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Einstein–Maxwell equations: Solution-generating methods as “coordinate” transformations in the solution spaces

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Abstract

The solution-generating methods discovered for integrable reductions of the Einstein and Einstein–Maxwell field equations (soliton-generating techniques, Bäcklund transformations, HKX transformations, Hauser–Ernst homogeneous Hilbert problem, and other group-theoretical methods) can be described explicitly as transformations of especially defined “coordinates ” in the infinite-dimensional solution spaces of these equations. In general, the role of such “coordinates ” for every local solution can be performed by monodromy data of fundamental solutions of the corresponding spectral problems. However, for large classes of fields, these can be the values of Ernst potentials on the boundaries that consist of degenerate orbits of the space–time isometry group such that space–time geometry and the electromagnetic fields behave regularly near these boundaries. In this paper, transformations of such “coordinates” corresponding to different known solution-generating procedures are described by relatively simple algebraic expressions that do not require any particular choice of the initial (background) solution. Explicit forms of these transformations allow us to find the interrelations between the sets of free parameters that arise in different solution-generating procedures and to determine some physical and geometrical properties of each generating solution even before a detail calculations of all its components.

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Notes

  1. In a later paper [10], Kinnersley and Chitre made an interesting comment: “Our inclusion of electromagnetism throughout this work has been an enormous help rather than a hindrance. It has revealed a striking interrelationship between electromagnetic and gravitational fields that could not possibly have been anticipated.”

  2. Harrison already mentioned Belinski and Zakharov’s results in that paper.

  3. Earlier, a construction of \(2\times 2\) matrix linear singular integral equation with the kernel of a Cauchy type, whose solutions determine the solutions of the appropriate Riemann–Hilbert problem on the spectral plane, was suggested for generating “nonsoliton” vacuum solutions in Belinski and Zakharov’s first paper [16]. However, in [30], the integral equation method of Hauser and Ernst was more elaborated and examples of the construction of exact solutions for the rational choice of arbitrary functions in the kernel were described.

  4. The relations between different matrix potentials suggested for the Einstein–Maxwell equations in different approaches were described by Kramer [37].

  5. The structure of this system with its supplementary conditions differs essentially from more simple structure of the spectral problem used for construction of solitons in [19], [20].

  6. The expression for conformal factor for electrovacuum solitons was found in [39].

  7. For a given local solution, the Ernst potentials are defined up to some gauge freedom, which, however, that does not change the geometry and physical parameters of the solution.

  8. These equations were originally derived by Ernst for stationary axisymmetric vacuum fields [45], and were then generalized to the case of stationary axisymmetric electrovacuum fields in [46]. In these equations, the Weyl cylindrical coordinates were used. For these coordinates, \(\alpha=\rho\) and \(\beta=z\). Similar equations can easily be derived in the hyperbolic case as well, and these are usually called the hyperbolic Ernst equations.

  9. The constant parameter \(\Omega_0\) can be made equal to zero by using appropriate linear transformations with constant coefficients of the Killing vectors \(\partial/\partial x^a\). However, if one of the Killing vectors corresponds to axial symmetry with a \(2\pi\)-periodic angle coordinate \(\varphi\), this transformation of Killing vectors is not an admissible global coordinate transformation and it should be regarded as some “cut-and-past” procedure changing the space–time manifold such that the role of a \(2\pi\)-periodic angle coordinate is played not by the old coordinate \(\varphi\) but by some new angular coordinate \(\varphi^\prime\). In addition, several regular intervals separated by the sources may exist on the symmetry axis, and expansions of type (15), (16) may be applicable near each of these intervals. In that case, the constants \(\Omega_0\) may be different on different intervals and we cannot make all of them equal to zero simultaneously by any global Killing vector transformation.

  10. Using the functions \(\mathcal{E}_0(\beta)\) and \(\Phi_0(\beta)\) as “coordinates,” we should take into account that the Ernst potential \(\mathcal{E}\) is defined up to an arbitrary additive imaginary constant and the potential \(\Phi\) up to an additive complex constant. Changes of these constants lead to “gauge” transformations of \(\mathcal{E}_0(\beta)\) and \(\Phi_0(\beta)\), which leave physical properties of the solutions unchanged.

  11. In the case of an odd number of solitons, the Belinski–Zakharov soliton-generating procedure leads in the elliptic case to the solutions whose metric signature changes in comparison with that of the initial solution; in the hyperbolic case, the corresponding generated solutions describe the waves that have singularities on the null wave fronts.

  12. Here and below, “NUT” means one of the parameters that characterizes the source in the Kerr–NUT solution (besides the mass \(m\) and angular momentum \(a\) parameters) and which was named after Newman, Tamburino, and Unti; see book [1] for the details.

  13. It seems useful to clarify here that the number of solitons means the number of simple poles in the dressing matrix \( {\large\boldsymbol{\chi}} \) on the spectral plane \(\lambda\) in the case of Belinski–Zakharov solitons (18) and on the spectral plane \(w\) in the case of electrovacuum solitons (26). Because of the obvious difference between these two techniques and of the structures of the “spectral” planes \(\lambda\) and \(w\), the vacuum part of solutions with \(N\) electrovacuum solitons should be compared with solutions with \(2N\) Belinski–Zakharov vacuum solitons. This comparison shows that in contrast to Belinski–Zakharov vacuum solitons, the number \(N\) of solitons (i.e., poles) in (26) can be not only even, but can also be odd. The vacuum restriction of electrovacuum \(N\)-soliton solution (26) coincides with the vacuum Belinski–Zakharov \(2N\)-soliton solution with complex conjugate pairs of poles, while the electrovacuum generalization of the Belinski–Zakharov vacuum solitons with pairs of real poles does not arise in this way. However, the electrovacuum solutions of soliton type with real poles can arise (at least for special choices of the background solutions) as a result of analytic continuations of electrovacuum soliton solutions with complex poles in the space of their constant parameters.

  14. We note that our notation introduced in the foregoing differs in some points from Kinnersley and Chitre’s notation. In particular, we use the metric signature \((-,+,+,+)\) instead of \((+,-,-,-)\) used in their papers. Our enumeration of coordinates and notation for indices in (1)–(4) are also different from those in [10]–[12]. As a result, for the metrics on orbits, we use \(h_{ab}\) and hence \(f_{AB}\to -h_{ab}\). In this section, however, in contrast to the rest of the paper, we let \(\ast\) denote complex conjugation, \(\nabla\) denote the gradient operator (instead of \(\partial_\mu\)) and \(\widetilde{\nabla}\) denote the dual operator (instead of our usual \(\varepsilon_\mu{}^\nu\partial_\nu\)). In addition, for stationary axisymmetric fields (to which Kinnersley and Chitre restricted themselves), we should set \({\epsilon=\epsilon_0=-1}\) and \(\epsilon_1=\epsilon_2=1\) in notation.

  15. We recall that \(G\) in (36) is the upper left entry of the matrix \(G_{ab}\).

  16. Some examples of such subspaces of solutions are cylindrical waves, stationary axisymmetric fields created by compact sources and considered near some intervals on the axis between or outside the sources, some cosmological-like solutions, plane waves near the “focusing singularities,” solutions with Killing horizons, and some others.

References

  1. H. Stephani, D. Kramer, M. MacCallum, C. Hoenselaers, and E. Herlt, Exact Solutions of Einstein’s Field Equations (Cambridge Monographs on Mathematical Physics), Cambridge Univ. Press, Cambridge (2003).

    Book  MATH  Google Scholar 

  2. G. A. Alekseev, “Integrable and non-integrable structures in Einstein–Maxwell equations with Abelian isometry group \(\mathcal G_2\),” Proc. Steklov Inst. Math., 295, 1–26 (2016).

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Ehlers, Konstruktionen und Charakterisierungen von Lösungen der Einsteinschen Gravitationsfeldgleichungen (Ph.D. thesis), University of Hamburg, Hamburg (1957).

    Google Scholar 

  4. J. Ehlers, Les theories relativistes de la gravitation, CNRS, Paris (1959).

    Google Scholar 

  5. B. K. Harrison, “New solutions of the Einstein–Maxwell equations from old,” J. Math. Phys., 9, 1744–1752 (1968).

    Article  ADS  MATH  Google Scholar 

  6. D. Kramer and G. Neugebauer, “Eine exakte stationäre Lösung der EINSTEIN– MAXWELL– Gleichungen,” Ann. Phys., 479, 59–61 (1969).

    Article  Google Scholar 

  7. W. Kinnersley, “Generation of stationary Einstein–Maxwell fields,” J. Math. Phys., 14, 651–653 (1973).

    Article  ADS  MathSciNet  Google Scholar 

  8. R. Geroch, “A method for generating solutions of Einstein’s equations. II,” J. Math. Phys., 13, 394–404 (1972).

    Article  ADS  MathSciNet  MATH  Google Scholar 

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

    Article  ADS  Google Scholar 

  10. W. Kinnersley and D. M. Chitre, “Symmetries of the stationary Einstein–Maxwell field equations. II,” J. Math. Phys., 18, 1538–1542 (1977).

    Article  ADS  Google Scholar 

  11. W. Kinnersley and D. M. Chitre, “Symmetries of the stationary Einstein–Maxwell field equations. III,” J. Math. Phys., 19, 1926–1931 (1978).

    Article  ADS  Google Scholar 

  12. W. Kinnersley and D. M. Chitre, “Symmetries of the stationary Einstein–Maxwell field equations. IV. Transformations which preserve asymptotic flatness,” J. Math. Phys., 19, 2037–2042 (1978).

    Article  ADS  Google Scholar 

  13. B. Julia, “Application of supergravity to gravitation theory,” in: Unified Field Theories of more than 4 Dimensions, Including Exact Solutions (Proceedings of the International School of Cosmology and Gravitation, Erice, Trapani, Sicily, May 20 – June 1, 1982, V. De Sabbata and E. Schmutzer, eds.), World Sci., Singapore (1983), pp. 215–233.

    Google Scholar 

  14. B. Julia, “Kac–Moody symmetry of gravitation and supergravity theories,” in: Applications of Group Theory in Physics and Mathematical Physics (Chicago, July, 1982, Lectures in Applied Mathematics, Vol. 21, M. Flato, P. Sally, and G. Zuckerman, eds.), AMS, Providence, RI (1985), pp. 353–373.

    Google Scholar 

  15. P. Breitenlohner and D. Maison, “On the Geroch group,” Ann. Inst. H. Poincaré Phys. Theor., 46, 215–246 (1987).

    MathSciNet  MATH  Google Scholar 

  16. 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,” Sov. Phys. JETP, 48, 985–994 (1978).

    ADS  Google Scholar 

  17. V. A. Belinskii and V. E. Zakharov, “Stationary gravitational solitons with axial symmetry,” Sov. Phys. JETP, 50, 1–9 (1979).

    ADS  Google Scholar 

  18. G. A. Alekseev, “On soliton solutions of Einstein’s equations in a vacuum,” Sov. Phys. Dokl., 28, 158–160 (1981).

    ADS  Google Scholar 

  19. G. A. Aleksejev, “Soliton configurations of Einstein–Maxwell fields,” in: \(9\)th International Conference on General Relativity and Gravitation. Abstracts of Contributed Papers, Vol. 1 (Friedrich Schiller University, Jena, German Democratic Republic, July 14–19, 1980, E. Schmutzer, ed.), International Society on General Relativity and Gravitation, Jena, DDR (1980), pp. 2–3.

    Google Scholar 

  20. G. A. Alekseev, “\(N\)-soliton solutions of Einstein–Maxwell equations,” JETP Lett., 32, 277–279 (1980).

    Google Scholar 

  21. B. K. Harrison, “Bäcklund transformation for the Ernst equation of general relativity,” Phys. Rev. Lett., 41, 1197–1200 (1978).

    Article  ADS  MathSciNet  Google Scholar 

  22. H. D. Wahlquist and F. B. Estabrook, “Bäcklund transformation for solutions of the Korteweg–de Vries equation,” Phys. Rev. Lett., 31, 1386–1390 (1973); “Prolongation structures of nonlinear evolution equations,” J. Math. Phys., 16, 1–7 (1975).

    Article  ADS  MathSciNet  Google Scholar 

  23. B. K. Harrison, “New large family of vacuum solutions of the equations of general relativity,” Phys. Rev. D, 21, 1695–1697 (1980).

    Article  ADS  MathSciNet  Google Scholar 

  24. B. K. Harrison, “Unification of Ernst-equation Bäcklund transformations using a modified Wahlquist–Estabrook technique,” J. Math. Phys., 24, 2178–2187 (1983).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  25. G. Neugebauer, “Bäcklund transformations of axially symmetric stationary gravitational fields,” J. Phys. A: Math. Gen., 12, L67–L70 (1979).

    Article  ADS  Google Scholar 

  26. G. Neugebauer, “Recursive calculation of axially symmetric statiomary Einstein fields,” J. Phys. A: Math. Gen., 13, 1737–1740 (1980).

    Article  ADS  Google Scholar 

  27. G. Neugebauer, “A general integral of the axially symmetric stationary Einstein equations,” J. Phys. A: Math. Gen., 13, L19–L21 (1980).

    Article  ADS  MathSciNet  Google Scholar 

  28. C. Hoenselaers, W. Kinnersley, and B. C. Xanthopoulos, “Generation of asymptotically flat, stationary space–times with any number of parameters,” Phys. Rev. Lett., 42, 481–482 (1979).

    Article  ADS  Google Scholar 

  29. C. Hoenselaers, W. Kinnersley, and B. C. Xanthopoulos, “Symmetries of the stationary Einstein–Maxwell equations. VI. Transformations which generate asymptotically flat spacetimes with arbitrary multipole moments,” J. Math. Phys., 20, 2530–2536 (1979).

    Article  ADS  Google Scholar 

  30. I. Hauser and F. J. Ernst, “Integral equation method for effecting Kinnersley–Chitre transformations,” Phys. Rev. D, 20, 362–369 (1979).

    Article  ADS  MathSciNet  Google Scholar 

  31. I. Hauser and F. J. Ernst, “A homogeneous Hilbert problem for the Kinnersley–Chitre transformations,” J. Math. Phys., 21, 1126–1140 (1980).

    Article  ADS  MathSciNet  Google Scholar 

  32. F. J. Ernst and I. Hauser, “On the transformation of one electrovac spacetime into another,” in: \(9\)th International Conference on General Relativity and Gravitation. Abstracts of Contributed Papers, Vol. 1 (Friedrich Schiller University, Jena, German Democratic Republic, July 14–19, 1980, E. Schmutzer, ed.), International Society on General Relativity and Gravitation, Jena, DDR (1980), pp. 84–85.

    Google Scholar 

  33. I. Hauser and F. J. Ernst, “A Fredholm equation for effecting Kinnersley–Chitre transformations,” in: \(9\)th International Conference on General Relativity and Gravitation. Abstracts of Contributed Papers, Vol. 1 (Friedrich Schiller University, Jena, German Democratic Republic, July 14–19, 1980, E. Schmutzer, ed.), International Society on General Relativity and Gravitation, Jena, DDR (1980), pp. 89–90.

    Google Scholar 

  34. I. Hauser and F. J. Ernst, “Integral equation method for effecting Kinnersley–Chitre transformations. II,” Phys. Rev. D, 20, 1783–1790 (1979).

    Article  ADS  MathSciNet  Google Scholar 

  35. I. Hauser and F. J. Ernst, “A homogeneous Hilbert problem for the Kinnersley–Chitre transformations of electrovac spacetimes,” J. Math. Phys., 21, 1418–1422 (1980).

    Article  ADS  MathSciNet  Google Scholar 

  36. D. Kramer and G. Neugebauer, “Prolongation structure and linear eigenvalue equations for Einstein–Maxwell fields,” J. Phys. A: Math. Gen., 14, L333–L338 (1981).

    Article  ADS  MathSciNet  Google Scholar 

  37. D. Kramer, “Equivalence of various pseudopotential approaches for Einstein–Maxwell fields,” J. Phys. A: Math. Gen., 15, 2201–2207 (1982).

    Article  ADS  MathSciNet  Google Scholar 

  38. G. Neugebauer and D. Kramer, “Einstein–Maxwell solitons,” J. Phys. A: Math. Gen., 16, 1927–1936 (1983).

    Article  ADS  MathSciNet  Google Scholar 

  39. G. A. Alekseev, “Exact solutions in the general theory of relativity,” Proc. Steklov Inst. Math., 176, 215–262 (1988).

    MATH  Google Scholar 

  40. A. Eriş, M. Gürses, and A. Karasu, “Symmetric Space property and an inverse scattering formulation of the SAS Einstein–Maxwell field equations,” J. Math. Phys., 25, 1489–1495 (1984).

    Article  ADS  MathSciNet  Google Scholar 

  41. C. M. Cosgrove, “Relationship between the group-theoretic and soliton-theoretic techniques for generating stationary axisymmetric gravitational solutions,” J. Math. Phys., 21, 2417–2447 (1980).

    Article  ADS  MathSciNet  Google Scholar 

  42. C. M. Cosgrove, “Bäcklund transformations in the Hauser–Ernst formalism for stationary axisymmetric spacetimes,” J. Math. Phys., 22, 2624–2639 (1981).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. C. M. Cosgrove, “Relationship between the inverse scattering techniques of Belinskii–Zakharov and Hauser–Ernst in general relativity,” J. Math. Phys., 23, 615–633 (1982).

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. G. A. Alekseev, “The method of the inverse problem of scattering and the singular integral equations for interacting massless fields,” Dokl. Math., 30, 565–568 (1985).

    Google Scholar 

  45. F. J. Ernst, “New formulation of the axially symmetric gravitational field problem,” Phys. Rev., 167, 1175–1177 (1968).

    Article  ADS  Google Scholar 

  46. F. J. Ernst, “New formulation of the axially symmetric gravitational field problem. II,” Phys. Rev., 168, 1415–1417 (1968).

    Article  ADS  Google Scholar 

  47. V. Belinski and E. Verdaguer, Gravitational Solitons, Cambridge Univ. Press, Cambridge (2001).

    Book  MATH  Google Scholar 

  48. G. A. Alekseev, “Gravitational solitons and monodromy transform approach to solution of integrable reductions of Einstein equations,” Phys. D, 152–153, 97–103 (2001); arXiv: gr-qc/0001012.

    Article  MathSciNet  MATH  Google Scholar 

  49. N. R. Sibgatullin, Oscillations and Waves in Strong Gravitational and Electromagnetic Fields, Springer, Berlin (1991).

    MATH  Google Scholar 

  50. G. A. Alekseev, “Explicit form of the extended family of electrovacuum solutions with arbitrary number of parameters,” in: General Relativity and Gravitation (Proceedings of the 13th International Conference, Córdoba, Argentina, June 28 – July 4, 1992, R. J. Gleiser, C. N. Kozameh, and O. M. Moreschi, eds.), IOP, London (1993), pp. 3–4.

    Google Scholar 

  51. G. A. Alekseev and J. B. Griffiths, “Infinite hierarchies of exact solutions of the Einstein and Einstein–Maxwell equations for interacting waves and inhomogeneous cosmologies,” Phys. Rev. Lett., 84, 5247–5250 (2000).

    Article  ADS  MathSciNet  Google Scholar 

  52. D. A. Korotkin and V. B. Matveev, “Algebro-geometric solutions of gravitation equations,” St. Petersburg Math. J., 1, 379–408 (1990).

    MathSciNet  MATH  Google Scholar 

  53. G. A. Alekseev, “Integrability of the boundary value problems for the Ernst equations,” in: Nonlinear Evolution Equations and Dynamical Systems (NEEDS’92) (Proceedings of the Eighth International Workshop, Dubna, July 6–17, 1992, V. Makhan’kov, I. Puzynin, and O. Pashaev, eds.), World Sci., Singapore (1993), pp. 5–10.

    Google Scholar 

  54. R. Meinel, M. Ansorg, A. Kleinwächter, G. Neugebauer, and D. Petroff, Relativistic Figures of Equilibrium, Cambridge Univ. Press, Cambridge (2008).

    Book  MATH  Google Scholar 

  55. G. A. Alekseev and J. B. Griffiths, “Collision of plane gravitational and electromagnetic waves in a Minkowski background: solution of the characteristic initial value problem,” Class. Quantum Grav., 21, 5623–5654 (2004); arXiv: gr-qc/0410047.

    Article  ADS  MathSciNet  MATH  Google Scholar 

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    G. A. Alekseev

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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2022, Vol. 211, pp. 502–534 https://doi.org/10.4213/tmf10228.

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Alekseev, G.A. Einstein–Maxwell equations: Solution-generating methods as “coordinate” transformations in the solution spaces. Theor Math Phys 211, 866–892 (2022). https://doi.org/10.1134/S0040577922060083

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