Gauge Theory of Gravitation

Transcription

Gauge Theory of Gravitation
A Solution to the Dark Matter, Dark Energy Problems
29 June 2012
Yi Yanga, Wai Bong Yeungb
aThe
Ohio State University
aIndiana University
bIoP Academia Sinica
References
  arXiv:1111.7062 (submitted to PLB) by Y. Yang & W.B. Yeung
Quadratic Gravitational Lagrangian with Torsion Can Give
Possible Explanations of the Form of Galactic Rotation Curves,
of the Amount of Intergalactic Lensings, and of the
Accelerating Expansion of the Universe
  arXiv:1205.2690 by Y. Yang & W.B. Yeung
Spontaneously Broken Erlangen Program Offers a Bridge
Between the Einstein and the Yang-Mills Theories
Yi Yang and Wai Bong Yeung 2012 June 29 Academia Sinica
2 / 30
Motivation
  General Relativity has tremendous success since 1916
  The deflection of starlight by the Sun
  Perihelion precession of Mercury
  Gravitational lensing
  Black hole
  …
 The foundation of Modern Cosmology
 But…
In the past decades, we are facing some
serious problems…
Most people believe…
3 / 30
The Dark Matter Problem
  Rotation curve of galaxy
  The observations can’t be described by Newtonian gravity with
luminous matter alone
Something extra must
Contribute to the
rotation velocities
Newtonian contribution
  v2 = GM/r
M: luminous matter
Postulate : M consists of
two components
Mlumi. And Mdark
4 / 30
The Dark Matter Problem
  Recently some very accurate data pin down the properties of Mdark
(if they exist) :
  Mdark must be more or less uniformly distributed, like a pitless peach
 This is in contrast of what is predicted for conventional cold
collision-less dark matter
5 / 30
The Dark Matter Problem – cont.
  Recently some very accurate data pin down the properties of Mdark
(if they exist) :
  Bullet Cluster : there are Two components of matter
 Rule out the MOND
6 / 30
An Empirical Formula
  Rotation curves show high regularity : some basic physical
principles waiting to be discovered
  In 2000, and later in 2008, P. Salucci and A. Burkert concluded a
formula to describe the Rotation Curves from 15,000 velocity
measurements of 1000 Rotation Curves
(Astro, Jour., 537:L9–L12, 2000 July 1 : DARK MATTER SCALING RELATIONS)
Total :
Regular matter :
Dark matter :
A sensible theory of dark matter should
be able to explain this empirical formula
7 / 30
The Dark Matter Problem – cont.
  Intergalactic gravitational lensing
  With GR, the angle bent is 4GM/r0
 It is an order of magnitude too small when the mass is taken as
the luminous mass
8 / 30
The Dark Energy Problem
  Observations since the 1990s indicate that the universe is
expanding at an accelerating rate (2011 Nobel Prize in physics)
9 / 30
Fundamental Interactions
  There are two different principles to describe the fundamental
forces in Nature
  Gauge theory
 
 Exchanging a gauge boson as the force carrier
  Electromagnetic : Photon
  Weak : W, Z bosons
  Strong : Gluons
Geometry effect
 Curved space-time
  Gravity
  A million-dollar-worth question :
Is there a way to describe all
fundamental interactions by
“one” principle
 Gauge Gravity
10 / 30
Einstein’s General Relativity
  Starts with the Riemann curvature tensor
Rλ σµν = ∂µ Γλσν − ∂ν Γλσµ + Γλκµ Γκσν − Γλκν Γκσµ
  Breaks the connections into their symmetric parts and anti-
symmetric parts
Γλσν = Γλ(σν) + Γλ[σν]
sets deliberately the anti-symmetric part (torsion) = 0
  Introduces a metric gµν into the spacetime
ds2 = gµν dxµ dxν
and requires that gµν be constant in parallel transportation, then
Γλ(σν)= Christoffel symbol
  Regards gµυ as the basic variables. Constructs the Einstein-Hilbert
Lagrangian
where
LEH =
√
gR
R = g σν Rλ σλν
11 / 30
Einstein’s General Relativity
  Due to the Birkhoff-Jebson theorm, Schwarzschild solution
(which gives inverse square force at long distance) is the only
spherical symmetry static solution, so we have :
  The relation
GM
r
and for a spiral galactic disk v 2 ∝ (I0 K0 − I1 K1 )
v2 =
  Light deflected by 4GM/r0
  Deceleration in the late time evolution of the Universe
  All going against General Relativity
12 / 30
What are we trying to do…
  Take the full connections both the symmetric parts and the
torsion parts, and treat both the connections and the metric a
independent variables
  Choose the Lagrangian recommended by Hermann Weyl as early
as 1919 (Note that in Weyl’s original notion, the connections are the
Christoffel symbols. And his theory is a theory with higher order derivatives)
L=
√
gRλσµν Rλσµν
  This Lagrangian (not protected by Birkhoff-Jebson theorem)
has two metrics as torsionless solutions for the spherical
symmetric situations :
  Schwarzschild metric
  Thompson-Pirani-Pavelle metric (Hsu and Yeung)
13 / 30
What are we trying to do…
  We postulate that there are two types of matter in Nature:
  Regular matter (m, M, G) : Schwarzschild metric
  Primed matter (m’, M’, G’) : Thompson-Pirani-Pavelle metric
  Other than the difference in the gravitational properties, we
assume that Primed matter has the same Eelectroweak, Strong
interactions as the regular matter
  If we calculate the respective acceleration produced by these
metrics with
then get
and
14 / 30
Rotation Curves
  For a object consisting both regular matter (m) and primed
matter (m’) sticking together, moving under the influence of a
source of regular matter (M) and primed matter (M’)

or
  We have used a fact :
  The regular matter couples to the regular matter
  The primed matter couples to the primed matter.
(This follows from the fact that regular matter and the primed
matter are separately covariantly conserved
)
15 / 30
Rotation Curves From Our Theory
  Because the relative rotation speed of the primed matter is
larger in compared with that of the regular matter, we expect
  What condensed mostly into stars is the regular matter
  The primed matter stays mostly in the halo.
Spiral galaxy
(regular matter)
Halo (primed matter)
  If the primed matter is uniformly distributing with density ρ’,
then M’ inside r will be (4π/3)ρ’r3
This is exactly the same
as the empirical formula
from Salucci et al.
16 / 30
Rotation Curves From Our Theory
  The new equation for rotation velocities agrees with observation
extremely well
For the Newtonian Gravity term
(aka Keplerian decline)
Ia, Ka are the modified Bessel functions
Contribution from
Primed Gravity
 Flatten the rotation curve
17 / 30
Rotation Curves – cont.
  Good agreement in our galaxy (The Milky Way), too
  Universal fit results :
  G* ~ 10-2
  m’/m ~ 10-9
 Don’t affect what we are seeing in
the solar system
18 / 30
Gravitational Lensing by Primed Matter
  It is easy to calculate the light deflection Δφ produced by the
Primed Matter in a galaxy cluster of size R and when the distance
of closest approach r0, as
3
3
�− 12 − 52
2
√
∆ϕ =
(R − r0 ) G
R
2
  Putting the data from Abell 1689, R = 300 kpc and (R – r0) = 30 kpc
and G* = 10-2 kpc-2 from the fits
 The angle of deflection is 4 x 10-3
 Many times larger than the prediction from GR
19 / 30
Accelerating Expanding Universe
  One more nice feature of the quadratic Lagrangian :
Having a cosmological solution with primodial torsion components
can be interpreted as accelerated expanding of the Universe
Torsion = ξ
 The Universe is twisted by the torsion
 Looks like a cosmological constant ξ from
the metrical point of view
  Our torsion selects the spatially flat metric
(κ = 0) as the only accompanying metric
 Agree with WMAP results
  This cosmic solution corresponds to
the case of Riemann tensor = 0
which is the pure gauge state from the point
of view of the gauge theory of gravitation
20 / 30
Geometries-Symmetries-Interactions
  Our chosen of quadratic Lagrangian seems to describe very well
the various gravitational phenomena. Is there something deep
inside this Lagrangian?
  Felix Klein
: Geometries classified by Symmetry groups
(The Erlangen Program)
  Einstein
: Geometry describes Interactions
(Gravity)
  Yang and Mills : Symmetries dictation interactions
(Gauge Theories)
 Geometries = Symmetries = Interactions
21 / 30
Our Way of Constructing a Gauge Theory of Gravity
  The most important assumption:
Metric is not a dynamical variable anymore, it is a background of
measuring clock and stick
  With a metric gµν, the corresponding vierbein eaλ satisfy
  On the local flat patch around any point of our spacetime, we
might have the freedom to choose our local coordinate system,
other than the local Minkowskian frame, subjected to the
requirements that the coordinate system won’t change
  The Law of Inertia
  The validity of Causality
  The idea of Parallelness
  The Affine Transformation (GL(4, R)) of Euler will leave the
above physical principles invariant
22 / 30
Our Way of Constructing a Gauge Theory of Gravity
  In other words
will leave physics invariant
 We take GL(4 R) as the characterizing symmetry
group of our spacetime ála Felix Klein
23 / 30
Full Implementation of Yang-Mills Doctrines
  Start with the Lie Algebra of GL(4 R) : 6 anti-symmetric
generators Jab, and 10 symmetric generators Tab following the
commutation relations :
  Define Mab = ½(Tab + Jab)
  Then, we can define the Yang-Mills gauge potential
and the Yang-Mills field strength tensor
24 / 30
  Yang-Mills Lagrangian :
1
1 m
m�
µν
LYM = Tr(Fµν F ) = F nµν F n�
2
2
µ� ν �
ηmm� ηnn�
  The Yang-Mills action, in a background metric gµν will look like
  Change of variables from Amnµ to Γρτµ by
  Miraculously the Yang-Mills field strength tensor can be re-
expressed in the Γ fields in a very single way :
25 / 30
  The Yang-Mills action will become
This is exactly the same as the quadratic Lagrangian that we
have chosen in the explanation of the dark matter, dark energy
problems!
  The GL(4 R) Yang-Mills theory of gravity
predicts the dark matter, dark energy
phenomena
26 / 30
Some Remarks
  According to Feynman :
a)  Have to vary gµν and Γσµν independently
b)  Extremize the action to get classical solutions
 Two equations of motion.
  The classical solutions retain only the local Lorentz symmetry,
 Classical gravitational phenomena is in a state of broken
symmetry : GL(4 R)  SO(3, 1).
( This solves the long standing problem that GL(4 R) has no
finite dimensional spinor representations while we are
seeing particles classified by the Lorentz group. )
27 / 30
Some Remarks – cont.
  In 1974, Prof. C. N. Yang tried to formulate a gauge theory of
gravity based on GL(4 R).
  For some reasons, he ignored the importance of torsion in his
formulation.
  He arrived at an equation by varying the connections, and
then identify the connections with the Christoffel symbol.
  There exists only one equation in his theory (the Yang’s
equation). The equation coming from the variation of gµν is
missing.
  The result is, numerous unphysical solutions (for example,
the Ni’s solution), that could have been eliminated by the
equation obtained by varying gµν, afflict the Yang’s theory.
  The Thompson-Pirani-Pavelle metric which satisfies both
equations was also a trouble.
  This bad feature now becomes a good feature, thanks to the
dark matter phenomena.
28 / 30
Some Remarks – cont.
  We speculate that the Standard Model for fundamental
interactions should be upgraded to include gravitational force as
 U(1) x SU(2) x SU(3) x GL(4 R)
29 / 30
‪Standing On The Shoulders Of Giants‬
Thank you!
30 / 30
Backup
31 / 30
32 / 30
Dark Matter Problem – cont.
  Supersymmetry provides a good dark matter candidate
 Lightest-supersymmetric particle (LSP)
  No hint from collider experiments : Tevatron and LHC
Lower limits for the mass of
the SUSY partners
Observed limits in the
CMSSM (m0, m1/2) plane
m( q~
)=
150
0
00
m( g~) = 1500
500
20
700
~) = 2
m( q
#" = LSP
800
Lint = 4.98 fb-1, s = 7 TeV
~q) =
m(
m1/2 [GeV]
CMS Preliminary
tan($)=10
A 0 = 0 GeV
µ>0
mt = 173.2 GeV
600
~±
LEP2 l
Razor
500
400
m( q~
)=
"±
LEP2 !
100
0
SS Dilepton
200
100
m( g~) = 1000
MT2
300
OS
Di
lepto
m( g~) = 500
n
Jets+MHT
1 Lepton
E's
RG
nt
Multi-Lepton
500
1000
1500
1
on
n-C
No
2000
ge
ver
2500
B
WS
E
No
3000
m0 [GeV]
33 / 30
Dark Matter Problem – cont.
  No evidence from direct search neither : CDMS I, II
34 / 30
Family Problem
  Why are there three families (generations) in Standard Model?
 One of the biggest mysteries in physics world
Between generations,
Particles have different
quantum numbers (flavor)
and masses,
but have identical interactions
Why??
35 / 30
Dark Energy Problem
36 / 30
Gauge Theory of Gravity
  Quick overview Einstein’s General Relativity :
  Einstein-Hilbert Lagrangian :
with R = Rαβ g αβ
LEH =
2√
−gR (first order of R)
κ2
  Hermann Weyl once remarked that the most natural
gravitational Lagrangian should be quadratic in the Riemann
curvature tensor
  Some works try to visualize gauge boson
interactions as geometrical manifestations
in a higher dimensional manifold with our
spacetime as a four dimensional sub-manifold.
Other works try to consider the geometrical
gravitation theory in the form of a local
gauge theory.  Nobody really succeed!
37 / 30
Rotation Curves – cont.
  Compare to the universal formula from P. Salucci and A. Burkert
  Universal formula
  Our formula
x3
a2 x + x3
 Surprisingly we get the same rotation curves from both the
theory and the observations
38 / 30

Gauge Theory of Gravitation

Transcription

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