Photonic Mass Model
Christoph SCHULTHEISS
Forschungszentrum Karlsruhe GmbH, Institut für Hochleistungsimpuls- und Mikrowellentechnik (IHM)
Postfach 3640, D-76021 Karlsruhe, FRG
Abstract:
A one dimensional model is presented, where a single photon, confined in
a cavity, gives a model of mass with respect to kinetic energy and inertia.
Then photon-mass interaction is reduced to photon-confined photon interaction,
where changes are exclusively Doppler based. However the model demands the
existence of a photon pool similar to the zero-point radiation as described by
H.B.G Casimir. However
this photon pool alone does not stabilize and conserve the distance of the
cavity mirrors. A thermal fraction must contribute in addition as it is
described in “Gravitational
Mass attraction caused by Ultra Long Wave Photons” A cavity in rest
or in translation will leave the photon pool unaffected. However during
accelerations of a cavity the model predicts a break of the isotropy of the
pool photon distribution such, that behind the cavity a repelling and before a
co-accelerating force on probes in first order of β is expected. Experimental
work on this subject are published in www.christoph-schultheiss.de/experiments_on_gravitational_forces.
A.
Introduction
The paper aims to model a mass built from a
“mass-less” confinement and photons inside the confinement which
reflect continuously back and forward along with 180°-
Since the
During reflection processes of photons with charged
particles the wavelength of interacting photons becomes shifted. This is a
consequence of energy- and momentum conservation. It is the conclusion of
several authors [1-5], that these shifts are identically to Doppler shifts,
i.e. after the interaction a particle moves with the velocity
β´ = v´/c and the interacting photon, coming back from the
particle, becomes Doppler red shifted with respect to the relative velocity b´c. This
is demonstrated next for an elastic 180°
With the dimensionless abbreviations α =
hν/mc2, γ = (1- β2)-½ , and
for an initial velocity b=0, by means of mutual addition and subtraction the
well known solutions of both equations:
(1) Energy:
α + 1 =
α´+ γ´
(2)
Momentum: α + 0 = - α´
+ β´γ´
are:
(3) α´
= α/(1 + 2α)
(4)
γ´ =
1+2α2/(1+2α)
The Doppler equation becomes visible, if Eq.1 and 2
are rewritten in:
(5)
γ´
= α – α´+ 1
(6)
β´γ´ = α +
α´
(7)
Subtraction:
γ´ - β´γ´ = 1 -
2α´
(8) Addition:
γ´
+ β´γ´ = 1+
2α
because of the definition
equation: γ´ - β´γ´
= (γ´ + β´γ´)-1
from Eq.7 and 8 it follows
(9) γ´
- β´γ´ = 1 - 2α´ =
(1 + 2α)-1 .
A comparison with Eq.3 gives:
(10)
γ´ - β´γ´ =
α´/α ,
which is the Doppler equation.
This result recommends the following interpretation:
The Compton Effect can be considered as a two step
event. First step is the reflection of the photon, where both mass and photon
change from the laboratory system S into the moved collided system S´.
Then in a second step the re-transformation of the photon into the laboratory
system takes place. In this step the Doppler red shift as formulated in Eq.10
occurs. This shift corresponds exactly to the velocity difference
ß´ energy and momentum law demands.
For an initial velocity b, the elastic
Analogous to the equations 1 and 2 the energy- and the
momentum law have the form:
(11)
α
+ γ = α´ + γ´
(12)
α
+ βγ = - α´+ β´γ´
With the abbreviated Doppler factors D = γ (1
– β), D-1 =
γ (1 + β) the solutions are:
(13)
α´=
αD2 (1 + 2αD)-1
(14)
D´ = D (1 +
2αD)-1,
D´ and α´ relate to the
observer. Kinetic energy (γ – 1) can be determined by D´,
using the definition equation γ´= ½ (1 + D´2)/
D´. This gives:
(15)
γ´-
1 = 2α2 D2 (1 + 2αD)-1
Together with equation 13 equation 14 makes visible
that:
(16) α´=
α DD´
energy- and momentum of the photon is changed
exclusively by products of Doppler shifts.
Before the collision in the system of the mass (MS)
the colliding photon has α D. After the collision the mass system
MS´ moves against MS with a velocity corresponding to
(17)
D =
D´/D = (1 + 2αD)-1
and the energy of the reflected photon with respect to
MS is α DD.
In this sense Tab.1 shortly summarizes this:
The photon α leaves the laboratory system (LS)
and changes into the moving mass system (MS) as αD.
Then it reflects elastically and returns from MS´ unchanged in momentum
back to MS as αD. On the way back to
LS a second Doppler shift occurs and αD
changes into αDD´, respectively
in αDDD (see Eq.17).
For the violet shift case, i.e. b
―› - β , D is replaced by
D ―› 1/D.
Table A.1: Change of energy- and momentum of a photon by Doppler shifts, during
inelastic 180° -
|
Comment |
b³ 0 |
|
LS |
a |
|
Transformation from LS to MS |
a D |
|
After collision in MS´ |
a D |
|
Back-transformed in MS |
a D D |
|
Back-transformed to LS |
a D DD = a D D´ |
Since a change of photon energy is exclusively subject
of Doppler shifts (see ref. 3-6), the
(18) Momentum α + 0 = - α´+ β´γ´
(19) Doppler α´ = α γ´(1 – β´)
The Doppler equation
gives
(20) β´ = (α2 – α´2)·(α2
+ α´2)-1 and γ´= (α2
+ α´2)/2αα´
Inserted in Eq.12 the solutions Eq.3 and 4 follow,
without taking direct use of the energy law. The elastic Compton Effect seems
to be reasonably described by momentum and Doppler law. The question arises,
why the energy law in the calculation Eq.1-10 describes the situation too?
To answer this question it is tried to proceed a step more. CCM gives the possibility to reduce the
photon-particle interaction solely to a photon-photon interaction theatre,
where the mass itself is modeled by photons. This means, photons are confined
in a cavity, reflect back- and forward between two mirrors and built rest
energy. Movements of the cavity imply Doppler shifts of the cavity photons and
besides the rest energy a kinetic energy term appears. Therefore a collision
with a free photon leads finally to Doppler shifts of all involved photons
– no matter free or confined. To simulate real elementary particles the
transition from a mass-loaded cavity to a mass-less cavity is unavoidable. A
non-closed physical system must be taken in account to stabilize the geometric
arrangement of the cavity mirrors, i.e. the mirror distance. This calls for the
cooperation of a surrounding photon radiation field quite similar as described
with the zero-point radiation of the space by Casimir
[6]. However, during acceleration of such a modeled mass, anisotropy effects in
the radiation field in the vicinity of the mass are the consequence.
It is not the intention to assert that the CCM frame
always reflects reality, but it could, because conservation of momentum and Lorentz transformation are doubtless the most basic
fundaments of nature. Therefore conclusions which result from of CCM are
worthwhile to be attended and discussed.
B
Confined light in a cavity which is in rest or in constant movement
A model on the basis of CCM is presented, where the
mass is assembled by a cavity, consisting of a pair of mirrors in a fixed
distance which contains confined photons. Photon – particle
interactions become reduced to free photon - confined photon
interactions, if the cavity mass is assumed to be zero. In the next chapters
the behavior of a cavity filled with photons is subject of investigations with
respect to rest, movement and acceleration.
It is well known that the reflection of moving waves
at a mirror build up a standing wave pattern in front of the mirror. In a
cavity, which consists of two parallel oriented ideal
conducting mirrors in a distance of L, only a selection of standing waves
following
(21) L
= ½ (j-1) λ, where j
= 2,3,4,….
is allowed.
These standing waves can be populated with an arbitrary number of confined
photons. In this one dimensional model spin effects will be neglected, only
momentum exchanges during collisions are subject of interest.
Figure 1, left graph, shows such a cavity in a Minkowsky space-time presentation. The mirrors built
vertical time-lines in a distance of L. The wave front - drawn as a dotted line
with arrow - and wave end - drawn as full line - move in a zigzag under
45° upward inclined with the speed of light. The wavelength (thick arrow)
is horizontal, i.e. purely space like. The length of the wavelength arrows in
fig.1 correspond to the case λ/L = 1. Additional fig.1, right, shows
a Lorentz transformed space time frame of the same
cavity in a movement to the right.
|
|
|
|
|
|
Fig.1: A
cavity in rest (left) and in movement (right) to the right in the frame of a Minkowski space-time presentation. The 45°-lines in
both presentations represent the zigzag spread of the wave-front (solid line)
and the wave-end (dotted line) for the case l /L = 1. The thick arrows are connection lines between wave-ends to
–fronts. In a system in rest they are designated with
„wavelength“. In a moving system (right) the wavelength becomes a
4-vector (see Appendix II).
On the basis of a common zero point respectively a space
time cross point the expressions of the Lorentz
transformation are:
ct´= γ (ct- βx)
(22)
x´ = γ (x- βct)
In fig.1 the observer moves to the left. Therefore b is
negative. The relevant transformed coordinates are:
(23)
(0,0)´= (0,0);
(24)
(L,0)´= (gL, bgL);
(25)
(0,L)´= (bgL, gL);
(26)
(L,L)´= [gL(1+b), gL(1+b)];
(27)
(0,nL)´= (nbgL, ngL);
(28)
(L, nL)´= [gL(1+nb), gL(n+b)];
The transformed wavelength is a 4-vector. Space- and
time interval of the wavelength is:
(29)
(0,0)´- (L,0)´ = [-gL, -bgL]
and for the
following point pairs:
(30)
(L,L)´- (0,L)´ = [gL, bgL]
and so on.
1. Real
cavity versus ideal cavity
For many aspects fig.1 does not fit for a “ free” particle, build solely from confined
photons in an at last mass less cavity. Assume two photons with a phase
difference of p in the cavity. At any time, the momentum transfer of
the left reflecting photons must exactly and instantaneously be equaled by the
momentum transfer of right reflecting one to hold the mirrors in rest.
Mass-less mirror-connection-rods for momentum transportation are physically not
available; instantaneous momentum transfer contradicts to the limitation of the
speed of light. In addition any mass-like rods lead to diameter vibrations and
increase the interior energy. In this case the energy law for inelastic
collisions is:
(31) α + 1 = α´ + δ´+
γ´
where d´
represents the vibration excitation. Nature tells that in elastic photon-elementary
particle interactions the term d´ is normally zero.
Therefore a model of mass, relaying on confined photons seems to be a
complicated or even an unrealistic undertaking. The problem is to find a
mechanism which efforts seeming instantaneous momentum transfer from one mirror
to the other.
Two solutions are offered next. The first one is a
dynamic mirror system, where nearly mass less mirrors experience a centripetal
force by means of elastic bands and are oscillatory pushed out by the collision
momenta of at least two confined photons with
π phase difference. The other solution, which is favored by the author, is
the action of a pool photon
theater where outside photons reflect simultaneously with confined photons
and compensate any mirror momentum (see Chap.3).
2.
Dynamics of nearly mass-less cavity walls
Concerning the dynamic mirror system one has to
imagine mirrors which are connected by elastic bands. The elastic bands are in
tension because of innumerous collisions of confined photons at the
mirrors. This means every time a confined photon reaches one mirror, the
mirror flies inward. After the collision the mirror flies outward with exactly
the same value of velocity.
The energy and momentum equations for the oscillation
are:
(32)
α + γ = α´ +
γ´
- α + βγ = α´- β´γ´
because of
the oscillatory boundary condition the momentum equation result in:
(33) α = β γ
respectively in
physical properties
(34) h/λ
= βγ
mc .
The interesting feature of course is the reduction of
mirror mass with the boundary that the photon momentum h/λ remains constant.
This means γ must
grow enormously i.e., ß → 1 must be
valid and therefore equation 34 becomes:
(35) γ mc ≈ h/λ respectively γ ≈
α
For very high values of γ the space time curve reminds to that of light.
Reflections i.e., reversal of movements are sharp corners and curves are
straight lines which meet themselves under the angel of 45° (see
fig.2).
Fig.2: Space
time graphs of a cavity in rest (left) and in movement, where nearly mass less
mirrors oscillate (grey area) by means of a centripetal force originated by
elastic bands and a centrifugal force generated by reflecting confined
photons
The oscillation area of the mirrors (grey marked in
fig.2) is ½ L for two confined photons which are in a phase
distance of π.
The energy of the whole system is distributed between
the mirrors and the confined photons:
(36)
Ehv/Emc2
= hν/γmc2 = α(1+α2)-½
= (1 + 1/α2)-½ ―› 1 for α ―› ∞
In the extreme case, where α and γ →
∞, ½ of the energy is distributed to
the nearly light fast mass and ½ to the confined photons.
At a first glance the light fast mirror behaves like a
photon. The question is, does it transform like a
photon or like a mass proportional with β2?
The definition equation for the relativistic mass
mγ1 for the high-gamma case is:
(37) mγ1 = m (1- β12)-½ ,
where β1
defers only marginally from 1. Now the cavity moves with a velocity β2 which
is small against 1. The increase of energy for the light fast mirrors can be
calculated by applying the relativistic addition law:
(38)
Using the definition equations for γ we have
(39) m γ´ = m γ1 γ2
(1 - β1 β2 ) = m γ1 D2
and the prove that
the light fast mirrors transform like photons is carried out. During a movement
of the cavity with ß2 and as a consequence of the oscillatory
movement of the mirrors with ß1 they experience red- and
violet shifts corresponding to β2 while the confined photons
experience violet-and red shifts in the same rhythm.
This Doppler shift theatre which comes up with the
dynamic mirror model can be also found in the mechanism shown next.
3.
External photon pool similar to Casmir’s zero point radiation defining
cavity walls built from mass-less mirrors
An alternative mechanism to
stabilize the mirrors could be an external photon theatre (pool) as can be seen
in fig.3. Mass-less mirrors are embedded in a
standing wave field of pool photons. Mirror reflections of confined
photons take place in phase with reflections of pool photons at the
rear mirror side – a process which is proposed to be called in-phase.
|
|
|
|
|
|
Fig.3: Space-time frame
of the compensation of mirror reflection momenta by
instantaneous reflections of external photons (thick arrows) coming from a
pool. This process will be called in-phase.
By the action of pool photons each mirror separately
remains in rest or in constant movement during the reflections. Of course the momenta of confined- and pool photons are
assumed to be equal.
With the tool of an external photon theatre the model
of a free particle build solely from confined photons can be
investigated. However serious demands meet the photon pool, which has to spend
a photon spectrum at any time at any place to conserve geometrical properties
of particles and to hold them stabilized. This will be a subject of the
next chapter too.
4.
Doppler shifts at a moved mirror during in-phase
If a cavity is in rest, an observer state, that confined
photons α* and
counter-flowing pool photons αp reflect
instantaneously, as shown in fig.4 left graph. If the cavity plus pool cavity or
the observer moves, all photons, which interact with the cavity mirrors,
underlie Doppler shifts. With the abbreviations D = γ (1 – β)
and D-1 = γ (1 + β) pool
photons moving to the right are transformed into (see fig.4):
(40)
αp ―› αp
D-1,
and after the
reflection, moving to the left, the transformation is:
(41)
αp ―› αp
D .
For confined photons in the cavity the shifts
are vice versa i.e., D has to be replaced by D-1 .
Figure 4 shows the reflection situation of at the left
mirror of a cavity in rest (left) and in movement (right). At the opposite
mirror the situation of pool- and confined photons is just vice
versa for both cases (not shown in fig.4).
Fig.4:
Space time presentation of confined- and pool photon reflections at
the left cavity mirror in rest (left) and in movement (right). Since momenta of photons are equal the mirror remains in rest or
in constant movement after reflection.
At a first glance the standing wave situation
in a moved mirror seems to be destroyed because of different wavelengths of
red- and violet shifted waves. However relativity contradicts which will be
demonstrated next:
A standing wave in a cavity is equivalent to a standing
wave in front of a mirror in rest. One wave moves to the right, the other
one to the left:
(42) F(x-ct) = eiω (t – x/c)
(43) F(-x-ct) = eiω (t + x/c)
Superposition of both waves gives (reflection at a
mirror):
(44) y
= F(x-ct)-F(-x-ct) = eiω (t - x/c) – e iω(t + x/c) = e iωt e –iωx/c -
e iωt e iωx/c
= e iωt (e-iωx/c
– e iωx/c ) = -2i eiωt sin(ωx/c)
a standing wave.
Lorentz
transformation: ct´= γ (ct- βx)
x´ = γ (x- βct)
(45) F´(x-ct)
= e iω/c [γ(ct-βx) – γ(x-βct)] = e
iω/c γ(1+β)·(ct-x) = e
iω/c·D · (ct-x)
wave violet shifted!
F´(-x-ct) = e iω/c [γ(ct-βx) + γ(x-βct)] = e
iω/c γ(1+β)·(ct+x) = e iω·D/c
· (ct-x) wave red shifted!
(46)
y = F(x-ct)-F(-x-ct) =
e iω/c γ(1+β)·(ct-x) - e
iω/c γ(1+β)·(ct+x) = e iω/c γ(ct+x) e iω/c γ(ct-x)β - e iω/c γ(ct+x) e iω/c γ(ct+x)β =
= e iω/c γct ·e
-iω/c γx·
e iω/c γctβ· e -iω/c γxβ - e iω/c γct ·e
iω/c γx·
e -iω/c γctβ· e -iω/c γxβ =
= e iω/c
γct ·e iω/c
γctβ· (e -iω/c γ(x-βct)
- e iω/c γ(x-βct))
=
= - e iω/c γ(ct-xβ) 2i·sin[ω/c γ(x - βct)]
or rewritten:
(47) y´ = - e iωt´
2i·sin(ω
x´/c)
gives the expression
of a moved standing wave.
A moved standing wave remains a standing
wave, although photons underlie Doppler shifts. Red- and violet shifts do
not imply a phase mismatch of the standing wave in a cavity, although
the lengths of the waves are different. Such conclusions will not appear if
consequently the real- and the complex term of the wave are considered in
calculation. This result is also in agreement with the well known invariance of
the wave number as well as with the invariance of the phase in SR [7].
In contrast to the behavior of free photons, confined
photons in a moved standing wave underlie length contraction
(transversal Doppler effect) and time dilatation just
as it is with moved masses. Therefore photons in a standing wave lose
characteristics like “free, non localized in an
area” as well as “undergoes longitudinal Doppler shifts”.
The over-all momentum- and energy situation in the
pool remains unchanged for a moved cavity. Reflected, Doppler-shifted pool
photons like αp D (see
in fig.4 right graph) have the same energy as if they approach from the
opposite side and penetrate the cavity (α*D represents this). Locally it
looks like a transfer from the pool- into confined photons and
then as a transfer back into pool photons:
(48) αp DLeft ← α*D ← αp DRight
αp D-1Left
→ α*D-1 → αp
D-1Right
Equations 48 give a proof, that the neutrality of the
pool distribution is not disturbed by the presence of a cavity, this is in rest
or in constant movement.
The demands on the quality of space, as sketched
shortly above, seems to be unrealistic. But in the field of QED, Casimir and Polder predict in 1949 zero point radiation in
space of a quality, which may be able to fulfill such demands [8]. The
space is filled with electromagnetic zero radiation of nearly unlimited energy.
Electrical conducting plates as shown in Fig.x
positioned in a distance L underlie apparent attraction forces,
resulting from the fact that inside the cavity only standing waves with λ
= 2L, L, ⅔L, ½L,… and outside in addition a pattern of
standing waves with λ = 4L, 6L, 8L, … is established. Therefore
the flux of photons outside the plates is larger then inside the cavity. The
computation gives:
(49) p = ħc π2/(240 L4) = hc π/(480 L4) = 1.3 10-3 Lμ-4 in N/m2,
where L
is the plate separation in Meter respectively Lµ in
Micrometer. Several authors could positively measure the so called
Casimir force at metallic foils positioned in a
mutual distance of micrometers [9].
|
|
|
|
|
|
Fig. 5: Two
conductive plates in a mutual distance L , where
the number of modes inside is lower then outside with the consequence that the
outside long wave modes generate a in-ward directed pressure to the plates
A cavity embedded in zero-point radiation will
experience a overshooting outer pressure coming from the long wave spectrum,
corresponding to L-4
, enormous at atomic dimensions.
The overshooting outer pressure may balance inside
reflecting thermal confined photons. The pressure of a confined
photon in a gap with distance L, wavelength λ = L/2, a non-relativistic
momentum transfer of
2h/λ during reflection and an area of approximately L2:
(50) pconfined =
½ hc/L4
A comparison with Eq.49 shows that the pre-factor in
Eq.50 is somewhat above 70, i.e. an expansion of
the cavity will take place. Roughly seen, a thermal fraction with wavelengths
above L must support the long wave zero point radiation:
(51) hc π/(480 L4) +
½ hc/L4 ·
(1-π/240) = ½ hc/L4
The stabilizing thermal fraction (second term in Eq.51)
could be the searched fraction of ultra long wavelength photon, which is a
possible candidate for mass attraction respectively gravitation see http://www.christoph-schultheiss.de/gravitational_mass_attraction
[10]. If this is true, a unified theory
between photonic mass and gravitation appears.
5. Behavior of confined photons and comparison with classical
mass
To close the aspects of a moved cavity the average
energy of one confined photon or of confined photons are
derived next:
Figure 4 demonstrates that confined photons in
a moved cavity underlie Doppler shifts, which are in equilibrium with Doppler
shifts of pool photons, so that the mirrors stay in rest or in constant
movement during reflections. Therefore the diameter of a cavity is an
invariant.
A confined photon has the momentum α*D-1 in
flight and α*D in counter flight direction. The behavior of
standing waves (see above) tells that this is true for a cavity in rest as well
as in movement. Thus the averaged energy contribution α* of
one confined photon is:
(52)
α* = ½ (α*D-1 +
α*D) = γ
α* ,
an expression,
which can be called as the kinetic energy of a confined photon.
Since there is no difference between momentum and
energy of a photon except of a constant, which is the speed of light, the
Einstein rest energy formula E = m c2 can be understood
as a scalar summation of all confined photon momenta
h/λ* .
The scalar momentum p* = m*c of such a mass is (n is the number of confined
photons):
(53)
m*c = ½ n
h/λ*·D-1 + ½ n h/λ*·D =
½ n h/λ*· (D + D-1) = n γ
h/λ*
Analogue to Eq.53 the resulting vectored momentum
of all confined photons give the resulting mass momentum:
(54)
m*ĉ = ½ n
h/λ*·D-1 - ½ n h/λ*·D =
½ n h/λ*· (D - D-1) = n βγ
h/λ*
The physical meaning of a scalar momentum m*c
of mass is: if all confined photons Σn h/λ*
escape in one direction, the resulting repulsion momentum is mc.
As can be seen in Eq.53 and 54, the left hand
properties like kinetic energy and momentum relate to a classical Newtonian
mass and the right hand terms relate to corresponding Doppler shifts of
counter-flowing confined photons in a cavity:
(55)
2γ = D-1 +
D
(56) 2βγ
= D-1
-
D
The expressions have the form of mass - confined
photon -equivalence equations.
Both models, namely the dynamic mirror mass model (see
derivation Eq.32 -39) as well as the above described pool photon model with
embedded cavity have charming aspects. However, later investigation shows that
only one confined photon per cavity is allowed in order to fulfill momentum
and energy law. This excludes the dynamic mirror model which demands minimum
two photons with a phase deviation of π.
In the next chapter it is assumed that a mass, modeled
by confined photons embedded in a photon pool, underlie a Compton Effect
with free photons. This leads to a complex interaction between three
light fast partners namely: free-, pool- and confined photons.
C.
Cavity velocity change and
discontinuous acceleration; polarization effect of pool
Let a free photon a collided with a
cavity in rest. This is a situation, which is described by a 180°
As can be seen from fig.1 and 3 a transition from rest
to movement is a transition from a cavity diameter L
to L/g. In
addition a phase relation for transition of a confined photon in rest
into one in movement has to be taken into consideration. A simple connection
frame is shown in fig. 5. It comes up, that after the start of the front mirror
the rear side of the cavity starts with the delay D with movement.
Fig.6:
Principle space-time frame of a discontinuous transition of a cavity in rest
into one of movement. Front and rear side of the cavity start movement at
different times. This generates relativistic length contraction of a moved
system.
By means of elementary geometry the transition time D can
be evaluated to:
(57) Δ
= L (γ-1)/βγ
In the transition from rest to velocity an already
mentioned phase correlation is necessary between the
Fig.7:
Phase conserving transition of confined photons of a cavity in
rest into a cavity in movement at a
The line through (½L-½L/γ,
½L) with the direction factor β-1 cross the
vertical line x = 0 at j:
(58) φ = ½ L - ½ Δct = ½ L[βγ
– (γ-1)]/βγ
This means, either the collision with free-photon only
takes place if the j -phase relation with confined photons fits
exactly accidentally, or confined photons leave the cavity and become
ex- changed by pool photons of equal momentum but with correct phase
relation. In fig. 7 the exchange of confined- with pool photons
in order to restore phase correlation is sketched. Of course leaving confined
photons convert into pool photons and entering pool photons
become confined one.
|
|
|
|
|
|
Fig.8:
The transition of a cavity in rest into a moved one demands an exchange of two confined-
with two pool photons in order to
conserve the mutual phase relation between confined photons in rest and
in movement
Let’s start the investigation with the
assumption that two confined photons α* will be emitted into
the pool and convert to pool photons. This emission does not change the
isotropy of pool photons. But, the absorption of a violet-shifted pool
photon αp D-1
and the red-shifted pool photon αp D by the cavity release corresponding pool photons
in counter flight direction. The consequences on the neutrality of the pool are
investigated next:
Equation 59 is the momentum frame before the
reflection. All involved photons are listed and ordered in free-, confined-
and pool photons. When a collision happens, one half of the confined
photon moves to the right (notation R) and one half to the left (notation
L). A rearrangement of the terms after the reflection is sketched in
Eq.60. If only one confined photon populates the cavity then n =
½ and one of the notations (left or right) must be cancelled. .
(59)
“momentum before”
Right Photons Left Photons Right “violet”
Left
“violet” Right “red”
Left “red”
←―→ ←—→ ←——―→
←——―→
←——―→ ←——―→
α + nα*-nαp - nα*+nαp + nαp
D-1-nαp D-1 + nαp
D-1 -nαp D-1
+ nαp
D -nαp D
+ nαp
D -nαp D
- αD
↔ ←—――—――→ ←——―→
←——―→
←——―→ ←——―→
Free Confined Photons 0
0
0
0
←—―—―—――—―—―——―—――———――――――—→
Pool Photons
(60)
“momentum after”
Right Photons
Left Photons
Right
Left
Right
Left
Right
Left
←——―→ ←——―→
←→ ←→ ←→ ←→ ←—→ ←—→
-αD + nαp
D-1 -nαp D – nαp D +nαp
D-1 + α - nαp
D-1 -nαp D-1
+ nαp D + nαp D-1 + nα* -nαp
- nα* + nαp
↔
←—―—―—―—―——―→
←———―—―→
Free
Confined Photons
0
←—―—―—――—―—―――——―—―—―—―→
Pool Photon
Since in Eq.59 and 60 momentum
law is conserved for free- and cavity photons, the terms of the pool have to be
equal too. The result is (α´ =
α D):
(61) α = - α´ + 2nα*(D-1-D)
←—→
2β´γ´
a momentum
equation with a pre factor of 4 n instead of 1 i.e., solutions with 2 (n=1), 4
(n=2) etc. confined photons contradicts to the momentum equation. However if
the cavity contains only one confined photon (n=½) we have:
(62) α = -α´ + 2 ½ α*
[2β´γ´]
α
= -α´ + β´γ´
which is the exact
momentum equation since for one confined photon . This means exactly one confined-
and the counterpart namely one pool photon are involvd
in the cavity frame.
This simple model tells that in the cavity two
half-waves move in opposite direction. The confined photon is in the
ground state respectively it represents the zero-point energy of the
cavity.
The appearance of a polarization effect in the pool
can be stated, depending on the flight direction of the confined photon during
collision. Terms of Eq.59 and 60 are sketched in fig.9.
Fig.9:
Anisotropic spread of pool photons after a
To close the consideration, the equations 61 and 62
can be written in the energy form. There all terms are positive.
(63) “before”
(64) “after”
Since in Eq.63 and 64 the energy law is conserved for free-
and confined photons, the terms of the pool have to be equal too. The
result is:
(65)
With the notations analogous to Eq.62 and n = ½
the energy law results
(66)
This result again confirms the validity of the single confined
photon frame (for n = 1 Eq.65 gives 
,
which is wrong!).
As can be seen in the pool term of Eq.59 the pool is
not neutral before the reflection, since it contains 
,
approaching from the right to the cavity. Fig.8 tells that after the reflection
for the case n = ½ (only one confined photon) the pool either
emit 
or 
,
depending on the flight direction of the confined photon. In a statistical
average – if locally many 
- terms
vanish and 
is valid.
The cavity seems to emit pool photons in both
directions (see fig.9) with a little increased strength in backward direction.
This can be seen for 
:
(67)
Therefore in summary on can state: during a
D.
Conclusion
As mentioned above and as can be seen in Fig.9 during
a
E.
Acknowledgement
I would like to express my gratitude to
Anselm Citron (University Karlsruhe, Germany), who
was very interested in the photonic model of mass and supported the development
of the theory during many discussions decisively.
[1] E. Schrödinger, Physik. Zeitschr. XXIII, 1922, p. 301, in German
[2] W. Cantor, Stroboscopy
Letters, 4 (3 & 4), 1971, p. 59
[3] R. Kidd, J. Ardini, A. Anton, Am. J. Phys. 53
(7) July 1985, p. 641
[4] D. S. Lemons, Am. J. Phys. 59 (11), November 1991
[5] D. Wilkins, Am. J. Phys. 60 (3), March 1992, p.221
[6] H. B. G. Casimir, Koinkl. Ned. Akyd. Wetenschap. Proc. 51, 793 (1948)
[7] M. Born, „Die Relativitätstheorie
Einsteins“, Heidelberger Taschenbücher, Springer Verlag, Berlin, Göttingen,
Heidelberg, 4. Auflage
1964, p. 257, 259, in German
[8] see ref.7
[9] S. K. Lamoreaux, Phys. Rev. Lett. 78, 5 (1996)
[10] Main subject of
the paper http://www.christoph-schultheiss.de/gravitational_mass_attraction
is the assumption that an intense isotropic ultra long wavelength (ULW) photons
distribution exists in space (a comparable small thermal spectrum over Casmir’s
zero point distribution) which causes virtual attraction forces by means of
electromagnetic scattering. However the intensity of the ULW electromagnetic
radiation in this theory can vary in a wide range. With the cavity
stabilisation formula (see Eq.50) an estimation is possible: The momentum per second (in 4π)
an elementary particle (proton) experience is about 3/2 · h/L4
· L/c ~ 104 . In the “mass
attraction” paper the momentum flow is given by: Photon flux j times
momentum of a ULW photon (10-60 kg m/s) i.e. j · 10-60.
The comparison gives: j is about 1064. Such a value indicates that
the cross section for
gravitational electromagnetic scattering is determined by the very tiny value
of the Planck length.