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!