Foundations of Physics

, Volume 39, Issue 3, pp 307–330 | Cite as

Gravitation, Electromagnetism and Cosmological Constant in Purely Affine Gravity

Article

Abstract

The Ferraris-Kijowski purely affine Lagrangian for the electromagnetic field, that has the form of the Maxwell Lagrangian with the metric tensor replaced by the symmetrized Ricci tensor, is dynamically equivalent to the metric Einstein-Maxwell Lagrangian, except the zero-field limit, for which the metric tensor is not well-defined. This feature indicates that, for the Ferraris-Kijowski model to be physical, there must exist a background field that depends on the Ricci tensor. The simplest possibility, supported by recent astronomical observations, is the cosmological constant, generated in the purely affine formulation of gravity by the Eddington Lagrangian. In this paper we combine the electromagnetic field and the cosmological constant in the purely affine formulation. We show that the sum of the two affine (Eddington and Ferraris-Kijowski) Lagrangians is dynamically inequivalent to the sum of the analogous (ΛCDM and Einstein-Maxwell) Lagrangians in the metric-affine/metric formulation. We also show that such a construction is valid, like the affine Einstein-Born-Infeld formulation, only for weak electromagnetic fields, on the order of the magnetic field in outer space of the Solar System. Therefore the purely affine formulation that combines gravity, electromagnetism and cosmological constant cannot be a simple sum of affine terms corresponding separately to these fields. A quite complicated form of the affine equivalent of the metric Einstein-Maxwell-Λ Lagrangian suggests that Nature can be described by a simpler affine Lagrangian, leading to modifications of the Einstein-Maxwell-ΛCDM theory for electromagnetic fields that contribute to the spacetime curvature on the same order as the cosmological constant.

Keywords

Purely affine gravity Einstein-Maxwell equations Cosmological constant Legendre transformation Eddington Lagrangian Ferraris-Kijowski Lagrangian 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Einstein, A.: Sitzungsber. Preuss. Akad. Wiss. (Berlin), p. 137 (1923) Google Scholar
  2. 2.
    Eddington, A.S.: The Mathematical Theory of Relativity. Cambridge (1924) Google Scholar
  3. 3.
    Schrödinger, E.: Space-Time Structure. Cambridge (1950) Google Scholar
  4. 4.
    Kijowski, J.: Gen. Relativ. Gravit. 9, 857 (1978) MATHCrossRefADSMathSciNetGoogle Scholar
  5. 5.
    Catto, D., Francaviglia, M., Kijowski, J.: Bull. Acad. Pol. Sci. 28, 179 (1980) MathSciNetGoogle Scholar
  6. 6.
    Palatini, A.: Rend. Circ. Mat. (Palermo) 43, 203 (1919) MATHCrossRefGoogle Scholar
  7. 7.
    Einstein, A.: Sitzungsber. Preuss. Akad. Wiss. (Berlin), p. 414 (1925) Google Scholar
  8. 8.
    Ferraris, M., Francaviglia, M., Reina, C.: Gen. Relativ. Gravit. 14, 243 (1982) MATHCrossRefADSMathSciNetGoogle Scholar
  9. 9.
    Einstein, A.: Sitzungsber. Preuss. Akad. Wiss. (Berlin), p. 844 (1915) Google Scholar
  10. 10.
    Hilbert, D.: Königl. Gesell. Wiss., Göttingen Nachr., p. 395 (1915) Google Scholar
  11. 11.
    Lorentz, H.A.: Koninkl. Akad. Wetensch. (Amsterdam) 24, 1389, 1759 (1916) Google Scholar
  12. 12.
    Lorentz, H.A.: Koninkl. Akad. Wetensch. (Amsterdam) 25, 468, 1380 (1916) Google Scholar
  13. 13.
    Landau, L.D., Lifshitz, E.M.: The Classical Theory of Fields. Pergamon, Oxford (1975) Google Scholar
  14. 14.
    Ferraris, M., Kijowski, J.: Gen. Relativ. Gravit. 14, 165 (1982) MATHCrossRefADSMathSciNetGoogle Scholar
  15. 15.
    Ferraris, M., Kijowski, J.: Rend. Sem. Mat. Univ. Polit. (Torino) 41, 169 (1983) MATHMathSciNetGoogle Scholar
  16. 16.
    Kijowski, J., Werpachowski, R.: Rep. Math. Phys. 59, 1 (2007) CrossRefADSMathSciNetGoogle Scholar
  17. 17.
    Ferraris, M., Kijowski, J.: Lett. Math. Phys. 5, 127 (1981) CrossRefADSMathSciNetGoogle Scholar
  18. 18.
    Kijowski, J., Pawlik, B., Tulczyjew, W.M.: Bull. Acad. Pol. Sci. 27, 163 (1979) MathSciNetGoogle Scholar
  19. 19.
    Kijowski, J., Magli, G.: Class. Quantum Gravity 15, 3891 (1998) MATHCrossRefADSMathSciNetGoogle Scholar
  20. 20.
    Jakubiec, A., Kijowski, J.: J. Math. Phys. 30, 1073 (1989) MATHCrossRefADSMathSciNetGoogle Scholar
  21. 21.
    Jakubiec, A., Kijowski, J.: J. Math. Phys. 30, 1077 (1989) MATHCrossRefADSMathSciNetGoogle Scholar
  22. 22.
    Einstein, A.: Ann. Phys. (Leipzig) 49, 769 (1916) ADSGoogle Scholar
  23. 23.
    Weyl, H.: Space, Time, Matter. Methuen (1922) Google Scholar
  24. 24.
    Schrödinger, E.: Proc. R. Ir. Acad. A 51, 163 (1947) Google Scholar
  25. 25.
    Einstein, A., Straus, E.G.: Ann. Math. 47, 731 (1946) CrossRefMathSciNetGoogle Scholar
  26. 26.
    Band, W.: Phys. Rev. 36, 1405 (1930) CrossRefADSGoogle Scholar
  27. 27.
    Eisenhart, L.P.: Proc. Natl. Acad. Sci. USA 42, 249 (1956) MATHCrossRefADSMathSciNetGoogle Scholar
  28. 28.
    Goenner, H.F.M.: Liv. Rev. Relativ. 7, 2 (2004) ADSMathSciNetGoogle Scholar
  29. 29.
    Martellini, M.: Phys. Rev. Lett. 51, 152 (1983) CrossRefADSGoogle Scholar
  30. 30.
    Martellini, M.: Phys. Rev. D 29, 2746 (1984) CrossRefADSMathSciNetGoogle Scholar
  31. 31.
    Ferraris, M., Kijowski, J.: Gen. Relativ. Gravit. 14, 37 (1982) MATHCrossRefADSMathSciNetGoogle Scholar
  32. 32.
    Chruściel, P.T.: Ann. Inst. Henri Poincarè 42, 329 (1985) MATHGoogle Scholar
  33. 33.
    Will, C.M.: Theory and Experiment in Gravitational Physics. Cambridge (1992) Google Scholar
  34. 34.
    Popławski, N.J.: J. Math. Phys. 47, 072501 (2006). arXiv:gr-qc/0503066 CrossRefADSMathSciNetGoogle Scholar
  35. 35.
    Riess, A.G., : Astron. J. 116, 1009 (1998) CrossRefADSGoogle Scholar
  36. 36.
    Perlmutter, S., : Astrophys. J. 517, 565 (1999) CrossRefGoogle Scholar
  37. 37.
    Spergel, D.N., : Astrophys. J. Suppl. 170, 377 (2007) CrossRefADSGoogle Scholar
  38. 38.
    Born, M., Infeld, L.: Proc. R. Soc. 144, 425 (1934) MATHCrossRefADSGoogle Scholar
  39. 39.
    Motz, L.: Phys. Rev. 89, 60 (1953) MATHCrossRefADSMathSciNetGoogle Scholar
  40. 40.
    Vollick, D.N.: Phys. Rev. D 72, 084026 (2005) CrossRefADSMathSciNetGoogle Scholar
  41. 41.
    Ferraris, M., Kijowski, J.: In: Kowalski, O. (ed.) Proceedings of Conference on Differential Geometry and its Applications, p. 167. Univerzita Karlova, Praha (1982) Google Scholar
  42. 42.
    Popławski, N.J., Phys, Mod.: Lett. A 22, 2701 (2007). arXiv:gr-qc/0610132 MATHGoogle Scholar
  43. 43.
    Schouten, J.A.: Ricci-Calculus. Springer, New York (1954) MATHGoogle Scholar
  44. 44.
    Kurşunoğlu, B.: Phys. Rev. 88, 1369 (1952) MATHCrossRefADSGoogle Scholar
  45. 45.
    Hély, J.: C. R. Acad. Sci. (Paris) 239, 385 (1954) MATHMathSciNetGoogle Scholar
  46. 46.
    Johnson, C.R.: Phys. Rev. D 31, 1236 (1985) CrossRefADSMathSciNetGoogle Scholar
  47. 47.
    Hehl, F.W., von der Heyde, P., Kerlick, G.D., Nester, J.M.: Rev. Mod. Phys. 48, 393 (1976) CrossRefADSGoogle Scholar
  48. 48.
    Hehl, F.W., Kerlick, G.D.: Gen. Relativ. Gravit. 9, 691 (1978) MATHCrossRefADSMathSciNetGoogle Scholar
  49. 49.
    Hehl, F.W., Lord, E.A., Smalley, L.L.: Gen. Relativ. Gravit. 13, 1037 (1981) MATHCrossRefMathSciNetGoogle Scholar
  50. 50.
    Sandberg, V.D.: Phys. Rev. D 12, 3013 (1975) CrossRefGoogle Scholar
  51. 51.
    Ponomarev, V.N., Obukhov, Yu.N.: Gen. Relativ. Gravit. 14, 309 (1982) CrossRefADSGoogle Scholar
  52. 52.
    Popławski, N.J.: arXiv:0705.0351
  53. 53.
    Popławski, N.J.: Int. J. Mod. Phys. A 23, 567 (2008). arXiv:gr-qc/0702129 MATHCrossRefADSGoogle Scholar
  54. 54.
    Landau, L.D., Lifshitz, E.M.: Mechanics. Pergamon, Oxford (1960) MATHGoogle Scholar
  55. 55.
    Popławski, N.J.: Int. J. Mod. Phys. A 23, 1891 (2008). arXiv:0706.4474 MATHCrossRefADSGoogle Scholar
  56. 56.
    Olmo, G.J.: Phys. Rev. Lett. 95, 261102 (2005) CrossRefADSGoogle Scholar
  57. 57.
    Puntigam, R.A., Lämmerzahl, C., Hehl, F.W.: Class. Quantum Gravity 14, 1347 (1997) MATHCrossRefADSGoogle Scholar
  58. 58.
    Hlavatý, V.: J. Ration. Mech. Anal. 3, 103 (1954) Google Scholar
  59. 59.
    Coley, A.: Gen. Relativ. Gravit. 16, 459 (1984) CrossRefADSMathSciNetGoogle Scholar
  60. 60.
    Anderson, J.D., Laing, P.A., Lau, E.L., Liu, A.S., Nieto, M.M., Turyshev, S.G.: Phys. Rev. Lett. 81, 2858 (1998) CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Department of PhysicsIndiana UniversityBloomingtonUSA

Personalised recommendations