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Transcript of 1 Laurent Nottale CNRS LUTH, Observatoire de Paris-Meudon luthier/nottale/ Paris, ENS, October 8,...
![Page 1: 1 Laurent Nottale CNRS LUTH, Observatoire de Paris-Meudon luthier/nottale/ Paris, ENS, October 8, EDU-2008.](https://reader033.fdocuments.in/reader033/viewer/2022061412/56649ebe5503460f94bc8001/html5/thumbnails/1.jpg)
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Laurent NottaleCNRS
LUTH, Observatoire de Paris-Meudon
http://www.luth.obspm.fr/~luthier/nottale/
Paris, ENS, October 8, EDU-2008
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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
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RELATIVITY
COVARIANCE EQUIVALENCE
weak / strong
Action Geodesical
CONSERVATIONNoether
FIRST PRINCIPLES
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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
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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…
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A
A
0
1X
t0 1
1. Continuity + nondifferentiability Scale dependence
0.01 0.11
Continuity + Non-differentiability implies Fractality
when
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Continuity + Non-differentiability implies Fractality
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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
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*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
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Origin
Orientation
Motion
Velocity
AccelerationScale
Resolution
Coordinate system
x
t
δ x
δ t
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FRACTALSFRACTALS
From fractal objectsFrom fractal objects
toto
Fractal space-timesFractal space-times
http://www.luth.obspm.fr/~luthier/nottale/
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Discrete zooms on a Discrete zooms on a fractal curvefractal curve
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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:
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Continuous zoom on a fractal Continuous zoom on a fractal curvecurve
Animation
QuickTime™ et undécompresseur Graphiquessont requis pour visionner cette image.
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Fractal geometry: space of positions and scales
© L. Nottale CNRS Observatoire de Paris-Meudon
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Curves of variable fractal dimension (in space)
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QuickTime™ et undécompresseur Animationsont requis pour visionner cette image.
Animation
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Laws of transformation of the scale variables
From scale invariance to scale covariance
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Dilatation operator (Gell-Mann-Lévy method):
First order scale differential First order scale differential equation:equation:
Taylor expansion:
Solution: fractal of constant dimension + transition:
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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
![Page 23: 1 Laurent Nottale CNRS LUTH, Observatoire de Paris-Meudon luthier/nottale/ Paris, ENS, October 8, EDU-2008.](https://reader033.fdocuments.in/reader033/viewer/2022061412/56649ebe5503460f94bc8001/html5/thumbnails/23.jpg)
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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)-
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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)
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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 »
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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'
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‘Scale dynamics’: scale dependence of the length and of the effective scale-dimension in the case of an harmonic oscillator ‘scale-potential’
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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.
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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)
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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
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Road toward Schrödinger Road toward Schrödinger (1): infinity of geodesics(1): infinity of geodesics
––> generalized « fluid » approach:
Differentiable Non-differentiable
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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:
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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 »)
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Covariant derivative operatorCovariant derivative operatorClassical(differentiable)part
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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…
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Newton
Schrödinger
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Application in Application in astrophysics: astrophysics: gravitational gravitational
structuresstructuresMacroscopic Macroscopic
Schrödinger equationSchrödinger equation
http://www.luth.obspm.fr/~luthier/nottale/
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Three representations
Geodesical (U,V) Generalized Schrödinger (P,)
Euler + continuity (P, V)
New « potential » energy:
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Gauge invariance of gravitationalSchrödinger equation
Gauge transformation of :case ofKepler potential --> dimensionless
One finds invariance under the transformation:
Provided
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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)
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Application to the Application to the formation pf planetary formation pf planetary
systemssystems
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Simulation of trajectorySimulation of trajectory
Kepler central potential GM/rState n = 3, l = m = n-1
Process:
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n=3
Solutions: Kepler potentialSolutions: Kepler potential
Generalized Laguerre polynomials
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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
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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
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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
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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
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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)
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Exoplanets (data 2006)
(P / M*)^(1/3)
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Exoplanets (data 2008, N=301)
(P / M*)^(1/3)
Num
ber
Predicted probability peaks
(main peak cut)
Proba = 5 x10-7
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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
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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 )
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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
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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 !
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Stars:Planetary nebulae
Da Rocha 2000, Da Rocha & Nottale 2003
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Stars:ejection and accretion
SN 1987A, deprojected angle : 41.2 ± 1.0 d° predeicted angle: (l=4, m=2): 40.89 d°
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Applications of scale Applications of scale laws in geosciences:laws in geosciences:
critical and log-periodic critical and log-periodic lawslaws
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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
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Arctic sea ice extent decrease(Mean August)
Confirmation: full melting one year later (2012)
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South California earthquake rate
Log-periodic deceleration from ~1796, g=1.27
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May 2008
SichuanSeism
Date (day, May 2008)
magnitude
rate
Log-periodic
deceleration
of
replicas
Mainearthquake
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Applications in physics Applications in physics and cosmologyand cosmology
Special scale relativity --> value of strong coupling
Scale-dependent vacuum --> value of cosmological constant
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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)
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Comparison between theoretical prediction and
experimental value of alphas(mZ)
Date prediction
prediction
Data: PDG 1992-2006
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Value of the Value of the cosmological constantcosmological constant
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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
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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
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Comparison prediction-observations
Gunn-Tinsley LN, Hubblediagram ofInfraredellipticals
LN, age problem
SNe,WMAP 3yrlensing
SNeI SNe,WMAP1yrlensing
prediction