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Digitization of the harmonic oscillator in Extended Relativity

Geometry Days in Novosibirsk 2013. Digitization of the harmonic oscillator in Extended Relativity. Yaakov Friedman Jerusalem College of Technology P.O.B. 16031 Jerusalem 91160, Israel email: friedman@jct.ac.il. Relativity principle  symmetry.

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Digitization of the harmonic oscillator in Extended Relativity

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  1. Geometry Days in Novosibirsk 2013 Digitization of the harmonic oscillator in Extended Relativity Yaakov Friedman Jerusalem College of Technology P.O.B. 16031 Jerusalem 91160, Israel email: friedman@jct.ac.il

  2. Relativity principle symmetry • Principle of Special Relativity for inertial systems • General Principle of relativity for accelerated system The transformation will be a symmetry, provided that the axes are chosen symmetrically.

  3. Consequences of the symmetry • If the time does not depend on the acceleration: and -Galilean • If the time depends also directly on the acceleration: (ER)

  4. Transformation between accelerated systems under ER • Introduce a metric on which makes the symmetry Sg self-adjoint or an isometry. • Conservation of interval: • There is a maximal acceleration , which is a universal constant with • The proper velocity-time transformation (parallel axes) • Lorentz type transformation with:

  5. The Upper Bound for Acceleration • If the acceleration affects the rate of the moving clock then: • there is a universal maximal acceleration (Y. Friedman, Yu. Gofman, PhysicaScripta, 82 (2010) 015004.) • There is an additional Doppler shift due to acceleration (Y. Friedman, Ann. Phys. (Berlin) 523 (2011) 408)

  6. Experimental Observations of the Accelerated Doppler Shift • Kündig's experiment measured the transverse Doppler shift (W. Kündig, Phys. Rev. 129 (1963) 2371) • Kholmetskii et al: The Doppler shift observed differs from the one predicted by Special Relativity. (A.L. Kholmetski, T. Yarman and O.V. Missevitch, PhysicaScripta 77 035302 (2008)) • This additional shift can be explained with Extended Relativity. Estimation for maximal acceleration (Y. Friedman arXiv:0910.5629)

  7. Further Evidence • DESY (1999) experiment using nuclear forward scattering with a rotating disc observed the effect of rotation on the spectrum. Never published. Could be explained with ER • ER model for a hydrogen and using the value of ionization of hydrogen leads approximately to the value of the maximal acceleration () • Thermal radiation curves predicted by ER are similar to the observed ones

  8. Classical Mechanics

  9. Classical Hamiltonian Which can be rewritten as • The two parts of the Hamiltonian are integrals of velocity and acceleration respectively.

  10. Hamiltonian System • The Hamiltonian System is symmetric in x and u as required by Born’s Reciprocity

  11. Classical Harmonic Oscillator (CHO) • The Hamiltonian • The kinetic energy and the potential energy are quadratic expressions in the variables u and ωx.

  12. Example: Thermal Vibrations of Atoms in Solids • CHO models well such vibrations and predicts the thermal radiation for small ω • Why can’t the CHO explain the radiation for large ω?

  13. CHO can not Explain the Radiation for Large ω. Plank introduced a postulate that can explain the radiation curve for large ω. Can Special Relativity Explain the Radiation for Large ω?

  14. Special Relativity • Rate of clock depends on the velocity • Magnitude of velocity is bounded by c • Proper velocity u and Proper time τ

  15. Special Relativity Hamiltonian Special Relativity Harmonic Oscillator (SRHO) • The kinetic energy is hyperbolic in ‘u’The potential energy is quadratic ‘ωx’Born’s Reciprocity is lost

  16. Can SRHO Explain Thermal Vibrations? • Typical amplitude and frequencies for Thermal Vibrations • Therefore SRHO can’t explain thermal vibrations in the non-classical region. • But

  17. Extended Relativity

  18. Extended Relativistic Hamiltonian Extends both Classical and Relativistic Hamiltonian • For Harmonic Oscillator • Born’s Reciprocity is restored • Both terms are hyperbolic

  19. Effective Potential Energy (a) (b) (c) (d) The effective potential is linearly confined The confinement is strong when is significantly large

  20. Harmonic Oscillator Dynamics for Extremely Large ω

  21. Harmonic Oscillator Dynamics for Extremely Large ω • Acceleration (digitized)

  22. Harmonic Oscillator Dynamics for Extremely Large ω • Velocity • The spectrum of ‘u’ coincides with the spectrum of energy of the Quantum Harmonic Oscillator

  23. Harmonic Oscillator Dynamics for Extremely Large ω • Position

  24. Transition between Classical and Extended Relativity

  25. Transition between Classical and Non-classical Regions (d) • Acceleration (c) (b) (a)

  26. Transition between Classical and Non-classical Regions • Velocity (a) (b) (c) (d)

  27. Comparison between Classical and Extended Relativistic Oscillations

  28. Comparison between Classical and Extended Relativistic Oscillations

  29. Comparison between Classical and Extended Relativistic Oscillations

  30. Comparison between Classical and Extended Relativistic Oscillations • Comparison between the ω and the effective ω.

  31. Acceleration for a given at different Amplitudes (Energies) (c) (d) (b) A=10^-10 A=10^-9 A=5*10^-9 A=10^-8 (a)

  32. Comparison between Classical and Extended Relativistic Oscillations

  33. Testing the Acceleration of a Photon • ER: • CL: ER CL

  34. The future of ER • More experiments • More theory: EM, GR, QM (hydrogen), Thermodynamics

  35. ThanksAny questions?

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