Digitization of the harmonic oscillator in Extended Relativity
Yaakov FriedmanJerusalem College of Technology
P.O.B. 16031 Jerusalem 91160, Israelemail: [email protected]
Geometry Days in Novosibirsk 2013
Relativity principle symmetry
• Principle of Special Relativity for inertial systems• General Principle of relativity for accelerated
systemThe transformation will be a symmetry, provided that the axes are chosen symmetrically.
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)
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:
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)
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)
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
Classical Mechanics
Classical Hamiltonian
Which can be rewritten as
• The two parts of the Hamiltonian are integrals of velocity and acceleration respectively.
𝐻 (𝑝 ,𝑥 )= 𝑝22𝑚+𝑉 (𝑥 )
1𝑚𝐻 (𝑥 ,𝑢)=∫
0
𝑢
𝑣𝑑𝑣−∫0
𝑥
𝑎 ( 𝑦 )𝑑𝑦
𝑢−𝑜𝑏𝑗𝑒𝑐 𝑡′ 𝑠 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦𝑎−𝑜𝑏𝑗𝑒𝑐𝑡 𝑠𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛
Hamiltonian System
• The Hamiltonian System is symmetric in x and u as required by Born’s Reciprocity
{ 𝑑𝑥𝑑𝑡 =𝑢
𝑑𝑢𝑑𝑡 = 𝐹
𝑚=𝑎
Classical Harmonic Oscillator (CHO)
• The kinetic energy and the potential energy are quadratic expressions in the variables u and ωx.
• The Hamiltonian
𝑎 (𝑥 )=− 𝑘𝑚 𝑥=−𝜔2𝑥
𝐻 (𝑥 ,𝑢)=𝑚∫0
𝑢
𝑣𝑑𝑣−𝑚∫0
𝑥
𝑎 (𝑦 )𝑑𝑦=𝑚∫0
𝑢
𝑣𝑑𝑣−𝑚∫0
𝜔𝑥
𝑦 𝑑𝑦
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?
Plank introduced a postulate that can explain the radiation curve for large ω.
CHO can not Explain the Radiation for Large ω.
Can Special Relativity Explain the Radiation for Large ω?
• Rate of clock depends on the velocity• Magnitude of velocity is bounded by c• Proper velocity u and Proper time τ
Special Relativity
𝑢=𝑑𝑥𝑑𝜏
Special Relativity Hamiltonian
𝐻 (𝑥 ,𝑢)=𝑚𝑐2𝛾 (𝑣 (𝑢) )+𝑉 (𝑥 )=𝑚𝑐2√1+𝑢2𝑐2+𝑉 (𝑥 )
Special Relativity Harmonic Oscillator (SRHO)
𝐻 (𝑥 ,𝑢)=𝑚𝑐2√1+𝑢2𝑐2 +𝑚𝜔2𝑥22
• The kinetic energy is hyperbolic in ‘u’The potential energy is quadratic ‘ωx’Born’s Reciprocity is lost
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
𝐴𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒− 𝐴 10− 9𝑐𝑚 𝜔 1015𝑠− 1
𝑣𝑚𝑎𝑥=𝐴𝜔 106 𝑐𝑚𝑠 ≪𝑐
𝑎𝑚𝑎𝑥=𝐴𝜔2 1021 𝑐𝑚𝑠2
Extended Relativity
Extended Relativistic Hamiltonian
• For Harmonic Oscillator
• Born’s Reciprocity is restored• Both terms are hyperbolic
Extends both Classical and Relativistic Hamiltonian
𝐻 (𝑥 ,𝑢)=𝑚∫0
𝑢 𝑣
√1+ 𝑣2𝑐2𝑑𝑣−𝑚∫
0
𝑥 𝑎(𝑦)
√1+𝑎(𝑦)2
𝑎𝑚2
𝑑𝑦
𝐻 (𝑥 ,𝑢)=𝑚𝑐2√1+𝑢2𝑐2 +𝑚 𝑎𝑚2
𝜔2 √1+𝜔4𝑥2
𝑎𝑚2
Effective Potential Energy
(a)
(b)(c)(d)
(𝑎 )𝜔=5∗1014𝑠−1(𝑏 )𝜔=7∗1014 𝑠− 1(𝑐 )𝜔=9∗1014𝑠−1
(𝑑)𝜔=1021𝑠− 1
The effective potential is linearly confined The confinement is strong when is significantly large
Harmonic Oscillator Dynamics for Extremely Large ω
Harmonic Oscillator Dynamics for Extremely Large ω
• Acceleration (digitized)
𝑉 𝑞 (𝑥 )=𝑎𝑚|𝑥|
𝑎 (𝑡 )= 𝑑𝑢𝑑𝑡 =− 𝜕𝐻𝜕𝑥 ={ 𝑎𝑚 𝑥<0
−𝑎𝑚𝑥>0
• Velocity
Harmonic Oscillator Dynamics for Extremely Large ω
• The spectrum of ‘u’ coincides with the spectrum of energy of the Quantum Harmonic Oscillator
𝑢 (𝑡 )=2𝑇 𝑎𝑚𝜋 2 ∑
𝑘=0
∞ (−1 )𝑘
(2𝑘+1 )2sin( 2𝜋 (2𝑘+1 ) 𝑡
𝑇 )
• Position
Harmonic Oscillator Dynamics for Extremely Large ω
𝑑𝑥𝑑𝑡 =𝜕𝐻
𝜕𝑢 =𝑢 (𝑡 )
√1+𝑢 (𝑡 )2
𝑐2
=𝑎𝑚𝑡
√1+ (𝑎𝑚𝑡 )2
𝑐2
Transition between Classical and Extended Relativity
• Acceleration
Transition between Classical and Non-classical Regions
(a)
(𝑑)𝜔=30∗1014 𝑠− 1
(𝑏)𝜔=9∗1014 𝑠− 1
(𝑐 )𝜔=15∗1014 𝑠−1
(b)
(c)
(d)
• Velocity
Transition between Classical and Non-classical Regions
(𝑑 )𝜔=30∗1014 𝑠− 1
(𝑏)𝜔=9∗1014 𝑠− 1
(𝑐 )𝜔=15∗1014𝑠−1(a)
(b)(c)
(d)
Comparison between Classical and Extended Relativistic Oscillations
Comparison between Classical and Extended Relativistic Oscillations
𝜔=1015𝑠− 1
Comparison between Classical and Extended Relativistic Oscillations𝜔=1016𝑠−1
Comparison between Classical and Extended Relativistic Oscillations
• Comparison between the ω and the effective ω.
0 50000000000000000
1000000000000000
2000000000000000
3000000000000000
4000000000000000
5000000000000000
6000000000000000
Clasical
ERD
ERD limit
ω
effec
tive
ω
Acceleration for a given at different Amplitudes (Energies)
(a) A=10^-10(b) A=10^-9(c) A=5*10^-9(d) A=10^-8
(a)
(d)
(c)
(b)
Comparison between Classical and Extended Relativistic Oscillations
Non Classical region Classical region
(slide 18) square wave Aω2cos(ωt) a(t)
triangle wave (slide 19) Aω sin(ωt) u(t)
(slide 20) -A cos(ωt) x(t)
2π/ω T
m0Aam m0A2ω2/2 E-E0
2π/T (2k+1) : k=0,1,2,3… {ω} spectrum
33
Testing the Acceleration of a Photon
• CL: • ER:
ERCL
34
The future of ER
• More experiments• More theory: EM, GR, QM (hydrogen),
Thermodynamics
Thanks
Any questions?
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