|
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
- 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).
- Einstein A., A generalization of the relativistic theory
of gravitation, Ann. of Math. (2) 46 (1945), 578-584.
- Eisenhart L.P., Generalized Riemann spaces, Proc.
Nat. Acad. Sci. USA 37 (1951), 311-314.
- Eisenhart L.P., Generalized Riemann spaces, II, Proc.
Nat. Acad. Sci. USA 38 (1952), 505-508.
- Moffat J.W., New theory of gravity, Phys. Rev. D 19 (1979), 3554-3558.
- Moffat J.W., A new nonsymmetric gravitational theory, Phys.
Lett. B 355 (1995), 447-452, gr-qc/9411006.
- 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.
- Moffat J.W., Nonsymmetric gravitational theory, J. Math.
Phys. 36 (1995), 3722-3232, Erratum, J. Math.
Phys. 36 (1995), 7128.
- Moffat J.W., Noncommutative quantum gravity, Phys. Lett.
B 491 (2000), 345-352, hep-th/0007181.
- 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.
- Prokopec T., Valkenburg W., The cosmology of the
nonsymmetric theory of gravitation, Phys. Lett. B 636 (2006), 1-4, astro-ph/0503289.
- Moffat J.W., Toth V.T., Testing modified gravity with
globular cluster veloscity dispersions, Astrophys. J. 680 (2008), 1158-1161,
arXiv:0708.1935.
- 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.
- Atanasiu Gh., Hashiguchi M., Miron R., Supergeneralized
Finsler spaces, Rep. Fac. Sci. Kagoshima Univ. Math. Phys. Chem.
(1985), no. 18, 19-34.
- 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).
- Vacaru S., Nonholonomic Ricci flows. I. Riemann metrics and
Lagrange-Finsler geometry, math.DG/0612162.
- Vacaru S., Nonholonomic Ricci flows. II. Evolution
equations and dynamics, J. Math. Phys. 49 (2008), 043504, 27 pages, math.DG/0702598.
- Vacaru S., Ricci flows and solitonic pp-waves, Internat. J.
Modern Phys. A 21 (2006), 4899-4912, hep-th/0602063.
- 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.
- 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.
- Vacaru S., Nonholonomic Ricci flows. IV. Geometric methods,
exact solutions and gravity, arXiv:0705.0728.
- Vacaru S., Nonholonomic Ricci flows. V. Parametric
deformations of solitonic pp-waves and Schwarzschild solutions, arXiv:0705.0729.
- 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.
- Vacaru S., Finsler and Lagrange geometries in Einstein and
string gravity, Int. J. Geom. Methods Mod. Phys. 5 (2008), 473-511, arXiv:0801.4958.
- Miron R., Anastasiei M., The geometry of Lagrange spaces:
theory and applications, Fundamental Theories of Physics, Vol. 59, Kluwer Academic Publishers, Dordrecht, 1994.
- Vacaru S., Spinor structures and nonlinear connections in
vector bundles, generalized Lagrange and Finsler spaces, J. Math. Phys.
37 (1996), 508-524.
- 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.
- Vacaru S., Clifford-Finsler algebroids and nonholonomic
Einstein-Dirac structures, J. Math. Phys. 47 (2006), 093504, 20 pages, hep-th/0501217.
- Vacaru S., Gauge and Einstein gravity from non-Abelian gauge
models on noncommutative spaces, Phys. Lett. B 498 (2001), 74-82, hep-th/0009163.
- Vacaru S., Exact solutions with noncommutative symmetries in
Einstein and gauge gravity, J. Math. Phys. 46 (2005), 042503, 47 pages, gr-qc/0307103.
- Vacaru S., Deformation quantization of nonholonomic almost Kähler models and Einstein gravity,
Phys. Lett. A 372 (2008), 2949-2955, arXiv:0707.1667.
- 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.
- 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.
- 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.
- Légaré J., Moffat J.W., Field equations and
conservation laws in nonsymmetric gravitational theory, Gen. Relativity Gravitation
27 (1995), 761-775, gr-qc/9412009.
- 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.
- Barcelo C., Liberaty S., Visser M., Analogue gravity,
Living Rev. Relativity 8 (2005), lrr-2005-12, 113 pages, gr-qc/0505065.
- Bejancu A., Farran H.R., Foliations and geometric
structures, Mathematics and Its Applications, Vol. 580, Springer, Dordrecht, 2005.
- Damour T., Deser S., McCarthy J.G., Nonsymmetric gravity
theories: inconsistencies and a cure, Phys. Rev. D 47 (1993), 1541-1556, gr-qc/9207003.
- Vacaru S., Einstein gravity in almost Kähler
variables and stability of gravity of nonholonomic distributions and
nonsymmetric metrics, arXiv:0806.3808.
- Castro C., On Born's deformed reciprocal complex
gravitational theory and noncommutative gravity, Phys.
Lett. B 668 (2008), 442-446.
- Vacaru S., Deformation quantization of almost Kähler
models and Lagrange-Finsler spaces, J. Math. Phys. 48 (2007), 123509, 14 pages, arXiv:0707.1519.
- Castro C., W-geometry from Fedosov's deformation quantization, J. Geom. Phys. 33 (2000), 173-190,
hep-th/9802023.
- 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.
- Vacaru S., Spectral functionals, nonholonomic Dirac
operators, and noncommutative Ricci flows, arXiv:0806.3814.
|
|