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
Geometry Days in Novosibirsk 2013
Relativity principle
ο
symmetry
β’
Principle
of Special Relativity for inertial systems
β’
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
β’
If the time does not depend on the
acceleration:
πΎ
=
1
and
π
=
0

Galilean
β’
If
the time depends also directly on the
acceleration
:
π
β
0
(
ER)
3
Transformation between accelerated
systems under ER
β’
Introduce a metric
π
(
π
,
β
1
,
β
1
,
β
1
)
on
(
;
)
which makes
the symmetry
S
g
self

adjoint
or an
isometry
.
β’
Conservation
of
interval:
2
=
π
2
β
2
β’
There
is a
maximal acceleration
π
=
π
,
which
is a
universal
constant
with
π
=
π
π
β’
The proper velocity

time transformation (parallel axes)
β’
Lorentz type
transformation with:
4
The Upper Bound for Acceleration
β’
If the acceleration affects the rate of the
moving clock then:
β
there is a universal maximal acceleration
(Y. Friedman, Yu. Gofman,
Physica
Scripta
,
82
(
2010
)
015004
.)
β
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
β’
KΓΌndig's experiment measured the transverse
Doppler shift
(W.
KΓΌndig
, Phys. Rev.
129
(
1963
)
2371
)
β’
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
))
β’
This additional shift can be explained with
Extended Relativity
. Estimation for maximal
acceleration
(Y. Friedman
arXiv:
0910.5629
)
6
π
=
10
21
/
2
Further Evidence
β’
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
β’
ER model for a hydrogen and using the value
of ionization of hydrogen leads approximately
to the value of the maximal acceleration (
)
β’
Thermal radiation curves predicted by
ER are similar to the observed ones
7
Classical Mechanics
8
Classical Hamiltonian
9
Which can be rewritten as
β’
The two parts of the Hamiltonian are integrals
of velocity and acceleration respectively.
π»
,
=
2
2
+
π
(
)
1
π»
,
=
π’
0
β
π₯
0
β
β²
β
`
Hamiltonian System
10
β’
The Hamiltonian System is symmetric in
x
and
u
as
required by
Bornβs
Reciprocity
=
=
πΉ
=
Classical Harmonic Oscillator (CHO)
11
β’
The kinetic energy and the potential energy are quadratic
expressions in the variables u and
Ο
x.
β’
The Hamiltonian
=
β
=
β
2
π»
,
=
π’
0
β
π₯
0
=
π’
0
β
π
π₯
0
Example: Thermal Vibrations of
Atoms in Solids
β’
CHO models well such vibrations and predicts
the thermal radiation for small
Ο
12
β’
Why canβt the CHO explain the radiation for large
Ο?
Plank introduced a postulate that can explain
the radiation curve for large
Ο.
13
CHO can not Explain the Radiation
for Large Ο.
Can Special Relativity Explain the
Radiation for Large Ο?
β’
Rate of clock depends on the velocity
β’
Magnitude of velocity is
b
ounded by c
β’
Proper velocity u and Proper time
Ο
14
Special Relativity
=
π
Special Relativity Hamiltonian
15
π»
,
=
2
πΎ
+
π
=
2
1
+
2
2
+
π
Special Relativity Harmonic Oscillator
(SRHO)
π»
,
=
2
1
+
2
2
+
2
2
2
β’
The kinetic energy is
hyperbolic
in βuβ
The potential energy is
quadratic
β
Ο
xβ
Bornβs
Reciprocity
is lost
Can SRHO Explain Thermal Vibrations?
β’
Typical amplitude and frequencies for Thermal
Vibrations
β’
Therefore SRHO canβt explain thermal
vibrations in the non

classical region.
β’
But
16
π΄
β
π΄
~
10
β
9
~
10
15
β
1
πππ₯
=
π΄
~
10
6
βͺ
πππ₯
=
π΄
2
~
10
21
2
Extended Relativity
17
Extended Relativistic Hamiltonian
18
β’
For Harmonic Oscillator
β’
Bornβs
Reciprocity is restored
β’
Both terms are hyperbolic
Extends both Classical and Relativistic Hamiltonian
π»
,
=
1
+
2
2
π’
0
β
(
)
1
+
(
)
2
π
2
π₯
0
π»
,
=
2
1
+
2
2
+
π
2
2
1
+
4
2
π
2
Effective Potential Energy
19
(a)
(b)
(c)
(d)
=
5
β
10
14
β
1
=
7
β
10
14
β
1
=
9
β
10
14
β
1
=
10
21
β
1
The effective potential is linearly confined
The confinement is strong when
is significantly large
20
Harmonic Oscillator Dynamics for
Extremely Large
Ο
Harmonic Oscillator Dynamics for Extremely Large
Ο
β’
Acceleration (digitized)
21
π
π
=
π
=
=
β
π»
=
π
<
0
β
π
>
0
β’
Velocity
22
Harmonic Oscillator Dynamics for Extremely Large
Ο
β’
The spectrum of βuβ coincides with the spectrum of
energy of the
Quantum
Harmonic Oscillator
=
2
π
π
π
2
β
1
π
2
+
1
2
sin
2
π
2
+
1
π
β
π
=
0
β’
Position
23
Harmonic Oscillator Dynamics for Extremely Large
Ο
=
π»
=
1
+
2
2
=
π
1
+
π
2
2
Transition
b
etween Classical and
Extended Relativity
24
β’
Acceleration
25
Transition between Classical and Non

classical
Regions
(a)
=
30
β
10
1
4
β
1
=
7
β
10
14
β
1
=
9
β
10
14
β
1
=
15
β
10
14
β
1
(b)
(c)
(d)
β’
Velocity
26
Transition between Classical and Non

classical
Regions
=
30
β
10
1
4
β
1
=
7
β
10
14
β
1
=
9
β
10
14
β
1
=
15
β
10
14
β
1
(a)
(b)
(c)
(d)
Comparison between Classical and
Extended Relativistic Oscillations
27
28
Comparison between Classical and Extended
Relativistic Oscillations
=
10
15
β
1
29
Comparison between Classical and Extended
Relativistic Oscillations
=
10
16
β
1
Comparison between Classical and
Extended Relativistic Oscillations
β’
Comparison between the
Ο
and the effective
Ο
.
30
0
1E+15
2E+15
3E+15
4E+15
5E+15
6E+15
0
5E+15
effective Ο
Ο
Clasical
ERD
ERD limit
Acceleration for a given
at different
Amplitudes (Energies)
31
(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
32
Non Classical region
Classical
region
(slide
18
)
square wave
?
A
Ο
2
cos(
Ο
t)
a(t)
triangle wave (slide
19
)
A
Ο
sin(
Ο
t)
u(t)
(slide
20
)

δ
cos
(
Ο
t)
x(t)
16
π΄
2
2
+
32
π΄
π
2
Ο
/
Ο
T
m
0
Aa
m
m
0
A
2
Ο
2
/
2
E

E
0
2
Ο
/T (
2
k+
1
)
: k=
0
,
1
,
2
,
3
β¦
{
?Λ
}
spectrum
Testing the Acceleration
of a Photon
33
β’
CL
:
π
=
π
π
β’
ER
:
π
=
π
+
πΆπ
π

β’
πΆ
=
π
π
π
β
ππ
ER
CL
The future of ER
β’
More experiments
β’
More theory: EM, GR, QM (hydrogen),
Thermodynamics
34
Thanks
Any questions?
35
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