1 laurent nottale cnrs luth, observatoire de paris-meudon luthier/nottale/ paris, ens, october 8,...
TRANSCRIPT
1
Laurent NottaleCNRS
LUTH, Observatoire de Paris-Meudon
http://www.luth.obspm.fr/~luthier/nottale/
Paris, ENS, October 8, EDU-2008
2
Scales in naturePlanck scale10 cm-33
10 cm-28
10 cm-16
3 10 cm-13
4 10 cm-11
1 Angstrom
40 microns
1 m
6000 km700000 km1 millard km
1 parsec
10 10
10 20
10 30
10 40
10 50
10 60
1
Grand Unification
accelerators: today's limitelectroweak unification
electron Compton lengthBohr radius
quarks
virus bacteries
human scale
Earth radiusSun radiusSolar System
distances to StarsMilky Way radius10 kpc
1 Mpc100 Mpc
Clusters of galaxiesvery large structuresCosmological scale10 cm28
atoms
3
RELATIVITY
COVARIANCE EQUIVALENCE
weak / strong
Action Geodesical
CONSERVATIONNoether
FIRST PRINCIPLES
4
Giving up the hypothesis of differentiability of
space-time
Explicit dependence of coordinates in terms of scale variables
+ divergence --> (theory : = dX ;experiment : = apparatus resolution)
Generalize relativity of motion ?
Transformations of non-differentiable coordinates ? ….
Theorem
FRACTAL SPACE-TIME
Complete laws of physics by fundamental scale laws
Continuity +SCALE RELATIVITY
5
Principle of scale relativity
Scale covarianceGeneralized principle
of equivalence
Linear scale-laws: “Galilean”self-similarity,
constant fractal dimension,scale invariance
Linear scale-laws : “Lorentzian”varying fractal dimension,
scale covariance,invariant limiting scales
Non-linear scale-laws: general scale-relativity,
scale dynamics,gauge fields
Constrain the new scale laws…
6
A
A
0
1X
t0 1
1. Continuity + nondifferentiability Scale dependence
0.01 0.11
Continuity + Non-differentiability implies Fractality
when
7
Continuity + Non-differentiability implies Fractality
8
Continuity + Non-differentiability implies Fractality
divergence
Lebesgue theorem (1903):« a curve of finite length is almost everywhere differentiable »
Since F is continuous and no where or almost no where differentiable
i.e., F is a fractal curve
2. Continuity + nondifferentiability
when
9
*Re-definition of space-time resolution intervals as characterizing the state of scale of the coordinate system
*Relative character of the « resolutions » (scale-variables):only scale ratios do have a physical meaning, never an absolute scale
*Principle of scale relativity: « the fundamental laws of nature are valid in any coordinate system, whatever its state of scale »
*Principle of scale covariance: the equations of physics keep their form (the simplest possible)* in the scale transformations
of the coordinate system
Weak: same form under generalized transformations
Strong: Galilean form (vacuum, inertial motion)
Principle of relativity of scales
10
Origin
Orientation
Motion
Velocity
AccelerationScale
Resolution
Coordinate system
x
t
δ x
δ t
11
FRACTALSFRACTALS
From fractal objectsFrom fractal objects
toto
Fractal space-timesFractal space-times
http://www.luth.obspm.fr/~luthier/nottale/
12
Discrete zooms on a Discrete zooms on a fractal curvefractal curve
13
von Koch von Koch curvecurve
F0
F1
F2
F3
F4
F∞
L0
L1 = L0 (p/q)
L2 = L0 (p/q)2
L3 = L0 (p/q)3
L4 = L0 (p/q)4
L∞ = L0 (p/q)∞
Generator:p = 4q = 3
Fractal dimension:
14
Continuous zoom on a fractal Continuous zoom on a fractal curvecurve
Animation
QuickTime™ et undécompresseur Graphiquessont requis pour visionner cette image.
15
Fractal geometry: space of positions and scales
© L. Nottale CNRS Observatoire de Paris-Meudon
16
Curves of variable fractal dimension (in space)
17
18
19
QuickTime™ et undécompresseur Animationsont requis pour visionner cette image.
Animation
20
Laws of transformation of the scale variables
From scale invariance to scale covariance
21
Dilatation operator (Gell-Mann-Lévy method):
First order scale differential First order scale differential equation:equation:
Taylor expansion:
Solution: fractal of constant dimension + transition:
22
ln L
ln ε
transitionfractal
scale -independent
ln ε
transitionfractal
delta
variation of the length variation of the scale dimension
"scale inertia"scale -independent
Case of « scale-inertial » laws (which are solutions of a first order
scale differential equation in scale space).
Dependence on scale of the length (=fractal coordinate)Dependence on scale of the length (=fractal coordinate) and of the effective fractal dimension and of the effective fractal dimension
= DF - DT
23
Asymptotic behavior:
Scale transformation:
Law of composition of dilatations:
Result: mathematical structure of a Galileo group ––>
Galileo scale transformation Galileo scale transformation groupgroup
-comes under the principle of relativity (of scales)-
24
ln L
ln ε
transition
fractal
ln ε
transitionfractal
delta special scale-relativity
Planck scale
scaleindependent
scaleindependentPlanck scale
variation of the scale dimensionvariation of the length
(Simplified case : )
Scale dependence of the length and of the Scale dependence of the length and of the effective scale dimension in special scale-effective scale dimension in special scale-
relativity (log-Lorentzian laws of scale relativity (log-Lorentzian laws of scale transformations)transformations)
25
Scale dynamics
Scale laws that are solutions of second order partial differential equations in the scale space
Least action principle in scale space ––> Euler Lagrange scale equations in terms of the « djinn »
Resolution identified as « scale velocity »:
Djinn (variable scale dimension) identified with « scale time »
26
ln L
ln ε
transitionfractal
ln ε
transitionfractal
delta constant "scale-force"
variation of the scale dimension
scaleindependent
scaleindependent
variation of the length
(asymptotic)
'Scale dynamics': scale dependence of the length and of the effective scale-dimension in the case of a constant 'scale-force'
27
‘Scale dynamics’: scale dependence of the length and of the effective scale-dimension in the case of an harmonic oscillator ‘scale-potential’
28
Scale dependence of the length and of the scale dimension in the case of a log-periodic behavior (discrete scale invariance) including a fractal / nonfractal transition.
29
Foundation of Foundation of quantum quantum
mechanicsmechanicsEffets on the motion equationsEffets on the motion equations
of the of the
fractal structures internal to geodesicsfractal structures internal to geodesics
http://www.luth.obspm.fr/~luthier/nottale/
Cf: Nottale Fractal Space-Time World Scientific (1993); Célérier Nottale J. Phys. A 37, 931 (2004); 39, 12565 (2006); Nottale Célérier J. Phys. A 40, 14471 (2007)
30
Fractality Discrete symmetry breaking (dt)
Infinity ofgeodesics
Fractalfluctuations
Two-valuedness (+,-)
Fluid-likedescription
Second order termin differential equations
Complex numbers
Complex covariant derivative
NON-NON-DIFFERENTIABILITYDIFFERENTIABILITY
31
Road toward Schrödinger Road toward Schrödinger (1): infinity of geodesics(1): infinity of geodesics
––> generalized « fluid » approach:
Differentiable Non-differentiable
32
Road toward Schrödinger (2): Road toward Schrödinger (2): ‘differentiable part’ and ‘fractal ‘differentiable part’ and ‘fractal
part’part’Minimal scale law (in terms of the space resolution):
Differential version (in terms of the time resolution):
Case of the critical fractal dimension DF = 2:
Stochastic variable:
33
Road toward Schrödinger (3): Road toward Schrödinger (3): non-differentiability ––> complex non-differentiability ––> complex
numbersnumbersStandard definition of derivative
DOES NOT EXIST ANY LONGER ––> new definition
TWO definitions instead of one: they transform one in another by the reflection (dt <––> -dt )
f(t,dt) = fractal fonction (equivalence class, cf LN93)Explicit fonction of dt = scale variable (generalized « resolution »)
34
Covariant derivative operatorCovariant derivative operatorClassical(differentiable)part
35
Covariant derivative operator
Fundamental equation of dynamics
Change of variables (S = complex action) and integration
Generalized Schrödinger equation
FRACTAL SPACE-TIME–>QUANTUM FRACTAL SPACE-TIME–>QUANTUM MECHANICSMECHANICS
Ref: LN, 93-04, Célérier & Nottale 04-07. See also works by: Ord, El Naschie, Hermann, Pissondes, Dubois, Jumarie, Cresson, Ben Adda, Agop, et al…
36
Newton
Schrödinger
37
Application in Application in astrophysics: astrophysics: gravitational gravitational
structuresstructuresMacroscopic Macroscopic
Schrödinger equationSchrödinger equation
http://www.luth.obspm.fr/~luthier/nottale/
38
Three representations
Geodesical (U,V) Generalized Schrödinger (P,)
Euler + continuity (P, V)
New « potential » energy:
39
Gauge invariance of gravitationalSchrödinger equation
Gauge transformation of :case ofKepler potential --> dimensionless
One finds invariance under the transformation:
Provided
40
n=0 n=1
n=2(2,0,0)
n=2(1,1,0)
E = (3+2n) mD
Hermite polynomials
Solutions: 3D harmonic oscillator potential 3D (constant Solutions: 3D harmonic oscillator potential 3D (constant density)density)
41
Application to the Application to the formation pf planetary formation pf planetary
systemssystems
42
Simulation of trajectorySimulation of trajectory
Kepler central potential GM/rState n = 3, l = m = n-1
Process:
43
n=3
Solutions: Kepler potentialSolutions: Kepler potential
Generalized Laguerre polynomials
44
Solar System :Solar System : inner and outer systems inner and outer systems
SI
J
S
U
N
P
m VT
M HunC
HHil
1
4
9
16
25
36
rank n101 2 3 4 5 6 7 8 9
√a (obs.)
7 49
1
2
3
4
5
6
SE
N
Ref: LN 1993, Fractal space-time and microphysics (World Scientific) Chap. 7.2
New predictions
(at that time)0.043 UA/Msol 0.17 UA/Msol
55 UA
45
Outer solar system:Outer solar system:Kuiper belt (SKBOs)Kuiper belt (SKBOs)
60 70 80 90 100 110 120 130
2
4
6
8
10
Semi-major axis (A.U.)
SKBO
7 8 9 10
10 20 30 40 500
2 3 4 65Rank n
1
Ref: Da Rocha Nottale 03
46
Outer Solar System:Outer Solar System:Kuiper belt (SKBOs)Kuiper belt (SKBOs)
60 70 80 90 100 110 120 130
2
4
6
8
10
Semi-major axis (A.U.)
SKBO
7 8 9 10
10 20 30 40 500
2 3 4 65Rank n
1
Ref: Da Rocha Nottale 03
2003 UB 313 (« Eris »)
Validation of predicted probability peak at 55 AU
47
New New planet:Sednaplanet:Sedna
2001
FP
185
Sed
na 2
003
VB
12
( a / 57 UA )1/2
SK
BO
s
nex=7
PredictePredicted,AUd,AU (57)(57) 228228 513513 912912 142142
5520520522
ObserveObservedd 5757 227227 509509
Num
ber
48
Solar System: Sun, solar Solar System: Sun, solar cyclecycle
If the Sun had kept its initial rotation: would then be the Kepler period,
But, like all stars of solar-type, the Sun has been subjected to an important loss of angular momentum since its formation (cf. Schatzman & Praderie, The Stars, Springer)
Wave function:
Fundamental period:
On the surface of the Sun:
(Pecker Schatzman)
Result: Observed period:11 ans
Ref: LN, Proceedings of CASYS’03, AIP Conf. Proc. 718, 68 (2004)
(equator)
49
Exoplanets (data 2006)
(P / M*)^(1/3)
50
Exoplanets (data 2008, N=301)
(P / M*)^(1/3)
Num
ber
Predicted probability peaks
(main peak cut)
Proba = 5 x10-7
51
Exoplanets (data 2008, N=301)Main peak
Predicted (1993) fundamental level, 0.043 AU/ Msol
mer
cury
Ven
us
Ear
th
Mar
s
Cer
es
Hyg
eia
52
Extrasolar planetary system:PSR B1257+12
25 2624 66 67 98 9997
10 20 30 40 50 60 70 80 90 1000 110
Period (days)
days days days
1 2 3 4 5 6 7 8
A B C
Refs: Nottale 96, 98, Da Rocha & Nottale 03
Data:Wolszczan 94, 00
Mpsr =1.4 ± 0.1 Msol --> w = (2.96 ± 0.07) x 144 km/s, i.e. 432 km/s = Keplerian velocity for Rsol
Proba < 10-5 of obtaining such an agreement by chance
Prediction of other orbits: P1=0.322 j, P2=1.958 j, P3=5.96 j
Residuals in Wolszczan’s data 00: P = 2.2 j (2.7 )
53
Comparison to the inner Solar System
m V T M
Distance to the star, normalized by its mass (MPSR=1.5 Msol). n^2 law
54
New comparison to the TSR prediction (improved observational data, Wolszczan et al 2003)
A B C
Base: planet C : aC = 68, nC = 8
Planet A: (aA)pred = 27.5 <--> (aA)obs = 27.503 ± 0.002
(nA)pred = 5 <--> (nA)obs = 5.00028 ± 0.00020
Planet B: (aB)pred = 52.5 <--> (aB)obs = 52.4563 ± 0.0001
(nB)pred = 7 <--> (nB)obs = 6.997 ± 0.00001
nA/nA = 5 x 10-5 Improvement by a factor 12 !
55
Stars:Planetary nebulae
Da Rocha 2000, Da Rocha & Nottale 2003
56
Stars:ejection and accretion
SN 1987A, deprojected angle : 41.2 ± 1.0 d° predeicted angle: (l=4, m=2): 40.89 d°
57
Applications of scale Applications of scale laws in geosciences:laws in geosciences:
critical and log-periodic critical and log-periodic lawslaws
58
Arctic sea ice extent decrease
Tc = 2012 --> free from ice in 2011 ! (possibly 2010: expected 1 M km2
(Minimum 15 september of each year)
Critical power lawy0-a (T-Tc)-g
2007 and 2008 values predicted before observation(Nottale 2007)
Constant rate
59
Arctic sea ice extent decrease(Mean August)
Confirmation: full melting one year later (2012)
60
South California earthquake rate
Log-periodic deceleration from ~1796, g=1.27
61
May 2008
SichuanSeism
Date (day, May 2008)
magnitude
rate
Log-periodic
deceleration
of
replicas
Mainearthquake
62
Applications in physics Applications in physics and cosmologyand cosmology
Special scale relativity --> value of strong coupling
Scale-dependent vacuum --> value of cosmological constant
63
Comparison to experimental data + extrapolation by renormalization group
10 1 10-3 3
106
109
1012
1018
1027
Energy (GeV)
10
20
30
40
50
4π 2
eWZt GUT
e
0 10 20 30 40 50
l
α1
α0
α2
α3
αg
∞-1
-1
-1
-1
-1
λln ( / r )
C ( )λ
QCD
p
r0
« Bare » (infinite energy) effective electromagnetic inverse coupling
Grand unification chromodynamics and gravitational inverse couplings
Mass-coupling relations(from scale-relativisticgauge theory)
New:E = 3.2 1020 eV
Electroweakunificationscale
Predicted strongcoupling at Z scale0.1173(4)
64
Comparison between theoretical prediction and
experimental value of alphas(mZ)
Date prediction
prediction
Data: PDG 1992-2006
65
Value of the Value of the cosmological constantcosmological constant
66
0 10 20 30 40 50 60
-140
-120
-100
-80
-60
-40
-20
0q ν
r
r
-4
-6
log (r / l )pl p L
Λ
Vac
uum
ene
rgy
dens
ity
Nottale L. 1993, Fractal Space-Time and Microphysics (World Scientific)
Nottale L., 2003, Chaos Solitons and Fractals, 16, 539. "Scale-relativistic cosmology" http://www.luth.obspm.fr/~luthier/nottale/NewCosUniv.pdf
5.3 x 10-3 eVe ?
Cosmological constant and vacuum energy density
67
Cosmological constant and vacuum energy density.
Value of r0 ? Conjecture: quark-hadron + electron-electron transition during primordial universe *Largest interquark distance: ––> Compton length of effective mass of quarks in pion:
*QCD scale for 6 quarks (extrapolation):
*Classical radius of the electron–––> e-e cross section re
2
–––> Result:
= 1.362 10-56 cm-2
h2= 0.38874(12)
H0=71 ± 3 km/s.Mpc, = 0.73 ± 0.04 (Wmap…)
Predicted (LN 93): Observed:
h2= 0.40 ± 0.03
68
Comparison prediction-observations
Gunn-Tinsley LN, Hubblediagram ofInfraredellipticals
LN, age problem
SNe,WMAP 3yrlensing
SNeI SNe,WMAP1yrlensing
prediction