Report

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: [email protected] 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. 2 Consequences of the symmetry • If the time does not depend on the acceleration: = 1 and = 0-Galilean • If the time depends also directly on the acceleration: ≠ 0 (ER) 3 Transformation between accelerated systems under ER • Introduce a metric (, −1, −1, −1) on (; ) which makes the symmetry Sg self-adjoint or an isometry. • Conservation of interval: 2 = 2 − 2 • There is a maximal acceleration = , which is a universal constant with = • The proper velocity-time transformation (parallel axes) • Lorentz type transformation with: 4 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, Physica Scripta, 82 (2010) 015004.) – There is an additional Doppler shift due to acceleration (Y. Friedman, Ann. Phys. (Berlin) 523 (2011) 408) 5 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, Physica Scripta 77 035302 (2008)) • This additional shift can be explained with Extended Relativity. Estimation for maximal acceleration (Y. Friedman arXiv:0910.5629) = 1021 / 2 6 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 7 Classical Mechanics 8 Classical Hamiltonian 2 , = + () 2 Which can be rewritten as 1 , = − ′ − 0 0 − ` • The two parts of the Hamiltonian are integrals of velocity and acceleration respectively. 9 Hamiltonian System = = = • The Hamiltonian System is symmetric in x and u as required by Born’s Reciprocity 10 Classical Harmonic Oscillator (CHO) = − = −2 • The Hamiltonian , = − 0 = 0 − 0 0 • The kinetic energy and the potential energy are quadratic expressions in the variables u and ωx. 11 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? 12 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? 13 Special Relativity • Rate of clock depends on the velocity • Magnitude of velocity is bounded by c • Proper velocity u and Proper time τ = 14 Special Relativity Hamiltonian , = 2 + = 2 2 1+ 2 + Special Relativity Harmonic Oscillator (SRHO) , = 2 2 2 2 1+ 2 + 2 • The kinetic energy is hyperbolic in ‘u’ The potential energy is quadratic ‘ωx’ Born’s Reciprocity is lost 15 Can SRHO Explain Thermal Vibrations? • Typical amplitude and frequencies for Thermal Vibrations − ~10−9 ~1015 −1 = ~10 ≪ 6 • Therefore SRHO can’t explain thermal vibrations in the non-classical region. • But = 2 ~1021 2 16 Extended Relativity 17 Extended Relativistic Hamiltonian , = 0 1+ 2 − 2 0 () ()2 1+ 2 Extends both Classical and Relativistic Hamiltonian • For Harmonic Oscillator , = 2 2 2 4 2 1+ 2 + 2 1+ 2 • Born’s Reciprocity is restored • Both terms are hyperbolic 18 Effective Potential Energy (a) = 5 ∗ 1014 −1 = 7 ∗ 1014 −1 = 9 ∗ 1014 −1 (b) (c) = 1021 −1 (d) The effective potential is linearly confined The confinement is strong when is significantly large 19 Harmonic Oscillator Dynamics for Extremely Large ω 20 Harmonic Oscillator Dynamics for Extremely Large ω = • Acceleration (digitized) = =− = − <0 >0 21 Harmonic Oscillator Dynamics for Extremely Large ω • Velocity 2 = 2 ∞ =0 −1 2 2 + 1 sin 2 2 + 1 • The spectrum of ‘u’ coincides with the spectrum of energy of the Quantum Harmonic Oscillator 22 Harmonic Oscillator Dynamics for Extremely Large ω • Position = = 2 1+ 2 = 1+ 2 2 23 Transition between Classical and Extended Relativity 24 Transition between Classical and Non-classical Regions • Acceleration (d) (c) = 7 ∗ 1014 −1 = 9 ∗ 1014 −1 = 15 ∗ 1014 −1 = 30 ∗ 1014 −1 (b) (a) 25 Transition between Classical and Non-classical Regions • Velocity = 7 ∗ 1014 −1 = 9 ∗ 1014 −1 = 15 ∗ 1014 −1 (a) (c) = 30 ∗ 1014 −1 (b) (d) 26 Comparison between Classical and Extended Relativistic Oscillations 27 Comparison between Classical and Extended Relativistic Oscillations = 1015 −1 28 Comparison between Classical and Extended Relativistic Oscillations = 1016 −1 29 Comparison between Classical and Extended Relativistic Oscillations • Comparison between the ω and the effective ω. 6E+15 effective ω 5E+15 4E+15 Clasical 3E+15 ERD 2E+15 ERD limit 1E+15 0 0 ω 5E+15 30 Acceleration for a given at different Amplitudes (Energies) (c) (b) (a) (d) (a) (b) (c) (d) A=10^-10 A=10^-9 A=5*10^-9 A=10^-8 31 Comparison between Classical and Extended Relativistic Oscillations Classical region Non Classical region a(t) Aω2cos(ωt) square wave (slide 18) u(t) Aω sin(ωt) triangle wave (slide 19) x(t) -A cos(ωt) (slide 20) T E-E0 spectrum 2π/ω 2 16 2 + 32 m0A2ω2/2 m0Aam {ω} 2π/T (2k+1) : k=0,1,2,3… 32 Testing the Acceleration of a Photon • CL: = • ER: = • = ER + | ≈ CL 33 The future of ER • More experiments • More theory: EM, GR, QM (hydrogen), Thermodynamics 34 Thanks Any questions? 35