Finsler Geometry
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Finsler Geometry vs. Dark Matter and Dark Energy Hypothesis CHANG Zhe Institute of High Energy Physics Chinese Academy of Sciences 17/11/2009 at ITP. I. Experimental observations. 1. Galactic rotation curves 2. Velocity dispersions of galaxies 3. Missing matter in clusters of galaxies

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I. Experimental observations

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Finsler Geometryvs.Dark Matter and Dark Energy HypothesisCHANG ZheInstitute of High Energy PhysicsChinese Academy of Sciences17/11/2009 at ITP

I. Experimental observations

1. Galactic rotation curves

2. Velocity dispersions of galaxies

3. Missing matter in clusters of galaxies

4. Large scale structure formation

5. Accelerated expanding Universe

6. Pioneer anomaly

7. Anomalous increase of the Astronomical Unit

8. Anomalous orbital-energy changes during spacecraft flyby of earth

9 ……

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. Missing matter in clusters of galaxies

X-ray measurements of hot intracluster gas correspond closely to Zwicky's observations of mass-to-light ratios for large clusters of nearly 10 to 1. Many of the experiments of the Chandra X-ray Observatory use this technique to independently determine the mass of clusters.

Strong gravitational lensing as observed by the Hubble Space Telescope in Abell 1689 indicates the presence of dark matter - Enlarge the image to see the lensing arcs.

4. Large scale structure formation

Observations suggest that structure formation in the universe proceeds hierarchically, with the smallest structures collapsing first and followed by galaxies and then clusters of galaxies. As the structures collapse in the evolving universe, they begin to "light up" as the baryonic matter heats up through gravitational contraction and the object approaches hydrostatic pressure balance.

5. Pioneer anomaly

Pioneer 10 (also called Pioneer F) was the first spacecraft to travel through the asteroid belt, which it entered on July 15, 1972, and to make direct observations of Jupiter, which it passed by on December 3, 1973. It was launched from Cape Canaveral Air Force Station's Launch Complex 36A on March 3, 1972 at 01:49:00 UTC. Pioneer 10 is heading in the direction of Aldebaran, located in Taurus. By some definitions, Pioneer 10 has become the first artificial object to leave the solar system. It is the first human-built object to have been set upon a trajectory leading out of the solar system.

  • The Pioneer anomaly is the observed deviation from predicted trajectories and velocities of various unmanned spacecraft visiting the outer solar system, most notably Pioneer 10 and Pioneer 11.

  • Both Pioneer spacecraft are escaping from the solar system, and are slowing down under the influence of the Sun's gravity. Upon very close examination, however, they are slowing down slightly more than expected.

6. Anomalous increase of the Astronomical unit

Transits of Venus across the face of the Sun were for long the best method of measuring the astronomical unit

The latest planetary ephemerides gave the accurate value of AU

Reports from Krasingky and Brumberg

Celest. Mech. Dyn. Astrn. 90, 267 (2004).


Proc. IAU Colloq. 196, 163 (2005)

7. Accelerated expanding Universe

  • Cosmology is the scientific study of the large scale properties of the Universe as a whole.

  • It endeavors to use the scientific method to understand the origin, evolution and ultimate fate of the entire Universe.

  • Cosmology involves the formation of theories or hypotheses about the universe which make specific predictions for phenomena that can be tested with observations.

  • Depending on the outcome of the observations, the theories will need to be abandoned, revised or extended to accommodate the data.

Ricci tensor

By making use of the energy-momentum tensor, the Einstein equation reads

Time-time component:

Space-space components:

The space-time components give 0=0

Friedmann equation

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 general case was revived in 1918 by Paul Finsler's thesis, written under the direction of Caratheodory.

Mathematical ProblemsLecture delivered before the International Congress of Mathematicians at Paris in 1900By Professor David Hilbert

  • 4. Problem of the straight line as the

    shortest distance between two points

  • 23. Further development of the methods

    of the calculus of variations

1926, L. Berwald: Berwald connection Torsion free: yes g-compatibility: no

1934, E. Cartan: Cartan connection

Torsion free: no g-compatibility: yes

1948, S. S. Chern: Chern connection

Torsion free: yes g-compatibility: no

Chern connection differs from that of Berwald's by an À term

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

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 Nijon 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 dyiterms in ij

together with the symmetry

Almost g-compatibility implies that



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.

III.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

Gravitational field equation on Berwald space

X. Li and Z. Chang, Toward a Gravitation Theory in Berwald--Finsler Space ,gr-qc/0711.1934.

To get a modified Newton's gravity, we consider a particle moving slowly in a week 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, Phys. Lett. B668, 453(2008).

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, 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


Universal Rotation Curves

IV.Possible model of accelerated expanding Universe

Z. Chang and X. Li,

Robertson-Walker metric

satisfies the requirements of

homogeneity, isotropy and closure

Modified Friedmann model in Randers-Finsler space of approximate Berwald type as a possible alternative to dark energy hypothesis, Phys. Lett. B676 (2009) 173,

arXiv 0901.1023

  • Ricci tensor

    By making use of the energy-momentum tensor, the Einstein equation reads

    Time-time component:

    Space-space components:

    The space-time components give 0=0

    Friedmann equation

The Randers-Finsler metric

The Friedmann equation

Let to be the Robertson-Walker type

Omitting the O(b2) term and combining with the the space-space component of the field equations, we obtain

One can see clearly that the accelerated expanding universe is guaranteed by the constraint

So that the complete constraint on Randers-Finsler structure to support accelerated expanding universe is

It means that a negative provides an effective repulsive force in the course of universe expanding.

V. Conlusions

A possible unified scenario for large structure

  • Secular increase of astronomical unit

  • Pioneer anomaly

  • Galactic rotation curves

  • Missing matter in clusters of galaxies

  • Large scale structure formation

  • Accelerated expanding Universe

    Finsler geometry may really supply a framework of astronomy and cosmology without invoking dark

    matter and dark energy hypothesis.

Thanks for your attention!

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