1 Laurent Nottale CNRS LUTH, Observatoire de Paris-Meudon luthier/nottale/ Paris, ENS, October 8, EDU-2008

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

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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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8


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


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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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18

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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

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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)

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 »

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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.

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


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


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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

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

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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

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)

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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

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

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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

Documents

1 Laurent Nottale CNRS LUTH, Observatoire de Paris-Meudon luthier/nottale/ Paris, ENS, October 8, EDU-2008