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PP-waves with torsion: a metric-affine model for the massless neutrino

  • Vedad Pasic
  • Elvis Barakovic
Research Article

Abstract

In this paper we deal with quadratic metric-affine gravity, which we briefly introduce, explain and give historical and physical reasons for using this particular theory of gravity. We then introduce a generalisation of well known spacetimes, namely pp-waves. A classical pp-wave is a 4-dimensional Lorentzian spacetime which admits a nonvanishing parallel spinor field; here the connection is assumed to be Levi-Civita. This definition was generalised in our previous work to metric compatible spacetimes with torsion and used to construct new explicit vacuum solutions of quadratic metric-affine gravity, namely generalised pp-waves of parallel Ricci curvature. The physical interpretation of these solutions we propose in this article is that they represent a conformally invariant metric-affine model for a massless elementary particle. We give a comparison with the classical model describing the interaction of gravitational and massless neutrino fields, namely Einstein–Weyl theory and construct pp-wave type solutions of this theory. We point out that generalised pp-waves of parallel Ricci curvature are very similar to pp-wave type solutions of the Einstein–Weyl model and therefore propose that our generalised pp-waves of parallel Ricci curvature represent a metric-affine model for the massless neutrino.

Keywords

Quadratic metric-affine gravity pp-Waves Torsion Massless neutrino Einstein–Weyl theory 

Notes

Acknowledgments

The authors are very grateful to D Vassiliev, J B Griffiths and F W Hehl for helpful advice and to the Ministry of Education and Science of the Federation of Bosnia and Herzegovina, which supported our research.

References

  1. 1.
    Adamowicz, W.: Plane waves in gauge theories of gravitation. Gen. Relativ. Gravit. 12, 677–691 (1980)MathSciNetCrossRefzbMATHADSGoogle Scholar
  2. 2.
    Alekseevsky, D.V.: Holonomy groups and recurrent tensor fields in Lorentzian spaces. In: Stanjukovich, K.P. (ed.) Problems of the Theory of Gravitation and Elementary Particles issue 5, pp. 5–17. Atomizdat, Moscow (in Russian) (1974)Google Scholar
  3. 3.
    Audretsch, J.: Asymptotic behaviour of neutrino fields in curved space-time. Commun. Math. Phys. 21, 303–313 (1971)MathSciNetCrossRefADSGoogle Scholar
  4. 4.
    Audretsch, J., Graf, W.: Neutrino radiation in gravitational fields. Commun. Math. Phys. 19, 315–326 (1970)MathSciNetCrossRefzbMATHADSGoogle Scholar
  5. 5.
    Babourova, O.V., Frolov, B.N., Klimova, E.A.: Plane torsion waves in quadratic gravitational theories in RiemannCartan space. Class. Quantum Grav. 16, 1149–1162 (1999). gr-qc/9805005
  6. 6.
    Berestetskii, V.B., Lifshitz, E.M., Pitaevskii, L.P.: Quantum Electrodynamics (Course of Theoretical Physics vol. 4) 2nd edn. Pergamon Press, Oxford (1982)Google Scholar
  7. 7.
    Blagojevic, M.: Gravitation and Gauge Symmetries. Institute of Physics Publishing, Bristol (2002)CrossRefzbMATHGoogle Scholar
  8. 8.
    Blagojević, M., Hehl, F.W.: Gauge Theories of Gravitation. A Reader with Commentaries. Imperial College Press, London (2013)CrossRefzbMATHGoogle Scholar
  9. 9.
    Brill, D.R., Wheeler, J.A.: Interaction of neutrinos and gravitational fields. Rev. Mod. Phys. 29, 465–479 (1957)MathSciNetCrossRefzbMATHADSGoogle Scholar
  10. 10.
    Brinkmann, M.W.: On Riemann spaces conformal to Euclidean space. Proc. Natl. Acad. Sci. USA 9, 1–3 (1923)CrossRefADSGoogle Scholar
  11. 11.
    Bryant, R.L.: Pseudo-Riemannian metrics with parallel spinor fields and vanishing Ricci tensor. Global Analysis and Harmonic Analysis (Marseille-Luminy, 1999) Sémin. Congr. 4 (Paris: Soc. Math. France), 53–94 (2000). math/0004073
  12. 12.
    Buchbinder, I.L., Kuzenko, S.M.: Ideas and Methods of Supersymmetry and Supergravity. Institute of Physics Publishing, Bristol (1998)zbMATHGoogle Scholar
  13. 13.
    Buchdahl, H.A.: Math. Rev. 20, 1238 (1959)Google Scholar
  14. 14.
    Collinson, C.D., Morris, P.B.: Space-time admitting neutrino fields with zero energy and momentum. J. Phys. A6, 915–916 (1972)ADSGoogle Scholar
  15. 15.
    Cotton, É.: Sur les varits a trois dimensions. Annales de la Facult des Sciences de Toulouse II 14(14), 385–438 (1899)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Davis, T.M., Ray, J.R.: Ghost neutrinos in general relativity. Phys. Rev. D 9, 331–333 (1974)CrossRefADSGoogle Scholar
  17. 17.
    Eddington, A.S.: The Mathematical Theory of Relativity, 2nd edn. The University Press, Cambridge (1952)Google Scholar
  18. 18.
    Esser, W.: Exact Solutions of the Metric-Affine Gauge Theory of Gravity. Diploma Thesis, University of Cologne, Cologne (1996)Google Scholar
  19. 19.
    Fairchild Jr, E.E.: Gauge theory of gravitation. Phys. Rev. D 14, 384–391 (1976)MathSciNetCrossRefADSGoogle Scholar
  20. 20.
    Fairchild Jr, E.E.: Erratum: gauge theory of gravitation. Phys. Rev. D 14, 2833 (1976)MathSciNetCrossRefADSGoogle Scholar
  21. 21.
    García, A., Macías, A., Puetzfeld, D., Socorro, J.: Plane fronted waves in metric affine gravity. Phys. Rev. D 62, 044021 (2000). gr-qc/0005038
  22. 22.
    García, A., Hehl, F.W., Heinicke, C., Macías, A.: The cotton tensor in Riemannian spacetimes. Class. Quantum Grav. 21, 10991118 (2004). gr-qc/0309008
  23. 23.
    Griffiths, J.B.: Colliding Plane Waves in General Relativity. Oxford University Press, Oxford (1991)zbMATHGoogle Scholar
  24. 24.
    Griffiths, J.B., Newing, R.A.: The two-component neutrino field in general relativity. J. Phys. A 3, 136–149 (1970)MathSciNetCrossRefADSGoogle Scholar
  25. 25.
    Griffiths, J.B., Newing, R.A.: Tetrad equations for the two-component neutrino field in general relativity. J. Phys. A 3, 269–273 (1970)MathSciNetCrossRefADSGoogle Scholar
  26. 26.
    Griffiths, J.B.: Some physical properties of neutrino-gravitational fields. Int. J. Theor. Phys. 5, 141–150 (1972)CrossRefGoogle Scholar
  27. 27.
    Griffiths, J.B.: Gravitational radiation and neutrinos. Commun. Math. Phys. 28, 295–299 (1972)CrossRefADSGoogle Scholar
  28. 28.
    Griffiths, J.B.: Ghost neutrinos in Einstein–Cartan theory. Phys. Lett. A 75, 441–442 (1980)CrossRefADSGoogle Scholar
  29. 29.
    Griffiths, J.B.: Neutrino fields in Einstein–Cartan theory. Gen. Relativ. Gravit. 13, 227–237 (1981)CrossRefzbMATHADSGoogle Scholar
  30. 30.
    Hehl, F.W.: Spin und Torsion in der Allgemeinen Relativitätstheorie oder die Riemann-Cartansche Geometrie der Welt. Technischen Universität Clausthal, Habilitationsschrift (1970)Google Scholar
  31. 31.
    Hehl, F.W.: Spin and torsion in general relativity I: foundations. Gen. Relativ. Gravit. 4, 333–349 (1973)MathSciNetCrossRefADSGoogle Scholar
  32. 32.
    Hehl, F.W.: Spin and torsion in general relativity II: geometry and field equations. Gen. Relativ. Gravit. 5, 491–516 (1974)MathSciNetCrossRefADSGoogle Scholar
  33. 33.
    Hehl, F.W., McCrea, J.D., Mielke, E.W., Ne’eman, Y.: Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance. Phys. Rep. 258, 1–171 (1995). gr-qc/9402012
  34. 34.
    Hehl, F.W., Macías, A.: Metric-affine gauge theory of gravity II. Exact solutions. Int. J. Mod. Phys. D 8, 399–416 (1999). gr-qc/9902076
  35. 35.
    Hehl, F.W., von der Heyde, P., Kerlick, G.D., Nester, J.M.: General relativity with spin and torsion: foundations and prospects. Rev. Mod. Phys. 48, 393–416 (1976)CrossRefADSGoogle Scholar
  36. 36.
    Higgs, P.W.: Quadratic Lagrangians and general relativity. Nuovo Cimento 11, 816–820 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    King, A.D., Vassiliev, D.: Torsion waves in metric-affine field theory. Class. Quantum Grav. 18, 2317–2329 (2001). gr-qc/0012046
  38. 38.
    Kramer, D., Stephani, H., Herlt, E., MacCallum, M.: Exact Solutions of Einstein’s Field Equations. Cambridge University Press, Cambridge (1980)zbMATHGoogle Scholar
  39. 39.
    Kröner, E.: Continuum theory of defects. In: Balian R., et al. (eds.) Physics of Defects, Les Houches, Session XXXV, 1980. North-Holland, Amsterdam (1980)Google Scholar
  40. 40.
    Kröner, E.: The continuized crystal—a bridge between micro- and macromechanics. Gesellschaft angewandte Mathematik und Mechanik Jahrestagung Goettingen West Germany Zeitschrift Flugwissenschaften, vol. 66 (1986)Google Scholar
  41. 41.
    Kuchowicz, C., Żebrowski, J.: The presence of torsion enables a metric to allow a gravitational field. Phys. Lett. A 67, 16–18 (1978)MathSciNetCrossRefADSGoogle Scholar
  42. 42.
    Lanczos, C.: A remarkable property of the Riemann–Christoffel tensor in four dimensions. Ann. Math. 39, 842–850 (1938)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Lanczos, C.: Lagrangian multiplier and Riemannian spaces. Rev. Mod. Phys. 21, 497–502 (1949)MathSciNetCrossRefzbMATHADSGoogle Scholar
  44. 44.
    Lanczos, C.: Electricity and general relativity. Rev. Mod. Phys. 29, 337–350 (1957)MathSciNetCrossRefzbMATHADSGoogle Scholar
  45. 45.
    Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields (Course of Theoretical Physics vol. 2) 4nd edn. Pergamon Press, Oxford (1975)Google Scholar
  46. 46.
    Mielke, E.W.: On pseudoparticle solutions in Yang’s theory of gravity. Gen. Relativ. Gravit. 13, 175–187 (1981)MathSciNetCrossRefzbMATHADSGoogle Scholar
  47. 47.
    Nakahara, M.: Geometry, Topology and Physics. Institute of Physics Publishing, Bristol (1998)Google Scholar
  48. 48.
    Obukhov, Y.N.: Generalized plane fronted gravitational waves in any dimension. Phys. Rev. D 69, 024013 (2004). gr-qc/0310121
  49. 49.
    Obukhov, Y.N.: Plane waves in metric-affine gravity. Phys. Rev. D 73, 024025 (2006). gr-qc/0601074
  50. 50.
    Olesen, P.: A relation between the Einstein and the Yang–Mills field equations. Phys. Lett. B 71, 189–190 (1977)MathSciNetCrossRefADSGoogle Scholar
  51. 51.
    Pasic, V.: New vacuum solutions for quadratic metric-affine gravity—a metric affine model for the massless neutrino? Math. Balk. New Ser. 24, Fasc 3–4, 329 (2010)Google Scholar
  52. 52.
    Pasic, V., Barakovic, E., Okicic, N.: A new representation of the field equations of quadratic metric-affine gravity. Adv. Math. Sci. J. 3(1), 33–46 (2014)Google Scholar
  53. 53.
    Pasic, V., Vassiliev, D.: PP-waves with torsion and metric-affine gravity. Class. Quantum Grav. 22, 3961–3975 (2005). gr-qc/0505157
  54. 54.
    Pauli, W.: Zur Theorie der Gravitation und der Elektrizität von Hermann Weyl. Physik. Zaitschr. 20, 457–467 (1919)zbMATHGoogle Scholar
  55. 55.
    Pavelle, R.: Unphysical solutions of Yang’s gravitational-field equations. Phys. Rev. Lett. 34, 1114 (1975)CrossRefADSGoogle Scholar
  56. 56.
    Penrose, O., Rindler, W.: Propagating modes in gauge field theories of gravity, vol. 2. Cambridge University Press, Oxford (1984, 1986)Google Scholar
  57. 57.
    Peres, A.: Some gravitational waves. Phys. Rev. Lett. 3, 571–572 (1959)CrossRefzbMATHADSGoogle Scholar
  58. 58.
    Peres, A.: PP—WAVES preprint (reprinting of [57]) (2002). hep-th/0205040
  59. 59.
    Pirani, F.A.E.: Introduction to Gravitational Radiation Theory. Lectures on General Relativity. Prentice-Hall, Inc. Englewood Cliffs, New Jersey (1964)Google Scholar
  60. 60.
    Singh, P.: On axial vector torsion in vacuum quadratic Poincaré gauge field theory. Phys. Lett. A 145, 7–10 (1990)MathSciNetCrossRefADSGoogle Scholar
  61. 61.
    Singh, P.: On null tratorial torsion in vacuum quadratic Poincaré gauge field theory. Class. Quantum Grav. 7, 2125–2130 (1990)CrossRefzbMATHADSGoogle Scholar
  62. 62.
    Singh, P., Griffiths, J.B.: On neutrino fields in Einstein–Cartan theory. Phys. Lett. A 132, 88–90 (1988)MathSciNetCrossRefADSGoogle Scholar
  63. 63.
    Singh, P., Griffiths, J.B.: A new class of exact solutions of the vacuum quadratic Poincaré gauge field theory. Gen. Relativ. Gravit. 22, 947–956 (1990)MathSciNetCrossRefzbMATHADSGoogle Scholar
  64. 64.
    Stephenson, G.: Quadratic Lagrangians and general relativity. Nuovo Cimento 9, 263–269 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  65. 65.
    Streater, R.F., Wightman, A.S.: PCT, spin and statistics, and all that. In: Princeton Landmarks in Physics, Princeton University Press, Princeton. ISBN:0-691-07062-8, corrected third printing of the 1978 edition (2000)Google Scholar
  66. 66.
    Thompson, A.H.: Yang’s gravitational field equations. Phys. Rev. Lett. 34, 507–508 (1975)CrossRefADSGoogle Scholar
  67. 67.
    Thompson, A.H.: Geometrically degenerate solutions of the Kilmister–Yang equations. Phys. Rev. Lett. 35, 320–322 (1975)MathSciNetCrossRefADSGoogle Scholar
  68. 68.
    Vassiliev, D.: Pseudoinstantons in metric-affine field theory. Gen. Relativ. Gravit. 34, 1239–1265 (2002). gr-qc/0108028
  69. 69.
    Vassiliev, D.: Pseudoinstantons in metric-affine field theory. In: Brambilla, N., Prosperi, G.M. (eds.) Quark Confinement and the Hadron Spectrum V, pp. 273–275. World Scientific, Singapore (2003)CrossRefGoogle Scholar
  70. 70.
    Vassiliev, D.: Quadratic non-Riemannian gravity. J. Nonlinear Math. Phys. 11(Supplement), 204–216 (2004)MathSciNetCrossRefADSGoogle Scholar
  71. 71.
    Vassiliev, D.: Quadratic metric-affine gravity. Ann. Phys. (Lpz.) 14, 231–252 (2005). gr-qc/0304028
  72. 72.
    Weyl, H.: Eine neue Erweiterung der Relativitätstheorie. Ann. Phys. (Lpz.) 59, 101–133 (1919)CrossRefzbMATHADSGoogle Scholar
  73. 73.
    Wilczek, F.: Geometry and interaction of instantons. In: Stump, D.R., Weingarten, D.H. (eds.) Quark Confinement and Field theory, pp. 211–219. Wiley-Interscience, New York (1977)Google Scholar
  74. 74.
    Yang, C.N.: Integral formalism for gauge fields. Phys. Rev. Lett. 33, 445–447 (1974)MathSciNetCrossRefADSGoogle Scholar

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© Springer Science+Business Media New York 2014

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TuzlaTuzlaBosnia and Herzegovina

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