Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 4 (2008), 071, 29 pages      arXiv:0806.3810      https://doi.org/10.3842/SIGMA.2008.071
Contribution to the Special Issue “Élie Cartan and Differential Geometry”

Einstein Gravity, Lagrange-Finsler Geometry, and Nonsymmetric Metrics

Sergiu I. Vacaru a, b
a) The Fields Institute for Research in Mathematical Science, 222 College Street, 2d Floor, Toronto, M5T 3J1, Canada
b) Faculty of Mathematics, University ''Al. I. Cuza'' Iasi, 700506, Iasi, Romania

Received June 24, 2008, in final form October 13, 2008; Published online October 23, 2008

Abstract
We formulate an approach to the geometry of Riemann-Cartan spaces provided with nonholonomic distributions defined by generic off-diagonal and nonsymmetric metrics inducing effective nonlinear and affine connections. Such geometries can be modelled by moving nonholonomic frames on (pseudo) Riemannian manifolds and describe various types of nonholonomic Einstein, Eisenhart-Moffat and Finsler-Lagrange spaces with connections compatible to a general nonsymmetric metric structure. Elaborating a metrization procedure for arbitrary distinguished connections, we define the class of distinguished linear connections which are compatible with the nonlinear connection and general nonsymmetric metric structures. The nonsymmetric gravity theory is formulated in terms of metric compatible connections. Finally, there are constructed such nonholonomic deformations of geometric structures when the Einstein and/or Lagrange-Finsler manifolds are transformed equivalently into spaces with generic local anisotropy induced by nonsymmetric metrics and generalized connections. We speculate on possible applications of such geometric methods in Einstein and generalized theories of gravity, analogous gravity and geometric mechanics.

Key words: nonsymmetric metrics; nonholonomic manifolds; nonlinear connections; Eisenhart-Lagrange spaces; generalized Riemann-Finsler geometry.

pdf (436 kb)   ps (257 kb)   tex (39 kb)

References

  1. Einstein A., Einheitliche Fieldtheorie von Gravitation and Electrizidät, Sitzungsberichte der Preussischen Akademie Wissebsgaften, Mathematischn-Naturwissenschaftliche Klasse, 1925, 414-419 (translated in English by A. Unzicker and T. Case, Unified Field Theory of Gravitation and Electricity, session report from July 25, 1925, 214-419, physics/0503046 and http://www.lrz-muenchen.de/~aunzicker/ae1930.html).
  2. Einstein A., A generalization of the relativistic theory of gravitation, Ann. of Math. (2) 46 (1945), 578-584.
  3. Eisenhart L.P., Generalized Riemann spaces, Proc. Nat. Acad. Sci. USA 37 (1951), 311-314.
  4. Eisenhart L.P., Generalized Riemann spaces, II, Proc. Nat. Acad. Sci. USA 38 (1952), 505-508.
  5. Moffat J.W., New theory of gravity, Phys. Rev. D 19 (1979), 3554-3558.
  6. Moffat J.W., A new nonsymmetric gravitational theory, Phys. Lett. B 355 (1995), 447-452, gr-qc/9411006.
  7. Moffat J.W., Review of nonsymmetric gravitational theory, in Proceedings of the Summer Institute on Gravitation (Banff Centre, Banff, Canada, 1990), Editors R.B. Mann and P. Wesson, World Sci. Publ., River Edge, NJ, 1991, 1991, 523-597.
  8. Moffat J.W., Nonsymmetric gravitational theory, J. Math. Phys. 36 (1995), 3722-3232, Erratum, J. Math. Phys. 36 (1995), 7128.
  9. Moffat J.W., Noncommutative quantum gravity, Phys. Lett. B 491 (2000), 345-352, hep-th/0007181.
  10. Moffat J.W., Late-time inhomogeneity and acceleration without dark energy, J. Cosmol. Astropart. Phys. 2006 (2006), no. 5, 001, 14 pages, astro-ph/0505326.
  11. Prokopec T., Valkenburg W., The cosmology of the nonsymmetric theory of gravitation, Phys. Lett. B 636 (2006), 1-4, astro-ph/0503289.
  12. Moffat J.W., Toth V.T., Testing modified gravity with globular cluster veloscity dispersions, Astrophys. J. 680 (2008), 1158-1161, arXiv:0708.1935.
  13. Miron R., Atanasiu Gh., Existence et arbitrariété des connexions compatibles à une structure Riemann généralisée du type presque k-horsympletique métrique, Kodai Math. J. 6 (1983), 228-237.
  14. Atanasiu Gh., Hashiguchi M., Miron R., Supergeneralized Finsler spaces, Rep. Fac. Sci. Kagoshima Univ. Math. Phys. Chem. (1985), no. 18, 19-34.
  15. Miron R., Anastasiei M., Vector bundles and lagrange spaces with applications to relativity, Geometry Balkan Press, Bucharest, 1997 (translation from Romanian of Editura Academiei Romane, 1987).
  16. Vacaru S., Nonholonomic Ricci flows. I. Riemann metrics and Lagrange-Finsler geometry, math.DG/0612162.
  17. Vacaru S., Nonholonomic Ricci flows. II. Evolution equations and dynamics, J. Math. Phys. 49 (2008), 043504, 27 pages, math.DG/0702598.
  18. Vacaru S., Ricci flows and solitonic pp-waves, Internat. J. Modern Phys. A 21 (2006), 4899-4912, hep-th/0602063.
  19. Vacaru S., Visinescu M., Nonholonomic Ricci flows and running cosmological constant. I. 4D Taub-NUT metrics, Internat. J. Modern Phys. A 22 (2007), 1135-1160, gr-qc/0609085.
  20. Vacaru S., Visinescu M., Nonholonomic Ricci flows and running cosmological constant: 3D Taub-NUT metrics, Romanian Rep. Phys. 60 (2008), 218-238, gr-qc/0609086.
  21. Vacaru S., Nonholonomic Ricci flows. IV. Geometric methods, exact solutions and gravity, arXiv:0705.0728.
  22. Vacaru S., Nonholonomic Ricci flows. V. Parametric deformations of solitonic pp-waves and Schwarzschild solutions, arXiv:0705.0729.
  23. Vacaru S., Nonholonomic Ricci flows, Exact solutions in gravity, and symmetric and nonsymmetric metrics, Internat. J. Theor. Phys., https://doi.org/10.1007/s10773-008-9841-8, to appear, arXiv:0806.3812.
  24. Vacaru S., Finsler and Lagrange geometries in Einstein and string gravity, Int. J. Geom. Methods Mod. Phys. 5 (2008), 473-511, arXiv:0801.4958.
  25. Miron R., Anastasiei M., The geometry of Lagrange spaces: theory and applications, Fundamental Theories of Physics, Vol. 59, Kluwer Academic Publishers, Dordrecht, 1994.
  26. Vacaru S., Spinor structures and nonlinear connections in vector bundles, generalized Lagrange and Finsler spaces, J. Math. Phys. 37 (1996), 508-524.
  27. Vacaru S., Spinors and field interactions in higher order anisotropic spaces, J. High Energy Phys. 1998 (1998), no. 9, 011, 50 pages, hep-th/9807214.
  28. Vacaru S., Clifford-Finsler algebroids and nonholonomic Einstein-Dirac structures, J. Math. Phys. 47 (2006), 093504, 20 pages, hep-th/0501217.
  29. Vacaru S., Gauge and Einstein gravity from non-Abelian gauge models on noncommutative spaces, Phys. Lett. B 498 (2001), 74-82, hep-th/0009163.
  30. Vacaru S., Exact solutions with noncommutative symmetries in Einstein and gauge gravity, J. Math. Phys. 46 (2005), 042503, 47 pages, gr-qc/0307103.
  31. Vacaru S., Deformation quantization of nonholonomic almost Kähler models and Einstein gravity, Phys. Lett. A 372 (2008), 2949-2955, arXiv:0707.1667.
  32. Kawaguchi A., Bezienhung zwischen einer metrischen linearen Ubertragung unde iener micht-metrischen in einem allemeinen metrischen Raume, Akad. Wetensch. Amsterdam Proc. 40 (1937), 596-601.
  33. Kawaguchi A., On the theory of non-linear connections. I. Introduction to the theory of general non-linear connections, Tensor (NS) 2 (1952), 123-142.
  34. Kawaguchi A., On the theory of non-linear connections. II. Theory of Minkowski spaces and of non-linear connections in a Finsler space, Tensor (NS) 6 (1956), 165-199.
  35. Légaré J., Moffat J.W., Field equations and conservation laws in nonsymmetric gravitational theory, Gen. Relativity Gravitation 27 (1995), 761-775, gr-qc/9412009.
  36. Miron R., Hrimiuc D., Shimada H., Sabau V.S., The Geometry of Hamilton and Lagrange spaces, Fundamental Theories of Physics, Vol. 118, Kluwer Academic Publishers Dordrecht, 2000.
  37. Barcelo C., Liberaty S., Visser M., Analogue gravity, Living Rev. Relativity 8 (2005), lrr-2005-12, 113 pages, gr-qc/0505065.
  38. Bejancu A., Farran H.R., Foliations and geometric structures, Mathematics and Its Applications, Vol. 580, Springer, Dordrecht, 2005.
  39. Damour T., Deser S., McCarthy J.G., Nonsymmetric gravity theories: inconsistencies and a cure, Phys. Rev. D 47 (1993), 1541-1556, gr-qc/9207003.
  40. Vacaru S., Einstein gravity in almost Kähler variables and stability of gravity of nonholonomic distributions and nonsymmetric metrics, arXiv:0806.3808.
  41. Castro C., On Born's deformed reciprocal complex gravitational theory and noncommutative gravity, Phys. Lett. B 668 (2008), 442-446.
  42. Vacaru S., Deformation quantization of almost Kähler models and Lagrange-Finsler spaces, J. Math. Phys. 48 (2007), 123509, 14 pages, arXiv:0707.1519.
  43. Castro C., W-geometry from Fedosov's deformation quantization, J. Geom. Phys. 33 (2000), 173-190, hep-th/9802023.
  44. Vacaru S., Stavrinos P., Gaburov E., Gon ta D., Clifford and Riemann-Finsler structures in geometric mechanics and gravity, Selected Works, Differential Geometry - Dynamical Systems, Monograph 7, Geometry Balkan Press, 2006, http://www.mathem.pub.ro/dgds/mono/va-t.pdf and gr-qc/0508023.
  45. Vacaru S., Spectral functionals, nonholonomic Dirac operators, and noncommutative Ricci flows, arXiv:0806.3814.

Previous article   Next article   Contents of Volume 4 (2008)