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
• 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: 𝛾 = 1 and 𝜅 = 0-Galilean
• If the time depends also directly on the
acceleration: 𝜅 ≠ 0 (ER)
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Transformation between accelerated
systems under ER
• Introduce a metric 𝑑𝑖𝑎𝑔(𝜇, −1, −1, −1) on
(𝑡; 𝑢) which makes the symmetry Sg self-adjoint or an
isometry.
• Conservation of interval: 𝑑𝑠 2 = 𝜇𝑑𝑡 2 − 𝑑𝑢 2
• There is a maximal acceleration 𝑎𝑚 = 𝜇, which is a
𝑔
universal constant with 𝜇 =
𝜅
• 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)
𝑎𝑚 = 1021 𝑐𝑚/𝑠 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 (𝑎𝑚)
• Thermal radiation curves predicted by
ER are similar to the observed ones
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Classical Mechanics
8
Classical Hamiltonian
𝑝2
𝐻 𝑝, 𝑥 =
+ 𝑉(𝑥)
2𝑚
Which can be rewritten as
1
𝐻 𝑥, 𝑢 =
𝑚
𝑢 − 𝑜𝑏𝑗𝑒𝑐𝑡 ′ 𝑠 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦
𝑢
𝑥
𝑣𝑑𝑣 −
0
𝑎 𝑦 𝑑𝑦
0
𝑎 − 𝑜𝑏𝑗𝑒𝑐𝑡`𝑠 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛
• The two parts of the Hamiltonian are integrals
of velocity and acceleration respectively.
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Hamiltonian System
𝑑𝑥
=𝑢
𝑑𝑡
𝑑𝑢 𝐹
= =𝑎
𝑑𝑡 𝑚
• The Hamiltonian System is symmetric in x and u as
required by Born’s Reciprocity
10
Classical Harmonic Oscillator (CHO)
𝑘
𝑎 𝑥 = − 𝑥 = −𝜔2 𝑥
𝑚
• The Hamiltonian
𝑢
𝐻 𝑥, 𝑢 = 𝑚
𝑥
𝑣𝑑𝑣 − 𝑚
0
𝑢
𝑎 𝑦 𝑑𝑦 = 𝑚
0
𝜔𝑥
𝑣𝑑𝑣 − 𝑚
0
𝑦𝑑𝑦
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
𝑑𝑥
𝑢=
𝑑𝜏
14
Special Relativity Hamiltonian
𝐻 𝑥, 𝑢 = 𝑚𝑐 2 𝛾 𝑣 𝑢
+ 𝑉 𝑥 = 𝑚𝑐 2
𝑢2
1+ 2 +𝑉 𝑥
𝑐
Special Relativity Harmonic Oscillator
(SRHO)
𝐻 𝑥, 𝑢 = 𝑚𝑐 2
𝑢2 𝑚𝜔2 𝑥 2
1+ 2 +
𝑐
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
𝐴𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 − 𝐴~10−9 𝑐𝑚
𝑣𝑚𝑎𝑥
𝜔~1015 𝑠 −1
𝑐𝑚
= 𝐴𝜔~10
≪𝑐
𝑠
6
• Therefore SRHO can’t explain thermal
vibrations in the non-classical region.
• But
𝑎𝑚𝑎𝑥 =
𝐴𝜔2 ~1021
𝑐𝑚
𝑠2
16
Extended Relativity
17
Extended Relativistic Hamiltonian
𝑢
𝐻 𝑥, 𝑢 = 𝑚
0
𝑥
𝑣
1+
𝑣2
𝑑𝑣 − 𝑚
𝑐2
0
𝑎(𝑦)
𝑑𝑦
𝑎(𝑦)2
1+ 2
𝑎𝑚
Extends both Classical and Relativistic Hamiltonian
• For Harmonic Oscillator
𝐻 𝑥, 𝑢 = 𝑚𝑐 2
2
𝑢2
𝑎𝑚
𝜔4𝑥 2
1+ 2 +𝑚 2 1+ 2
𝑐
𝜔
𝑎𝑚
• Born’s Reciprocity is restored
• Both terms are hyperbolic
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Effective Potential Energy
(a)
𝑎 𝜔 = 5 ∗ 1014 𝑠 −1
𝑏 𝜔 = 7 ∗ 1014 𝑠 −1
𝑐 𝜔 = 9 ∗ 1014 𝑠 −1
(b)
(c)
𝑑 𝜔 = 1021 𝑠 −1
(d)
The effective potential is linearly confined
The confinement is strong when 𝜔 is significantly large
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Harmonic Oscillator Dynamics for
Extremely Large ω
20
Harmonic Oscillator Dynamics for Extremely Large ω
𝑉𝑞 𝑥 = 𝑎𝑚 𝑥
• Acceleration (digitized)
𝑑𝑢
𝜕𝐻
𝑎𝑚
𝑎 𝑡 =
=−
=
−𝑎𝑚
𝑑𝑡
𝜕𝑥
𝑥<0
𝑥>0
21
Harmonic Oscillator Dynamics for Extremely Large ω
• Velocity
2𝑇𝑎𝑚
𝑢 𝑡 =
𝜋2
∞
𝑘=0
−1 𝑘
2𝜋 2𝑘 + 1 𝑡
sin
2
2𝑘 + 1
𝑇
• The spectrum of ‘u’ coincides with the spectrum of
energy of the Quantum Harmonic Oscillator
22
Harmonic Oscillator Dynamics for Extremely Large ω
• Position
𝑑𝑥 𝜕𝐻
=
=
𝑑𝑡 𝜕𝑢
𝑢 𝑡
𝑢 𝑡 2
1+ 2
𝑐
=
𝑎𝑚 𝑡
𝑎𝑚 𝑡
1+
𝑐2
2
23
Transition between Classical and
Extended Relativity
24
Transition between Classical and Non-classical
Regions
• Acceleration
(d)
(c)
𝑎 𝜔 = 7 ∗ 1014 𝑠 −1
𝑏 𝜔 = 9 ∗ 1014 𝑠 −1
𝑐 𝜔 = 15 ∗ 1014 𝑠 −1
𝑑 𝜔 = 30 ∗ 1014 𝑠 −1
(b)
(a)
25
Transition between Classical and Non-classical
Regions
• Velocity
𝑎 𝜔 = 7 ∗ 1014 𝑠 −1
𝑏 𝜔 = 9 ∗ 1014 𝑠 −1
𝑐 𝜔 = 15 ∗ 1014 𝑠 −1
(a)
(c)
𝑑 𝜔 = 30 ∗ 1014 𝑠 −1
(b)
(d)
26
Comparison between Classical and
Extended Relativistic Oscillations
27
Comparison between Classical and Extended
Relativistic Oscillations
𝜔 = 1015 𝑠 −1
28
Comparison between Classical and Extended
Relativistic Oscillations
𝜔 = 1016 𝑠 −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
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Acceleration for a given 𝜔 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
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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
𝐴
16
𝑐2
+ 32 𝐴 𝑎𝑚
m0A2ω2/2
m0Aam
{ω}
2π/T (2k+1) : k=0,1,2,3…
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Testing the Acceleration of a Photon
• CL: 𝒕 =
𝒙
𝒄
• ER: 𝒕 =
• 𝜶=
ER
𝒄𝟐
𝒂𝒎
𝒙𝟐 +𝟐𝜶𝒙
𝒄
|
≈ 𝟏𝒄𝒎
CL
33
The future of ER
• More experiments
• More theory: EM, GR, QM (hydrogen),
Thermodynamics
34
Thanks
Any questions?
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