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The equivalence problem for generic four-dimensional metrics with two commuting Killing vectors

  • D. Catalano FerraioliEmail author
  • M. Marvan
Article
  • 21 Downloads

Abstract

We consider the equivalence problem of four-dimensional semi-Riemannian metrics with the two-dimensional Abelian Killing algebra. In the generic case we determine a semi-invariant frame and a fundamental set of first-order scalar differential invariants suitable for solution of the equivalence problem. Genericity means that the Killing leaves are not null, the metric is not orthogonally transitive (i.e., the distribution orthogonal to the Killing leaves is non-integrable), and two explicitly constructed scalar invariants \(C_\rho \) and \(\ell _{{\mathcal {C}}}\) are nonzero. All the invariants are designed to have tractable coordinate expressions. Assuming the existence of two functionally independent invariants, we solve the equivalence problem in two ways. As an example, we invariantly characterize the Van den Bergh metric. To understand the non-generic cases, we also find all \(\varLambda \)-vacuum metrics that are generic in the above sense, except that either \(C_\rho \) or \(\ell _{{\mathcal {C}}}\) is zero. In this way we extend the Kundu class to \(\varLambda \)-vacuum metrics. The results of the paper can be exploited for invariant characterization of classes of metrics and for extension of the set of known solutions of the Einstein equations.

Keywords

Differential invariants Metric equivalence problem Kundu class 

Mathematics Subject Classification

83C20 35Q76 

Notes

References

  1. 1.
    Alekseev, G.A.: Thirty years of studies of integrable reductions of Einstein’s field equations. In: Damour, T., Jantzen, R.T. (eds.) The Twelfth Marcel Grossmann Meeting, pp. 645–666. World Scientific, Singapore (2012)CrossRefGoogle Scholar
  2. 2.
    Alekseevsky, D.V., Vinogradov, A.M., Lychagin, V.V.: Basic Ideas and Concepts of Differential Geometry, Encyclopaedia Mathematics Science, Vol. 28 (Springer, Berlin, 1991)Google Scholar
  3. 3.
    Åman, J.E., Karlhede, A.: An algorithmic classification of geometries in general relativity. In: Proceedings of the Fourth ACM Symposium on Symbolic and Algebraic Computation, Snowbird, Utah, USA (ACM), pp. 79–84 (1981)Google Scholar
  4. 4.
    Belinskiĭ, V.A., Zakharov, V.E.: Integration of the Einstein equations by means of the inverse scattering problem. Sov. Phys. JETP 75(6), 1955–1971 (1978)Google Scholar
  5. 5.
    Besse, A.L.: Einstein Manifolds. Springer, Berlin (1987)CrossRefGoogle Scholar
  6. 6.
    Bradley, M., Marklund, M.: Finding solutions to Einstein’s equations in terms of invariant objects. Class. Quantum Gravity 13, 3021–3037 (1996)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Carminati, J., McLenaghan, R.G.: Algebraic invariants of the Riemann tensor in a four-dimensional Lorentzian space. J. Math. Phys. 32, 3135–3140 (1991)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Carter, B.: Killing horizons and orthogonally transitive groups in space-time. J. Math. Phys. 10, 70–81 (1969)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Coley, A., Hervik, S., Pelavas, N.: Spacetimes characterized by their scalar curvature invariants. Class. Quantum Gravity 26, 025013 (2009). (33pp)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Cosgrove, C.M.: A new formulation of the field equations for the stationary axisymmetric gravitational field: I. General theory. J. Phys. A Math. Gen. 11, 2389–2404 (1978)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Cosgrove, C.M.: A new formulation of the field equations for the stationary axisymmetric gravitational field: II. Separable solutions. J. Phys. A Math. Gen. 11, 2405–2430 (1978)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ferrando, J.J., Sáez, J.A.: An intrinsic characterization of the Kerr metric. Class. Quantum Gravity 26, 075013 (2009). (13pp)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Gaffet, B.: The Einstein equations with two commuting Killing vectors. Class. Quantum Gravity 7, 2017–2044 (1990)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Geroch, R.: A method for generating solutions of Einstein’s equations. J. Math. Phys. 12, 918–924 (1971)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Geroch, R.: A method for generating new solutions of Einstein’s equations. II. J. Math. Phys. 13, 394–404 (1972)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Karlhede, A.: A review of the geometrical equivalence of metrics in general relativity. Gen. Rel. Gravit. 12, 693–707 (1980)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Karlhede, A.: The equivalence problem. Gen. Rel. Gravit. 38, 1109–1114 (2006)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Karlhede, A., MacCallum, M.A.H.: On determining the isometry group of a Riemannian space. Gen. Rel. Gravit. 14, 673–682 (1982)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Klein, C., Richter, O.: Ernst Equation and Riemann Surfaces. Springer, Berlin (2005)zbMATHGoogle Scholar
  20. 20.
    Kolassis, C.A., Santos, N.O.: Spacetimes with a preferred null direction and a two-dimensional group of isometries: the null dust case. Class. Quantum Gravity 4, 599–618 (1987)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Kundu, P.: Class of “noncanonical” vacuum metrics with two commuting Killing vectors. Phys. Rev. Lett. 42, 416–417 (1979)CrossRefGoogle Scholar
  22. 22.
    Lewis, T.: Some special solutions of the equations of axially symmetric gravitational fields. Proc. R. Soc. Lond. A 136, 176–192 (1932)CrossRefGoogle Scholar
  23. 23.
    Lychagin, V., Yumaguzhin, V.: Differential invariants and exact solutions of the Einstein–Maxwell equation. Anal. Math. Phys. 7, 19–29 (2017)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Lychagin, V., Yumaguzhin, V.: Differential invariants and exact solutions of the Einstein equations. Anal. Math. Phys. 7, 107–115 (2017)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Marvan, M., Stolín, O.: On local equivalence problem of spacetimes with two orthogonally transitive commuting Killing fields. J. Math. Phys. 49(2), 022503 (2008). 17 ppMathSciNetCrossRefGoogle Scholar
  26. 26.
    Milson, R., McNutt, D., Coley, A.: Invariant classification of vacuum pp-waves. J. Math. Phys. 54, 022502 (2013)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Olver, P.J.: Equivalence, Invariants and Symmetry. Cambridge University Press, New York (1995)CrossRefGoogle Scholar
  28. 28.
    O’Neill, B.: The fundamental equations of a submersion. Mich. Math. J. 13, 459–469 (1966)MathSciNetCrossRefGoogle Scholar
  29. 29.
    Ovsiannikov, L.V.: Group Analysis of Differential Equations. Academic Press, New York (1982)zbMATHGoogle Scholar
  30. 30.
    Petrov, A.Z.: Einstein Spaces. Pergamon, New York (1969)CrossRefGoogle Scholar
  31. 31.
    Papapetrou, A.: Champs gravitationnels stationnaires à symétrie axiale. Ann. Inst. H. Poincaré A 4(2), 83–105 (1966)zbMATHGoogle Scholar
  32. 32.
    Pollney, D., Skea, J.E.F., d’Inverno, R.A.: Classifying geometries in general relativity: III. Classification in practice. Class. Quantum Gravity 17, 2885–2902 (2000)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Pravda, V., Pravdová, A., Coley, A., Milson, R.: All spacetimes with vanishing curvature invariants. Class. Quantum Gravity 19, 6213–6236 (2002)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Skea, J.E.F.: A spacetime whose invariant classification requires the fifth covariant derivative of the Riemann tensor. Class. Quantum Gravity 17, L69–L74 (2000)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Stephani, H., Kramer, D., MacCallum, M., Hoenselaers, C., Herlt, E.: Exact Solutions of Einstein’s Field Equations, 2nd edn. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  36. 36.
    Verdaguer, E.: Soliton solutions in spacetime with two spacelike Killing fields. Phys. Rep. 229(1–2), 1–80 (1993)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Van den Bergh, N.: A class of inhomogeneous cosmological models with separable metrics. Class. Quantum Gravity 5, 167–177 (1988)MathSciNetCrossRefGoogle Scholar
  38. 38.
    Whelan, J.T., Romano, J.D.: Quasistationary binary inspiral. I. Einstein equations for the two Killing vector spacetime. Phys. Rev. D 60, 084009 (1999)MathSciNetCrossRefGoogle Scholar
  39. 39.
    Lim, W.C.: Non-orthogonally transitive G2 spike solution. Class. Quantum Gravity 32, 162001 (2015)CrossRefGoogle Scholar
  40. 40.
    Żorawski, K.: On deformation invariants. An application of Lie’s theory of groups. Acta Math. 16, 1–64 (1892) (in German) Google Scholar

Copyright information

© Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Instituto de Matemática e EstatísticaUniversidade Federal da Bahia, Campus de OndinaSalvadorBrazil
  2. 2.Mathematical Institute in OpavaSilesian University in OpavaOpavaCzech Republic

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