### The Harmonic Oscillator in Extended Relativistic Dynamics

```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
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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
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
```