Digitization of the harmonic oscillator in Extended Relativity

Yaakov FriedmanJerusalem College of Technology

P.O.B. 16031 Jerusalem 91160, Israelemail: friedman@jct.ac.il

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?