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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 LiInstitute 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
1. Galactic rotation curves
2. Velocity dispersions of galaxies
3. GZK cutoff in ultra-high energy cosmic
ray
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.
Rotation curve of a typical spiral galaxy: predicted (A) and observed (B).
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.
3. GZK cutoff in ultra-high energy cosmic ray
HiRes observes the ankle;
Has evidence for GZK suppression;
Can not claim the second knee.
DIP and DISCREPANCY between AGASA and HiRes DATA(energy calibration by dip)
II.Finsler geometryIn 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.
Finsler structure of M
.
with the following properties:(i) 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
The symmetric Cartan tensor
Cartan tensor Aijk=0 if and only if gij has no y-
dependence
A measurement of deviation from Riemannian
Manifold
Euler's theorem on homogenous function gives
Where li=yi/F
1. Chern connection
transform like
The nonlinear connection Nij on TM\0
where ijk is the formal Christoffel symbols of the second kind
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
Torsion freeness is equivalent to the absence of dyi terms in i
j
together with the symmetry
Almost g-compatibility implies that
where
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.
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)
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
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
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
Any coordinate transformations should in general take the form
If we require that
the matrix is the same with the usual one
F=0 presents invariant speed of light and arrow of cosmological time
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
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 i
jk in natural coordinates have no y dependence. A direct proposition on Berwald space is that hv--part of the Chern curvature vanishes identically
X. Li and Z. Chang, Toward a Gravitation Theory in Berwald--Finsler Space ,gr-qc/0711.1934.
Gravitational field equation on Berwald space
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
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
A modified Newton's gravity is obtained as the weak field
approximation of the Einstein's equation
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 ,
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
M. Milgrom, Astrophys. J. 270, 365 (1983).
G. Gentile, MOND and the universal rotation curve: similar phenomenologies, astro-ph/0805.1731
The MOND
M. Milgrom, The MOND paradigm, astro-ph/0801.3133.
Universal Rotation Curves
V. Conlusions1.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
Thanks for your attention!