i. experimental observations
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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 - PowerPoint PPT PresentationTRANSCRIPT
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Finsler Geometry vs.
Dark Matter and Dark Energy Hypothesis
CHANG ZheInstitute of High Energy Physics
Chinese Academy of Sciences
17/11/2009 at ITP
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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
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6. Pioneer anomaly
7. Anomalous increase of the Astronomical Unit
8. Anomalous orbital-energy changes during spacecraft flyby of earth
9 ……
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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.
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Rotation curve of a typical spiral galaxy: predicted (A) and observed (B).
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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.
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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.
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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.
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4. Large scale structure formationObservations suggest that structure formation i
n 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.
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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.
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– The Pioneer anomaly is the observed deviation from pre
dicted trajectories and velocities of various
unmanned spacecraft visiting the outer solar system, mos
t notably Pioneer 10 and Pioneer 11.
– Both Pioneer spacecraft are escaping from the solar sys
tem, and are slowing down under the influence of the S
un's gravity. Upon very close examination, however, the
y are slowing down slightly more than expected.
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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
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The latest planetary ephemerides gave the accurate value of AU
Reports from Krasingky and Brumberg Celest. Mech. Dyn. Astrn. 90, 267 (2004).
Standish Proc. IAU Colloq. 196, 163 (2005)
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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.
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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=0Friedmann equation
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II.Finsler geometryIn 1854 Riemann saw the difference between the q
uadratic 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.
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Mathematical ProblemsLecture delivered before the International
Congress of Mathematicians at Paris in 1900
By 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
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1926, L. Berwald: Berwald connection Torsion f
ree: 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
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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
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The symmetric Cartan tensor
Cartan tensor Aijk=0 if and only if gij has no y-de
pendence
A measurement of deviation from Riemannian M
anifold
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Euler's theorem on homogenous function gives
Where li=yi/F
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1. Chern connection
transform like
The nonlinear connection Nij on TM\0
where ijk is the formal Christoffel symbols of the second kind
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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
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Torsion freeness is equivalent to the absence of dyi terms in i
j
together with the symmetry
Almost g-compatibility implies that
where
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2.CurvatureThe 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.
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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 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
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X. Li and Z. Chang, Toward a Gravitation Theory in Berwald--Finsler Space ,gr-qc/0711.1934.
Gravitational field equation on Berwald space
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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).
To get a modified Newton's gravity, we consider a particle moving slowl
y in a week 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
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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 ,
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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
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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
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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
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• 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=0Friedmann equation
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The Randers-Finsler metric
The Friedmann equation
Let to be the Robertson-Walker type
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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
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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.
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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.
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Thanks for your attention!