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Finsler 几何及其相关的修正的色散关系和修正的牛顿引力 Xin Li Institute of High Energy Physics Chinese Academy of Sciences. Outline. Observational evidences Finsler geometry Local symmetry and violation of Lorentz invariance Gravity and large scale structure. I. Observational evidences.

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  1. Finsler几何及其相关的修正的色散关系和修正的牛顿引力Xin LiInstitute of High Energy PhysicsChinese Academy of Sciences

  2. Outline • Observational evidences • Finsler geometry • Local symmetry and violation of Lorentz invariance • Gravity and large scale structure

  3. I. Observational evidences 1. Galactic rotation curves 2. Velocity dispersions of galaxies 3. GZK cutoff in ultra-high energy cosmic ray

  4. 1. Galactic rotation curves In the late 1960s and early 1970s V. Rubin from Carnegie Institution of Washington presented that most stars in spiral galaxies orbit at roughly the same speed.

  5. Rotation curve of a typical spiral galaxy: predicted (A) and observed (B).

  6. 2. Velocity dispersions of galaxies Rubin's pioneering work has stood the test of time. Measurements of velocity curves in spiral galaxies were soon followed up with velocity dispersions of elliptical galaxies. While sometimes appearing with lower mass-to-light ratios, measurements of ellipticals still indicate a relatively high dark matter content.

  7. 3. GZK cutoff in ultra-high energy cosmic ray

  8. HiRes observes the ankle; Has evidence for GZK suppression; Can not claim the second knee.

  9. DIP and DISCREPANCY between AGASA and HiRes DATA(energy calibration by dip)

  10. II.Finsler geometry In 1854 Riemann saw the difference between the quadratic differential form--Riemannian geometry and the general case. The study of the metric which is the Fourth root of a quartic differential form is quite time--consuming and does not throw new light to the problem." Happily, interest in the generalcase was revived in 1918 by Paul Finsler's thesis, written under the direction of Caratheodory.

  11. Finsler structure of M . • with the following properties: • Regularity: F is C on the entire slit tangent bundle TM\ 0 • (ii) Positive homogeneity : F(x,  y)=  F(x,y), for all  >0 • (iii) Strong convexity: the Hessian matrix • Is positive-definite at every point of TM\0

  12. The symmetric Cartan tensor Cartan tensor Aijk=0 if and only if gij has no y-dependence A measurement of deviation from Riemannian Manifold

  13. Euler's theorem on homogenous function gives Where li=yi/F

  14. 1. Chern connection transform like The nonlinear connection Nijon TM\0 where ijk is the formal Christoffel symbols of the second kind

  15. Chern Theorem guarantees the uniqueness of Chern connection. S. S. Chern, Sci. Rep. Nat. Tsing Hua Univ. Ser. A 5, 95 (1948); or Selected Papers, vol. II, 194, Springer 1989. Torsion freeness Almost g-compatibility

  16. Torsion freeness is equivalent to the absence of dyiterms in ij together with the symmetry Almost g-compatibility implies that where

  17. 2.Curvature The curvature 2-forms of Chern connection are The expressionof ijin terms of the natural basis is of the form where R, P and Q are the hh-, hv-, vv-curvature tensors of the Chern connection, respectively.

  18. Very special relativity and Neutrino mass S.R. Coleman and S.L. Glashow, Phys. Lett. B405, 249 (1997). S.R. Coleman and S.L. Glashow, Phys. Rev. D59, 116008 (1999). A perturbative framework of QFT with Violation of the LI A.G. Cohen and S.L. Glashow, Phys. Rev. Lett. 97, 021601 (2006). Exact symmetry group of nature DISIM(2)

  19. III.Local symmetry and violation of Lorentz invariance G.Y.Bogoslovsky, Some physical displays of the space anisotropy relevant to the feasibility of its being detected at a laboratory ,gr-qc/0706.2621. G.W.Gibbons, J. Gomis and C.N.Pope, General Very Special Relativity is Finsler Geometry, hep-th/0707.2174 . Finslerian line element DISIMb(2) symmetry

  20. Randers sapce: a very interesting class of Finsler manifolds. G. Randers, Phys. Rev. 59, 195 (1941). Z.Chang and X.Li, Phys. Lett. B663,103(2008) The Randers metric The action of a free moving particle Canonical momentum pi Euler'stheorem for homogeneous functions guarantees the mass-shell condition

  21. Einstein's postulate of relativity: the law of nature and results of all experiments performed in a given frame of reference are independent of the translation motion of the system as a whole. This means that the Finsler structure F should be invariant undera global transformation of coordinates on the Randers spacetime

  22. Any coordinate transformations should in general take the form If we require that the matrix is the same with the usual one

  23. F=0 presents invariant speed of light and arrow of cosmological time

  24. UHECR threshold anomaly Z.Chang and X. Li, Cosmic ray threshold anomay in Randers space (2008). Head-on collision between a soft photon of energy and a high energy particle From the energy and momentum conservation laws, we have

  25. IV.Gravity and large scale structure The tangent spaces (TxM, Fx) of an arbitrary Finsler manifolds typically not isometric to each other. Given a Berwald space, all its tangent spaces are linearly isometric to a common Minkowski space A Finsler structure F is said to be of Berwald type if the Chern connection coefficients ijk in natural coordinates have no y dependence. A direct proposition on Berwald space is that hv--part of the Chern curvature vanishes identically

  26. Gravitational field equation on Berwald space X. Li and Z. Chang, Toward a Gravitation Theory in Berwald--Finsler Space ,gr-qc/0711.1934.

  27. To get a modified Newton's gravity, we consider a particle moving slowly in a weak stationary gravitational field. Suppose that the metric is close to the locally Minkowskian metric Z. Chang and X. Li, Modified Newton’s gravity in Finsler space as a possible alternative to dark matter hypothesis, astro-ph/ 0806.2184, to be published in Phys. Lett. B A modified Newton's gravity is obtained as the weak field approximation of the Einstein's equation

  28. Limit the metric to be the form a0is the deformation parameter of Finsler geometry The deformation of Finsler space should have cosmological significance. One wishes naturally the deformation parameter relates with the cosmological constant ,

  29. The geometrical factor of the density of baryons In the zero limit of the deformation parameter, familiar results on Riemann geometry are recovered The acceleration a of a particle in spiral galaxiesis

  30. M. Milgrom, The MOND paradigm, astro-ph/0801.3133. M. Milgrom, Astrophys. J. 270, 365 (1983). G. Gentile, MOND and the universal rotation curve: similar phenomenologies, astro-ph/0805.1731 The MOND Universal Rotation Curves

  31. V. Conlusions 1.Special relativity in Finsler space The threshold of the ultra-high energy cosmic ray in Finsler space is consistent with observation 2.General relativity in Finsler space In good agreement with the MOND, and can be used to describe the rotation curves of spiral galaxies without invoking dark matter

  32. Thanks for your attention!

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