2. 1 Laurent Nottale CNRS LUTH, Observatoire de Paris-Meudon http://www.luth.obspm.fr/~luthier/nottale/ Paris, ENS, October 8, EDU-2008

3. 2 Scales in nature

4. 3 RELATIVITY COVARIANCEEQUIVALENCE weak / strong ActionGeodesical CONSERVATION Noether FIRST PRINCIPLES

5. 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

6. 5 Principle of scale relativity Scale covariance Generalized 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…

7. 6 1. Continuity + nondifferentiabilityScale dependence Continuity + Non-differentiability implies Fractality when

8. 7 Continuity + Non-differentiability implies Fractality

9. 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

10. 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

11. 10 Origin Orientation Motion Velocity Acceleration Scale Resolution Coordinate system

12. 11 FRACTALS From fractal objects to to Fractal space-times http://www.luth.obspm.fr/~luthier/nottale/

13. 12 Discrete zooms on a fractal curve

14. 13 von Koch curve F0F0 F1F1 F2F2 F3F3 F4F4 F∞F∞ L0L0 L 1 = L 0 (p/q) L 2 = L 0 (p/q) 2 L 3 = L 0 (p/q) 3 L 4 = L 0 (p/q) 4 L ∞ = L 0 (p/q) ∞ Generator: p = 4 q = 3 Fractal dimension:

15. 14 Continuous zoom on a fractal curve Animation

16. 15 Fractal geometry: space of positions and scales

17. 16 Curves of variable fractal dimension (in space)

18. 17

19. 18

20. 19 Animation

21. 20 Laws of transformation of the scale variables From scale invariance to scale covariance

22. 21 Dilatation operator (Gell-Mann-Lévy method): First order scale differential equation: Taylor expansion: Solution: fractal of constant dimension + transition:

23. 22 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) and of the effective fractal dimension = D F - D T

24. 23 Asymptotic behavior: Scale transformation: Law of composition of dilatations: Result: mathematical structure of a Galileo group ––> Galileo scale transformation group -comes under the principle of relativity (of scales)-

25. 24 (Simplified case : ) Scale dependence of the length and of the effective scale dimension in special scale-relativity (log- Lorentzian laws of scale transformations)

26. 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 »

27. 26 (asymptotic) 'Scale dynamics': scale dependence of the length and of the effective scale-dimension in the case of a constant 'scale-force'

28. 27 ‘Scale dynamics’: scale dependence of the length and of the effective scale-dimension in the case of an harmonic oscillator ‘scale-potential’

29. 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.

30. 29 Foundation of quantum mechanics Effets on the motion equations of the of the fractal 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)

31. 30 FractalityDiscrete symmetry breaking (dt) Infinity of geodesics Fractal fluctuations Two-valuedness (+,-) Fluid-like description Second order term in differential equations Complex numbers Complex covariant derivative NON-DIFFERENTIABILITY

32. 31 Road toward Schrödinger (1): infinity of geodesics ––> generalized « fluid » approach: DifferentiableNon-differentiable

33. 32 Road toward Schrödinger (2): ‘differentiable part’ and ‘fractal part’ Minimal scale law (in terms of the space resolution): Differential version (in terms of the time resolution): Case of the critical fractal dimension D F = 2: Stochastic variable:

34. 33 Road toward Schrödinger (3): non-differentiability ––> complex numbers Standard 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 »)

35. 34 Covariant derivative operator Classical (differentiable) part

36. 35 Covariant derivative operator Fundamental equation of dynamics Change of variables (S = complex action) and integration Generalized Schrödinger equation FRACTAL SPACE-TIME–>QUANTUM MECHANICS 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…

37. 36 Newton Schrödinger

38. 37 Application in astrophysics: gravitational structures Macroscopic Schrödinger equation http://www.luth.obspm.fr/~luthier/nottale/

39. 38 Three representations Geodesical (U,V) Generalized Schrödinger (P,  ) Euler + continuity (P, V) New « potential » energy:

40. 39 Gauge invariance of gravitational Schrödinger equation Gauge transformation of  : case of Kepler potential -->  dimensionless One finds invariance under the transformation: Provided

41. 40 n=0n=1 n=2 (2,0,0) n=2 (1,1,0) E = (3+2n) mD  Hermite polynomials Solutions: 3D harmonic oscillator potential 3D (constant density)

42. 41 Application to the formation pf planetary systems

43. 42 Simulation of trajectory Kepler central potential GM/r State n = 3, l = m = n-1 Process:

44. 43 n=3 Solutions: Kepler potential Generalized Laguerre polynomials

45. 44 Solar System : inner and outer systems Ref: LN 1993, Fractal space-time and microphysics (World Scientific) Chap. 7.2 New predictions (at that time) 0.043 UA/M sol 0.17 UA/M sol 55 UA

46. 45 Outer solar system: Kuiper belt (SKBOs) Ref: Da Rocha Nottale 03

47. 46 Outer Solar System: Kuiper belt (SKBOs) Ref: Da Rocha Nottale 03 2003 UB 313 (« Eris ») Validation of predicted probability peak at 55 AU

48. 47 New planet:Sedna 2001 FP 185 Sedna 2003 VB 12 ( a / 57 UA ) 1/2 SKBOs n ex =7 Predicted,AU (57) (57) 228 228 513 513 912 91214252052 Observe d 57 57 227 227 509 509 Number

49. 48 Solar System: Sun, solar cycle 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)

50. 49 Exoplanets (data 2006) (P / M * )^(1/3)

51. 50 Exoplanets (data 2008, N=301) (P / M * )^(1/3) Number Predicted probability peaks (main peak cut) Proba = 5 x10 -7

52. 51 Exoplanets (data 2008, N=301) Main peak Predicted (1993) fundamental level, 0.043 AU/ M sol mercury Venus Earth Mars Ceres Hygeia

53. 52 Extrasolar planetary system: PSR B1257+12 Refs: Nottale 96, 98, Da Rocha & Nottale 03 Data: Wolszczan 94, 00 M psr =1.4 ± 0.1 M sol --> w = (2.96 ± 0.07) x 144 km/s, i.e. 432 km/s = Keplerian velocity for R sol Proba < 10 -5 of obtaining such an agreement by chance Prediction of other orbits: P 1 =0.322 j, P 2 =1.958 j, P 3 =5.96 j Residuals in Wolszczan’s data 00: P = 2.2 j (2.7  )

54. 53 Comparison to the inner Solar System mVTM Distance to the star, normalized by its mass (M PSR =1.5 M sol ). n^2 law

55. 54 New comparison to the TSR prediction (improved observational data, Wolszczan et al 2003) ABC Base: planet C : a C = 68, n C = 8 Planet A: (a A ) pred = 27.5 (a A ) obs = 27.503 ± 0.002 (n A ) pred = 5 (n A ) obs = 5.00028 ± 0.00020 Planet B: (a B ) pred = 52.5 (a B ) obs = 52.4563 ± 0.0001 (n B ) pred = 7 (n B ) obs = 6.997 ± 0.00001  n A /n A = 5 x 10 -5 Improvement by a factor 12 !

56. 55 Stars: Planetary nebulae Da Rocha 2000, Da Rocha & Nottale 2003

57. 56 Stars: ejection and accretion SN 1987A, deprojected angle : 41.2 ± 1.0 d° predeicted angle: (l=4, m=2): 40.89 d°

58. 57 Applications of scale laws in geosciences: critical and log-periodic laws

59. 58 Arctic sea ice extent decrease T c = 2012 --> free from ice in 2011 ! (possibly 2010: expected 1 M km 2 (Minimum 15 september of each year) Critical power law y 0 -a (T-T c ) -g 2007 and 2008 values predicted before observation (Nottale 2007) Constant rate

60. 59 Arctic sea ice extent decrease (Mean August) Confirmation: full melting one year later (2012)

61. 60 South California earthquake rate Log-periodic deceleration from ~1796, g=1.27

62. 61 May 2008 Sichuan Seism Date (day, May 2008) magnitude rate Log-periodic deceleration of replicas Main earthquake

63. 62 Applications in physics and cosmology Special scale relativity --> value of strong coupling Scale-dependent vacuum --> value of cosmological constant

64. 63 Comparison to experimental data + extrapolation by renormalization group « Bare » (infinite energy) effective electromagnetic inverse coupling Grand unification chromodynamics and gravitational inverse couplings Mass-coupling relations (from scale-relativistic gauge theory) New:E = 3.2 10 20 eV Electroweak unification scale Predicted strong coupling at Z scale 0.1173(4)

65. 64 Comparison between theoretical prediction and experimental value of alpha s (m Z ) Date prediction prediction Data: PDG 1992-2006

66. 65 Value of the cosmological constant

67. 66 Vacuum energy density 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 eV e ? Cosmological constant and vacuum energy density

68. 67 Cosmological constant and vacuum energy density. Value of r 0 ? 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  r e 2 –––> Result:  = 1.362 10 -56 cm -2   h 2  = 0.38874(12) H 0 =71 ± 3 km/s.Mpc,   = 0.73 ± 0.04 (Wmap…) Predicted (LN 93): Observed:   h 2  = 0.40 ± 0.03

69. 68 Comparison prediction-observations Gunn-Tinsley LN, Hubble diagram of Infrared ellipticals LN, age problem SNe, WMAP 3yr lensing SNeI SNe, WMAP1yr lensing prediction