electrovacuum Einstein–Maxwell fields (with a cosmological constant) for which the metrics admit an Abelian two-dimensional isometry group G 2 with nonnull orbits and electromagnetic fields possess the same symmetry. For the fields with such symmetries, we describe the structures of the so-called non-dynamical degrees of freedom, whose presence, just as the presence of a cosmological constant, changes (in a strikingly similar way) the vacuum and electrovacuum dynamical equations and destroys their well-known integrable structures. We find modifications of the known reduced forms of Einstein–Maxwell equations, namely, the Ernst equations and the self-dual Kinnersley equations, in which the presence of non-dynamical degrees of freedom is taken into account, and consider the following subclasses of fields with different non-dynamical degrees of freedom: (i) vacuum metrics with cosmological constant; (ii) space–time geometries in vacuum with isometry groups G 2 that are not orthogonally transitive; and (iii) electrovacuum fields with more general structures of electromagnetic fields than in the known integrable cases. For each of these classes of fields, in the case when the two-dimensional metrics on the orbits of the isometry group G 2 are diagonal, all field equations can be reduced to one nonlinear equation for one real function α(x1, x2) that characterizes the area element on these orbits. Simple examples of solutions for each of these classes are presented.
Similar content being viewed by others
References
G. A. Aleksejev, “Soliton configurations of Einstein–Maxwell fields,” in 9th Int. Conf. on General Relativity and Gravitation: Abstr. Contrib. Pap., Friedrich Schiller Univ., Jena, 1980, Vol. 1, pp. 2–3.
G. A. Alekseev, “N-soliton solutions of Einstein–Maxwell equations,” Pis’ma Zh. Eksp. Teor. Fiz. 32(4), 301–303 (1980) [JETP Lett. 32, 277–279 (1980)].
G. A. Alekseev, “On soliton solutions of the Einstein equations in a vacuum,” Dokl. Akad. Nauk SSSR 256(4), 827–830 (1981) [Sov. Phys., Dokl. 26(2), 158–160 (1981)].
G. A. Alekseev, “Soliton configurations of interacting massless fields,” Dokl. Akad. Nauk SSSR 268(6), 1347–1351 (1983) [Sov. Phys., Dokl. 28(2), 133–135 (1983)].
G. A. Alekseev, “The method of the inverse problem of scattering and the singular integral equations for interacting massless fields,” Dokl. Akad. Nauk SSSR 283(3), 577–582 (1985) [Sov. Phys., Dokl. 30(7), 565–568 (1985)].
G. A. Alekseev, “Exact solutions in the general theory of relativity,” Tr. Mat. Inst. im. V.A. Steklova, Akad. Nauk SSSR 176, 211–258 (1987) [Proc. Steklov Inst. Math. 176, 215–262 (1988)].
G. A. Alekseev, “Explicit form of the extended family of electrovacuum solutions with arbitrary number of parameters,” in 13th Int. Conf. on General Relativity and Gravitation: Abstr. Contrib. Pap., Huerta Grande, Cordoba, Argentina, 1992, Ed. by P.W. Lamberty and O. E. Ortiz, pp. 3–4.
G. A. Alekseev, “Gravitational solitons and monodromy transform approach to solution of integrable reductions of Einstein equations,” Physica D 152–153, 97–103 (2001); arXiv: gr-qc/0001012.
G. A. Alekseev, “Thirty years of studies of integrable reductions of Einstein’s field equations,” in Proc. Twelfth Marcel Grossmann Meeting on General Relativity, Part A: Plenary and Review Talks, Ed. by T. Damour, R. T. Jantzen, and R. Ruffini (World Scientific, New Jersey, 2012), pp. 645–666; arXiv: 1011.3846v1 [gr-qc].
G. A. Alekseev, “Monodromy transform and the integral equation method for solving the string gravity and supergravity equations in four and higher dimensions,” Phys. Rev. D 88 (2), 021503(R) (2013); arXiv: 1205.6238v1 [hep-th].
G. Alekseev, “Travelling waves in expanding spatially homogeneous space–times,” Class. Quantum Grav. 32(7), 075009 (2015); arXiv: 1411.3023v1 [gr-qc].
G. A. Alekseev, “Collision of strong gravitational and electromagnetic waves in the expanding universe,” Phys. Rev. D 93 (6), 061501(R) (2016); arXiv: 1511.03335v2 [gr-qc].
G. A. Alekseev and V. A. Belinski, “Equilibrium configurations of two charged masses in general relativity,” Phys. Rev. D 76 (2), 021501(R) (2007); arXiv: 0706.1981v1 [gr-qc].
G. A. Alekseev and A. A. Garcia, “Schwarzschild black hole immersed in a homogeneous electromagnetic field,” Phys. Rev. D 53(4), 1853–1867 (1996).
V. A. Belinskii and V. E. Zakharov, “Integration of the Einstein equations by means of the inverse scattering problem technique and construction of exact soliton solutions,” Zh. Eksp. Teor. Fiz. 75(6), 1953–1971 (1978) [Sov. Phys., JETP 48(6), 985–994 (1978)].
V. A. Belinskii and V. E. Zakharov, “Stationary gravitational solitons with axial symmetry,” Zh. Eksp. Teor. Fiz. 77(1), 3–19 (1979) [Sov. Phys., JETP 50(1), 1–9 (1979)].
B. Carter, “Black hole equilibrium states,” in Black Holes: Les Houches 1972, Ed. by C. DeWitt and B. S. DeWitt (Gordon and Breach, New York, 1973), pp. 57–214.
F. J. Ernst, “New formulation of the axially symmetric gravitational field problem,” Phys. Rev. 167(5), 1175–1178 (1968).
B. Gaffet, “The Einstein equations with two commuting Killing vectors,” Class. Quantum Grav. 7(11), 2017–2044 (1990).
R. Geroch, “A method for generating new solutions of Einstein’s equation. II,” J. Math. Phys. 13(3), 394–404 (1972).
E. Gourgoulhon and S. Bonazzola, “Noncircular axisymmetric stationary spacetimes,” Phys. Rev. D 48(6), 2635–2652 (1993).
J. B. Griffiths and J. Podolský, Exact Space–Times in Einstein’s General Relativity (Cambridge Univ. Press, Cambridge, 2009), Cambridge Monogr. Math. Phys.
I. Hauser and F. J. Ernst, “Integral equation method for effecting Kinnersley–Chitre transformations,” Phys. Rev. D 20(2), 362–369 (1979).
I. Hauser and F. J. Ernst, “Integral equation method for effecting Kinnersley–Chitre transformations. II,” Phys. Rev. D 20(8), 1783–1790 (1979).
I. Hauser and F. J. Ernst, “On the transformation of one electrovac spacetime into another,” in 9th Int. Conf. on General Relativity and Gravitation: Abstr. Contrib. Pap., Friedrich Schiller Univ., Jena, 1980, Vol. 1, pp. 84–85.
W. Kinnersley, “Generation of stationary Einstein–Maxwell fields,” J. Math. Phys. 14(5), 651–653 (1973).
W. Kinnersley, “Symmetries of the stationary Einstein–Maxwell field equations. I,” J. Math. Phys. 18(8), 1529–1537 (1977).
W. Kinnersley and D. M. Chitre, “Symmetries of the stationary Einstein–Maxwell field equations. II,” J. Math. Phys. 18(8), 1538–1542 (1977).
W. Kinnersley and D. M. Chitre, “Symmetries of the stationary Einstein–Maxwell field equations. III,” J. Math. Phys. 19(9), 1926–1931 (1978).
W. Kinnersley and D. M. Chitre, “Symmetries of the stationary Einstein–Maxwell field equations. IV: Transformations which preserve asymptotic flatness,” J. Math. Phys. 19(10), 2037–2042 (1978).
T. Lewis, “Some special solutions of the equations of axially symmetric gravitational fields,” Proc. R. Soc. London A 136, 176–192 (1932).
D. Maison, “Are the stationary, axially symmetric Einstein equations completely integrable?,” Phys. Rev. Lett. 41(8), 521–522 (1978).
A. Papapetrou, “Champs gravitationnels stationnaires `a symétrie axiale,” Ann. Inst. Henri Poincaré, Phys. Théor. 4(2), 83–105 (1966).
N. R. Sibgatullin, Oscillations and Waves in Strong Gravitational and Electromagnetic Fields (Nauka, Moscow, 1984; Springer, Berlin, 1991).
H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt, Exact Solutions of Einstein’s Field Equations, 2nd ed. (Cambridge Univ. Press, Cambridge, 2009).
Author information
Authors and Affiliations
Corresponding author
Additional information
Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2016, Vol. 295, pp. 7–33.
Rights and permissions
About this article
Cite this article
Alekseev, G.A. Integrable and non-integrable structures in Einstein-Maxwell equations with Abelian isometry group G 2 . Proc. Steklov Inst. Math. 295, 1–26 (2016). https://doi.org/10.1134/S0081543816080010
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0081543816080010