**Journal of Modern Physics
**Vol.5 No.9(2014), Article ID:47366,16 pages
DOI:10.4236/jmp.2014.59092

The Light as Composed of Longitudinal-Extended Elastic Particles Obeying to the Laws of Newtonian Mechanics

Alfredo Bacchieri

University of Bologna, Bologna, Italy

Email: abacchieri@libero.it

Copyright © 2014 by author and Scientific Research Publishing Inc.

This work is licensed under the Creative Commons Attribution International License (CC BY).

http://creativecommons.org/licenses/by/4.0/

Received 16 April 2014; revised 12 May 2014; accepted 7 June 2014

ABSTRACT

It is shown that the speed of longitudinal-extended elastic particles, emitted during an emission time T by a source S at speed u (escape speed toward the infinity due to all the masses in space), is invariant for any Observer, under the Newtonian mechanics laws. It is also shown that a cosmological reason implies the light as composed of such particles moving at speed u (function of the total gravitational potential). Compliance of c with Newtonian mechanics is shown for Doppler effect, Harvard tower experiment, gravitational red shift and time dilation, highlighting, for each of these subjects, the differences versus the relativity.

**Keywords:**
Escape Speed, Harvard Tower Experiment, Time Dilation, Redshift, Doppler Effect,
Compton Effect

1. Introduction

Here we present a solution, in accordance to the Newtonian mechanics, to the apparent constancy of c, based on following assumptions:

1) Gravity fields fixed to their related masses (intending that each field is moving together with its generating mass).

2) Finite mass of the universe, implying a finite value of U (total gravitational potential) and therefore of u (escape speed from the universe due to all the masses in space).

3) Light composed of longitudinal-extended elastic particles (as defined on §4) moving at speed c = u. This equality is supported by a cosmological reason, see §2.

On above bases (including, needless to say, Newton’s absolute time and space) we find:

a) The relation between u (total escape speed) and U (total gravitational potential), giving to the speed of light the cosmological reason of its value.

b) On Earth, the variation of u, (and therefore of c as per assumption III), due
to the variation of U (mainly caused by the variable distance Earth-Sun) is, during
one year, Δu (=Δc) ≤ ±0.05 m×s^{-1}, hence within the accuracy of the measured
value of c.

c) The invariance of the measure of c for any Reference frame under the Newtonian mechanics laws.

d) The longitudinal, generic and transverse Doppler effect for longitudinal-extended elastic particles, as defined, and their physical characterization.

e) As for the Harvard tower experiment [1] -[3] , regarding the variation of frequency (or wavelength) between a source (of gamma rays) and an absorber at different height, our relations give a shift equal to the observed and also predicted by the Relativity. Anyhow, with the source on the base (of the tower) the light arriving to the top has, as for the GR, a lower frequency, whereas on our bases, is the length of our particles which decreases (together with c); on the contrary, with source on the top, GR predicts an increase of the frequency of the light arrived to the base, whereas we show that, during the same path (top-base), is the length of these particles which increases (together with c), giving a red shift. Moreover, as for the value of the compensating speed source-absorber, (necessary to restore their resonance), we point out that the experiment did not give any clear indication about the effective direction of this speed. Indeed, scope of that experiment was to “establish the validity of the predicted gravitational red shift” [2] , hence the only value of this speed was taken in consideration; here, on §6, we show that, on our bases, the effective direction of this speed is contrary in both cases (source on top or base), to the one predicted (but not verified) by the Relativity.

f) As for the gravitational time dilation, on §6, it is shown that taking a source (of light) in altitude, it yields a negative variation of c as well as a negative variation of the frequency n inducing atomic clocks to run faster; moreover, through our Equation (29) regarding a source circling (around the Earth), we obtain, see (46), the exact variation of the ticking time of GPS system.

g) As for high red shifts related to far sources, we show that, disregarding the relative motion Earth-source, they depend on the increase of c (as well as the increase of the length of the said “longitudinal-extended elastic particles”) during the path of light toward higher (in absolute value) potential; on §7, Table 1, we give the values of c (on these far sources) related to the observed red shifts.

h) Our Equation (17), (regarding our Doppler effect for the light), applied to the Compton effect (indubitable Doppler effect), gives, see Appendix A, the Compton equation, which cannot be obtained through the relativistic Doppler effect equations.

2. Total Escape Speed (from a Point toward the Infinity) Due to All the Masses in Space

As known, considering in space one only mass M (regarded as a point-like), the gravitational
potential U acting on a particle having mass m = M, assuming U_{¥}
= 0, with s the distance M-m, is U = −MG/s; this relation, according to our
first assumption (I), is always valid in spite of any reciprocal motion between
M and m. The related Conservation of Energy (CoE), E = U + K, (where K =
represents the unitary, i.e. for unit of masskinetic energy of our particle arriving
from the infinity, where u_{¥} = 0), for
E = 0 gives U = −K, leading to

(1)

which is a scalar, (called escape speed), representing (in the considered point) the value of the velocity u, any massive particle, under a potential U, needs to reach the infinity, so u (escape velocity)must be referred to M.

Considering now two masses M_{1} and M_{2}, having, at a given time,
distances s_{1} and s_{2} from a considered point (we may call it
Emission point E_{p}), the potential U_{1,2} in E_{p} becomes

(2)

Now, the escape speed from two masses can be written

(3)

which is the value, in the considered point E_{p} of the (escape) velocity
u_{1,2} which has to be referred (at the considered time), to the point,
we may call it Centre of potential (C_{p}), where |U_{1,2}| has
the max value. Then, as

and we also get

(4)

therefore the escape speed due to all the n masses in space becomes

(5)

with
the universe mass,
the total gravitational potential in the considered
point E_{p}, and where u (function of U in E_{p}) can be called
as total escape speed (toward the infinity), while the escape velocity u is referred
to the centre C_{p}. Indeed, any unitary massive particle during its path
toward the infinity, has to comply with the CoE, U + K = 0, where K =
^{ }giving to this particle a speed u

(which depends on the location of the source) and yielding, for all the masses, the total energy equal to zero [Compliance of light with above relation E = U + K, is shown on Appendix B].

We assume now the equality c = u, hereafter supported by the estimated mass of universe and also by a cosmological reason: in fact, if c > u the energy of light will be lost forever and furthermore the observable masses, following the always increasing mass of light going toward the infinity, will also tend to the infinity moving away from each other. On the contrary, if c < u, all the masses in space (having speed lower than u), will tend to a gravitational collapse, whereas for c = u, the mass of light, tending to the infinity in an unlimited time, will avoid the two said events (collapse or dispersion).

Now the mass of universe, by some authors, is estimated [4]
-[6] to be
kg; the same order of magnitude is given through the number () of observable stars
[7] [8] , and since
from Earth the distribution of the masses appears to be homogeneous and isotropic,
under our assumption U_{¥} = 0, we may assume
their density as decreasing toward the infinity like a function ρ = ρ_{c}e^{−}^{as}
with
^{ }kg/m^{3} the critical
density [9] . So the mass of universe can be written

(6)

yielding

(7)

On Earth, the variation of potential due to an increase of the distance ds, can
be written as dU = −dmG/s where dm = ρ4πs^{2}ds with ρ = ρ_{c}e^{−}^{as},
hence the potential on Earth becomes

(8)

Now, according to (5), on Earth it is

(9)

Therefore, on Earth, u_{o} = c_{o}, so that
and, in general we may argue

(10)

The equality c = u, which implies the massiveness of light, means that, along any free path, the speed of light only depends on the value of the potential along that path.

[As for the relation, the Harvard tower experiment has shown that the fractional change in energy

(of light) is given by δE/E = −gh/c^{2}, and since the term gh is
the variation of potential from the ground to the height h, we may guess that c^{2}
has to be related to the total gravitational potential, as also shown on §6].

3. Annual Variation, on Earth, of the Total Escape Speed

On Earth a small variation of the total escape speed u_{o}, from (9), can
be written as

(11)

where ΔU is the variation of the total potential on Earth, mainly due to the variable distance Earth-Sun.

So considering the eccentricity e (=0.0167) of Earth’s orbit around the Sun, between
their average distance d (=1.5 ´ 10^{11} m) and their shortest distance
(Perihelion) p = (1 – e)d, and with u_{0} = 3 ´ 10^{8} m×s^{-1},
the (11) gives

(12)

with Δu_{e} the variation of u due to Earth’s orbit eccentricity, ΔU_{S}
the variation of potential on Earth due to Sun between the two said distances, with
M_{S} the mass of Sun. Hence from Aphelion to Perihelion, one should find
Δu_{AP} (=Δc_{AP}) = +0.10 m×s^{-1}and we note that
this variation is compatible with the accuracy of the measured value of c = 299792458
m×s^{-1}. Due to Earth’s rotation, there is also a daily variation which,
from midnight to noon, is of the order of; so, on Earth, u_{o}
is practically constant during one year, as it is for the measurements of the speed
of light.

4. Invariance of c for a Particular Particle, Here Defined, and Related Doppler Effect

Here we show that the Galileo’s velocities composition law, (related to point-particles), cannot be correctly applied to a particle, (hereafter called photon), defined as follows:

“Longitudinally-extended, elastic non divisible particle emitted at speed u by a source during an emission time T, and moving along one ray (continuous succession of photons), where two consecutive photons cannot be separated along a free path (constraint of non separation)”.

Of course, more photons emitted during an emission time T need an equal number of rays.

Calling front and tail the extremities of a photon, the constraint of non separation implies that, along a ray, any tail corresponds to the front of the next photon.

Referring to Figure 1(a) (where E_{p} is
the location of S at t = 0 and S_{T} its location at t = T), since the escape
velocity c (=u) of an emitted photon (AB) is referred to the Centre of potential
C_{p}, during its emission time (0 ≤ t ≤ T), the term v_{CpA} =
u should appear as the velocity of its front (A) from C_{p}.

The source S may have a velocity v_{CpS} from C_{p}, thus writing
v_{CpA} = v_{CpS}+ v_{SA} we should find v_{SA}
= u − v_{CpS}; this means that each photon emitted around the source
should have a length λ' = |v_{SA}T| = |(u − v_{CpS})|T depending
on v_{CpS}, but this is contrary to the experience showing that if the source
is fixed to its initial Emission point E_{p}_{ }(that is the point
where S is located at the start of the emission) the emitted photons, referring
to E_{p}, have equal characteristics. Thus, during the emission of a photon,
the velocity of its front, (to comply with these equal characteristics), has to
be referred to the initial Emission point E_{p}, therefore, see Figure 1(b), where E_{p} is our reference frame,
as for the front A, for definition, we have

(13)

[This condition also allows the whole photon to have a velocity u referred to C_{p},
as shown on Figure 1(d)].

Now the velocity of the front A, with respect to S, from (13), becomes

(14)

and still referring to Figure 1(a), (where S_{T}
is the location of S at t = T), should S be fixed to E_{p} (that is v_{EpS}
= 0), the length λ of each photon, after the emission time T, from (14) becomes
λ = v_{SA}T = uT, while, in general, it is

(15)

where
is the photon AB emitted with the source in motion from E_{p}.

Referring now to Figure 1(c), if a generic Observer O is our Reference frame, we can write

(a)(b)(c)(d)

Figure 1. (a) Photon AB
emitted under the supposed condition v_{CA }= u; (b) Emission of a photon
AB referred to the initial Emission point E_{p}; (c) Emission of a photon
AB referred to the generic Observer O; (d) Measurement of the speed of a photon
(AB) reflected by O_{1}.

(16)

where
is the photon emitted while the source is in motion, with velocity v_{OS},
from the Observer, and once more, if v_{EpS} = 0 (S fixed to E_{p}),
we find λ' = λ = uT (If S is now our Reference frame, and v_{EpS} is the
velocity of S from E_{p}, we still have the (15)).

Thus, after the emission time T, as for a source receding from the front of the considered photon, as in Figure 1(b) (or Figure 1(c)), the length λ' (for any Observer) turns out to be

(17)

where v (=|v_{EpS}|) is the speed (referred to E_{p}) of the source
S (along the direction E_{p}S), Δλ (=vT) is the path covered by S during
T, and where β = v/u, and we point out that the length λ' may change, along a free
path, and under constant potential, only during its emission.

Now, the speed of a point-particle is defined through two Observers, while the speed u' of a photon, because of its variable length during its emission, does not correspond to the speed of any point of it, hence we must consider its length referred to the time T' (transit time) the photon (front to tail)needs to cross one Observer, so it has to be defined

(18)

[As for this definition, let us consider a system composed of two balls connected through an elastic thread and let them fall in vertical line: during the fall, each part of the system has different speed, so we define the speed of the whole system according to Equation (18)].

Returning now to Figure 1(c), for the Observer
O, the transit time T' of the photon AB is given by the time the front A spend to
cover the path λ, that is T(λ/u), plus the time the tail B needs to cover the path
S_{T} − E_{P} = Δλ; now, once the photon AB has been emitted
(at t = T), the velocity of the front A has to be the same as any other part of
the emitted photon, hence the time needed by B to cover the path Δλ is ΔT = vT/u,
giving

(19)

Now, according to (18), the speed of the photon AB, referred to O, becomes

(20)

showing that the speed of photons emitted by a source S is invariant for any Observer, in spite of any speed of S with respect to the Observer [After the emission, each part of the photon has same velocity u, meaning that, during the emission, it is the velocity of its inner part to vary in order to change its length in the given time T].

As for an emitted photon, the measurement of c (through the method d/t) implies
its absorption and reflection by an Observer. In this way, the Observer becomes
the source of a new photon, with the Observer/Source located in the Emission point
E_{p}, so we may refer to Figure 1(b),
with the source fixed in E_{p}, finding u' = λ/T = u.

[Anyhow, we may obtain the same result (u' = λ'/T' = λ/T = u) as follows:_{}

the measurement of c (through the method d/t) implies two Observers at a constant
relative distance O_{1}O_{2}; on these bases, see
Figure 1(d) where C_{p} is now our Reference frame, after the reflection
of the photon from O_{1}, at t = T, the path covered by the front A to reach
O_{2T} is given by O_{1}O_{2T} that is λ¢ = λ
+ Δλ where λ is the length of the emitted photon AB and where Δλ = v_{Cp}T/c,
with v_{Cp} the speed of our frame O_{1}O_{2 }with respect
to C_{p}, yielding λ' = λ(1 + β) where β = v_{Cp}/c. The time needed
by the front A to cover the distance O_{1}O_{2T} is T' = T + Δλ/c
= T(1 + β), thus the measured speed (referred to the two Observers) becomes c' =λ'/T'
= λ/T = c in spite of any velocity of the co-moving Observers O_{1}_{
}O_{2}_{ }with respect to C_{p} (Anyhow, the Observer
O_{1} could state, for the front A, a velocity
different from u, if he could measure such a speed)].

For any Observer, the frequency of photons of the same ray has to be defined as n' = n/t with n the number of photons crossing the Observer during a time t; for t = T' (transit time of one photon), it is n = 1, thus n' = 1/T', so from (19) we get

(21)

showing that for v = 0, that is β = 0, we have n' = n, which is also valid if the
Observer (O) and the source (S) belongs to different potential: in fact, for O and
S at reciprocal rest, the number of photons emitted by S in a unit time has to be
equal, in the same time, to the number of them crossing O (like, for instance, the
number of balls falling from the top of a tower with respect to an Observer at the
tower base), and this implies n_{s} = n_{o}.

Now, the Figure 1(c), where a source emits a photon while it is in motion from the Observer O, also represents a longitudinal Doppler effect, which, in general, can be written a

(22)

with the sign + for S receding from the Observer, while the sign – is for S approaching it.

Hereafter we get our equations regarding both the generic and the transverse Doppler effect, followed by our relations regarding a source (of light) circling around an Observer.

To get a general relation for the Doppler effect, let us consider, see Figure 2(b), referring to the Observer O, a source S, located
in E_{p} (at t < 0), at rest with O. During this time let S emit photons
having length λ (=uT) and let E_{p}O = λ. Then, at t = 0, let S start to
move from E_{p} toward S_{T} (reached at t = T), with velocity v
(referred to O) along the generic direction a-a. Now, during the path E_{p}S_{T},
let S emit a photon λ¢ toward O. (On Figure 2(a),
the small arrow inside the triangle E_{p}OS_{T} represents the partial
λ¢ during its emission.) At t = T (end of emission), according to (16) we have
λ¢ = λ ‒ vT, thus the length of λ¢, assuming v = u, so to consider
E_{p}O = NO, with E_{p}N _{┴} S_{T}O, becomes

(23)

As for the transit time T', as before, we can write.

which can also be obtained considering that the front of λ¢, following
the tail of λ (thus directed toward O), takes a time T to reach O from E_{p},
while the tail of λ¢, emitted in S_{T}, has to cover the path S_{T}O
= S_{T}N + NO, spending the time ΔT = (vTcosα)/u for the path S_{T}N,
plus the time T for the path NO (equal to E_{p}O for v = u), giving

(24)

yielding

(25)

thus (For an opposite direction of S we get λ' = λ(1 ‒ βcosα) and T' = T(1 ‒ βcosα).

[Ray S_{2T}FO: referring now to Figure 2(b),
if S, between T and 2T, is still moving with same velocity v, the emitted photon
will have same length as, thus its front (at t
= 2T) will reach a point F, (corresponding at the same time to the tail of), at a distance
from O, so the ray Source-Observer, at t

= 2T, becomes the line S_{2T}FO].

Transverse Doppler effect: referring to Figure 3(a) left part, where, should S start to move at t

= 0 from E_{p} to S_{T} while emitting the photon, according to (16) we
can write

(valid for S approaching O) (26)

(a) (b)

Figure 2. Doppler effect for a photon, general case.

(a) (b)

Figure 3. (a) Transverse Doppler effect; (b) Source circling around the Observer O.

where
is the length corresponding to S_{T}O, while λ corresponds to E_{p}O.
On the contrary, see Figure 3(a) right part (where
the source is receding from the Observer), it will be

(valid for S receding from O) (27)

Then, hence.

Regarding a source circling around an Observer O, on Figure
3(b) the line E_{p}O represents a succession of photons λ already
emitted when S is fixed in E_{p}, while E_{p}F represents the last
of them (or it could represent the last photon emitted by S when reaching E_{p}).
Then, at t = 0 let S start to move from E_{p} with velocity v toward S_{T}.

Now, because of the constraint of non separation, the front of the first photon
emitted when S is moving between E_{p} and S_{T}, has to reach,
in F, the tail of previous photon, so, according to (16) the length of every photon
(emitted while S is moving along the orbit r) will be

(28)

thus

(29)

with r the orbit radius, ω the angular speed, giving to any whole photon the speed c' = c.

Figure 3(b) also shows a path (λ_{1}-λ_{4})
of a ray directed toward O (the lines connecting the photons λ_{2} and λ_{3}
to the orbit give the point where the source is located at the end of their emission).

5. Physical Characterization of These Photons

Now, similarly to a fluid flowing in a pipe (whose kinetic energy is K =
mv^{2} with m the mass passing in 1 s)the kinetic
energy of light flowing along one ray (according to our definition, photons are
also massive), has to be expressed with K_{c} =
mc^{2} with m the mass of the particles
passing in 1 s along one ray. Anyhow, the total energy of light flowing along one
ray is E = mc^{2} as also proved by the evidences of nuclear reactions like
n + p ® d + γ: indeed, in this reaction [10] , the lost mass, known through mass spectrometers, corresponds
to the value m = E/c^{2} where E (=hc/λ), (as λ is measured), is also known,
so E = mc^{2} represents the total energy of light flowing along one ray
(λ_{meas} is obtained [11] through the
value λ_{meas}/(d_{220}) given at pag. 369, where (d_{220})
is given at page 410).

So, writing E =
mc^{2} +
mc^{2} we may infer that each of these
particles is provided with an internal energy (K_{i} =
mc^{2}) equal to its kinetic energy.
Now, equating mc^{2} to
we get

(30)

where m written as

(31)

is the mass of light, with frequency, passing along one ray in 1 s, while the constant

(32)

is the mass of light passing along one ray during T, we may call it “mass of one photon”; so one finds

(33)

and therefore the Planck’s constant represents the energy of one photon. The energy of these particles passing in 1 s along one ray (energy of one ray of light) can now be written as

(34)

On the above bases, the total energy of light emitted by a source is given by n_{r}mc^{2}
with n_{r} the number of rays, and since m is the mass of light passing
along one ray in 1 s, this unitary (for unit of time) energy shall be equal to the
supplied power P during 1 s, thus n_{r}mc^{2} = P, hence the total
mass lost per second m_{T} (ºn_{r}m) by a source of light becomes

(35)

So, for a 1 W lamp, we get m_{T} = P/c^{2} @ 1.1 ´ 10^{-}^{17}
kg×s^{-}^{1}, while the number n_{r} of rays is

(36)

in our case, n_{r} @ 3 ´ 10^{18} rays. We point out that for
a given power P, the higher is the frequency, the lower is the number of rays, as
shown by (36) written as
= P/h. The number of photons emitted in 1 s becomes:

(37)

which, for P = 1 W, gives n_{γ} = h^{‒}^{1} (=1.5
´ 10^{33} photons/s), so the inverse of Planck constant corresponds
to the number of photons emitted in 1 s by a source of unitary power (This great
number of photons (having emission time T at speed c) can be regarded as a wave
function).

Now the momentum of the photons passing along one ray in 1s, considering their kinetic
energy only, that is K_{c} =
mc^{2}, according to Newtonian mechanics
should be written as

(38)

but considering both their kinetic and their internal energy, that is E = mc^{2}
we obtain

(39)

6. Revisitation of the Harvard Tower Experiment and Time Dilation

Referring to Harvard tower experiment [1]
-[3] , simply
represented on Figure 4, where h is the tower height,
calling c_{0} the value of c on Earth’s surface at the tower base and c_{h}
its value on its top, the variation c_{h} ‒ c_{o}) from the
tower base to its top, from (11), for c = u, becomes

(40)

where
is the variation of the total gravitational potential U, due to Earth, from the
tower base to its top. As
and
where M_{E} is the Earth’s mass and r its radius, we get
= M_{E}Gh/r^{2} where h (=r_{h} ‒ r_{o}) is
the tower height, yielding

(41)

showing that, on the top, where |U_{h}| < |U_{o}|, it is c_{h}_{
}< c_{o}, with.

(a) (b)

Figure 4.
Harvard tower experiment scheme, with the source at the base. (a) S and A at rest
at a different level h. In A, λ_{h} < λ_{o}, so the detector observes
a gravitational blue-shift; (b) Source and absorber relative motion (v) to compensate
the gravitational blue-shift through the Doppler effect.

Now, let S be a Mossbauer source and A an appropriate absorber; if they are close to each other (for instance, at the tower base), the absorber is in resonance with the source.

Then, see Figure 4(a), with S at the tower base
and taking A to its top, while S and A are at rest, the frequency of the emitted
photons (i.e. the number of photons emitted along the direction SA per unit of time)
has to be equal to the photons received by A, that is n_{h} = n_{o}
and since c_{h}_{ }< c_{o}, it must be λ_{h} < λ_{o}
(indeed Δλ/λ_{o} = Δc/c_{o}), so, contrary to ToR, a blue-shift
effect for A.

[On Figure 4(a) (photons arrived to the top), according
to n_{h} = n_{o}, it seems to be E_{h}/E_{o} = hn_{h}/hn_{o}
= 1 (here h is the Planck constant), but the (33) shows that h = γc^{2}
with γ (representing the mass of light passing during T along one ray), an effective
constant, so that we get E_{h}/E_{o} = (c_{h}/c_{o})^{2}
which shows a decrease of the energy of light from S to A].

Indeed, with S on the base emitting toward A on top, A goes out of resonance and
since on our bases n_{h} = n_{o}, the non-resonances physically
related to a variation of λ, whereas in the Harvard tower experiment [3] , “no mention has been made
of frequency or wavelength”.

Thus, to restore the resonance through the Doppler effect (i.e. to increase the
photon length fromits value λ_{h} in A to its initial value λ_{o}
in S), since λ_{h} < λ_{o}, A and S, see
Figure 4(b), have to recede from each other with speed v complying with
(17), here written λ_{o} = λ_{h} + vT, giving (λ_{h} ‒
λ_{o})/λ_{o} = ‒vT/λ_{o}.

Therefore, since Δλ/λ_{o} = Δc/c_{o} (as n_{o} = n_{h}),
we find Δc/c_{o} = ‒vT/λ_{o} and comparing to (41) we get, so the relative speed between S and A becomes

(42)

[This value is also predicted by General Relativity (GR)which, implying a decrease of v for light moving from the base to the top, predicts an opposite direction of v with respect to the one shown on Figure 4(b); at this regard, Pound-Rebka [3] operated in order to determine (through the value of v, obtained moving the source sinusoidally) the variation of energy of a beam on the upward and downward path, without any indication (because of the low value of v), about the direction of the compensating speed].

Now, if we take S to the tower top, with A located on its base (see Figure 5 which is referred on our bases), the experiment shows that the absorber goes out of resonance.

Now, according to Relativity, taking S to the top, the initial frequency of the
light should be n_{h} = n_{o}, which, on our bases, is wrong: with
S on the top, see Figure 5(a), the (10) written
as c^{2} + 2U = 0, between top and base gives, where U_{h} ‒ U_{0
}= gh, giving, then, and since 2gh

= c^{2} we can write c_{o} = c_{h}(1 + gh/c^{2}),
that is c_{h} < c_{o} as showed by (41), but what about n_{h}
and λ_{h}?

Well, referring to previous Figure 4(a), with source
S on the base, the length of photons arriving to the to pvaries from λ_{o}
to λ_{h} (with λ_{h} < λ_{o}), therefore if S has been taken
now to the top, should their initial length be λ_{h}, at their arrival to
the base, their length should be λ_{o}, and since the resonance, as seen,
depends on λ, the Absorber A (on the base, see Figure 5(a)),
should be now in resonance. Thus we can argue that taking the source on top, the
photons initial length has to be λ_{h} (=λ_{o}); then, as c_{h}
< c_{o} as shown by (41), it must be n_{h} < n_{o},
and in particu lar, according to (41) we get, giving
which is the same as the one predicted by GR, but on our bases, this variation is
due to a different initial frequency (as the source is now on the top), whereas
for GR this variation is related to the path of the emitted light between two different
levels. (Indeed on our bases the frequency remains constant during any path, should
source and observer be at recipro-

(a) (b) (c)

Figure 5.
Harvard tower experiment scheme, with the source on the top. (a) S on top, S and
A at rest: c_{h}, n_{h}, λ_{h} are the photons initial parameters
on the tower top; (b) S and A at rest. When photons reach the base, λ_{h-o }
> λ_{o}, so A observes a g-redshift; (c) S and A relative motion to compensate
the g-redshift via Doppler effect.

cal rest).

Thus, see Figure 5(b), with S emitting from the
top, S and A at rest, when the photons reach the base, as their final frequency
(n_{h-}_{0}) will remain the same as the initial one (that is n_{h-}_{o}
= n_{h}) and since c_{o} > c_{h}, it turns out that, along
the path SA, λ will increase, and its variation, opposite to the one given by (41),
yields now

, giving.

Now, as λ_{h-}_{o} > λ_{o}, the absorber, on the base, will
observe a gravitational red-shift so, to compensate it via Doppler shift, see Figure 5(c), S and A have now to move relative
to each other in order to decrease the final length λ_{h-}_{o} to
the resonance value λ_{o}; on the contrary, according to ToR, A and S should
recede from each other.

[Still referring to Figure 5, taking S to the tower
top, we have c_{h} < c_{o} and n_{h} < n_{o} implying,
contrary to GR, a decrease of the energy of light to be emitted by S,
, see (34), and therefore when these photons reach
the base (where c_{h-}_{o} = c_{o}) their energy becomes
giving a reason to the loss of energy as for light arriving to Earth coming from
sources located in points where |U_{S}| < |U_{o}|, and since, as
seen,

, we also give a cosmological reason to the high
redshift of sources where |U_{S}| = |U_{o}|.

Time Dilation

Well, the experience shows that, on board of GPS satellites, the atomic clocks run faster by about 38 μs/day than the ones on ground, meaning that, in altitude, their ticking time, (or interval time, intending the minimum time counted), is shorter than the one on ground.

Now, the ticking time t of atomic clocks is proportional to their frequency, so
on ground we can write t_{0} = kv_{0} while in altitude t_{h}
= kn_{h} yielding Δn/n_{o} = Δt/t_{o} where Δt (=t_{h}
‒ t_{o}) is the ticking time variation from ground to height h, with
Δn (=n_{h} ‒ n_{o}) representing their frequency variation
due to the gravitational potential variation.

Now, taking the sources (clocks) from ground to height h, the length of their photons,
at emission, remains constant, (λ_{h} = λ_{o}), thus, because of
the variation of c (from ground to height h), it has to correspond an equal variation
of n, so that the (40) can be written as

(43)

Now, GPS satellites have an orbit of r_{h} 26,600 km, that is an altitude
h 20,200 km, as r_{o} 6400 km is the Earth’s radius. Hence, the (43), because
of the variation of the potential, the variation of the counted time during one
day (Δt_{1d}), since in one day t_{1d} = 86,400 s, gives

(44)

where the sign means that the ticking time is decreasing, inducing the clocks to run faster. Then we have to take into account that the parameters of the photons emitted by atomic clocks on board of GPS satellites are changing because they are circling around the Earth.

Therefore, according to (29), that is T' = T(1 + β^{2})^{1/2} where
T' is the time a photon needs to cross the Observer, during one day (86,400 s),
since the orbital speed corresponds to two orbits every day (giving v = 2(2πr_{h}/86,400)
= 3870 m×s^{-}^{1}, and considering that for v = c we can write
(1 + β^{2})^{1/2} @ 1 +β^{2}/2, we get

(45)

representing the variation of the counted time in one day due to the orbital speed of GPS satellite, and since this variation is positive, it has to be deducted from the negative one due to the potential variation, thus the total variation of the counted time on GPS satellites, in one day, becomes

(46)

as observed. This equality also confirms that λ_{h} = λ_{o} as for
sources in altitude.

7. Red Shift

According to the Relativity, the gravitational red shift of light coming from the
Sun, with M_{S}_{ }and R_{S} its mass and radius, is. Now, for
Mpc, with s the distance Earth-source, the observed shifts are in the range
[12] , while from
to
Mpc they tend to become always positive, and between
and
Mpc (Bly) the red shifts (here in the range), practically follow the empirical
Hubble’s law z = H_{o}s/c; hence, since the value of the gravitational red
shifts of a typical galaxy, intended to be
should be of the order of 10^{-}^{9}, the Doppler effect appears
to be (as for the Relativity), the only satisfactory way to explain the observed
blue shifts and also the high (cosmological) red shifts.

On the contrary, on our basis, disregarding any motion between a source and an Observer
on Earth, which implies (as showed on §4) n = n_{o}, we get, where n_{o},
c_{o} and λ_{o} are observed on Earth, showing that for c_{o}
> c, it has to be λ_{o} > λ. Hence, the blue/red shifts observed on Earth
can be expressed as

(47)

where
is the potential on Earth, while U_{s} the one on the source (at distance
s). Thus the shift of a far source, disregarding the motion source-Earth, turns
out to be the variation of c (as well as λ) during the path of light toward a different
potential; for instance, going from Earth to Sun, and considering that along this
path the main variation of potential (ΔU_{(s)}) is due to the Sun only,
we can write U_{S} = U_{o} + ΔU_{(s)} where

(48)

With d the distance Earth-Sun; so on the Sun, and then

(49)

giving, while on the opposite path, Sun to Earth, it is, hence a blueshift (contrary to a red shift of the same value predicted by the Relativity).

As for
Mpc, according to (47), if U_{s} (potential on the source) is, in absolute
value, higher than the potential on Earth U_{o}, we get, on Earth, z < 0
(blue shift), and vice versa for |U_{s}| < |U_{o}|, hence, apart
Doppler effects, these red/blue shifts indicate that the potential, from Earth to
the sources in this space, may increase or decrease (and since for
Mpc, z is positive, we may also argue that our galaxy is close to the middle of
the masses of universe); then, over this distance, it turns out that, on the related
sources, U_{S} is (in absolute value), always lower than U_{o},
and also tending to zero for z → ∞.

In the range (where z follows the Hubble’s law), the (47), written as

(50)

for z = 1 gives yielding (through a simple artifice) to

(valid for z = 1) (51)

which shows that, for z = 1, U_{s} depends linearly on z; in particular,
Table 1 shows that, in the said range

Table 1 . Calculated values of U and c related to
the observed redshifts on Earth. The 4^{th} column is referred to Equation
(50); the 5^{th} to (51); the 6^{th} to (47).

Figure 6. Length of day
since 1969, when Earth-Moon laser ranging data started to be collected. The dotted
line shows the 2.3 msec∙cy^{−}^{1} trend expected as
a consequence of momentum conservation, when it is assumed that, using laser ranging,
an actual increase of Earth-Moon distance is measured. LOD data comes from the EOP
05. C04 series (Bizouard and Gambis 2009), as provided by the Earth Orientation
Centre (http://hpiers.obspm.fr). The Figure
has been taken from the paper of Sanejouand, Y.H., Empirical evidences in favour
of avarying-speed-of-light, arXiv 0908.0249.

(), the values given by (51), are practically the same as the ones given by (50). Table 1 also shows the values of U and c for various values of z.

8. Conclusions

We showed that, on our basis, c corresponds to the total escape speed u which is practically constant on Earth (see §3), while the annual variations of c, due to the eccentricity of the Earth’s orbit, are well shown on following Figure 6.

In fact, when the Earth (on perihelion) approaches the Sun, because of the distortion of the shape of Earth, the angular speed of Earth should decrease, hence, the length of day (LOD)should increase; moreover, during the approach Earth-Sun, due to the increase of the absolute value (on Earth) of U, the speed of light (on Earth) has also to increase, thus the LOD and c should show, at the same time, annual peaks, as shown on Figure 1, which may also represent, in another scale, the annual variations of the speed of light, since 1969.

Now we point out our attention on Compton effect: indeed, through the relativistic Doppler effect equations, one cannot get the Compton equation (which can be found in other ways), whereas through our Doppler effect Equation (17), we can obtain it; well, one can observe that the Compton effect is not a Doppler effect, but in this case why we get the Compton equation? Is it a coincidence or the relativistic Doppler effect equations are not correct?

Finally, regarding the Harvard tower experiment, the Relativity, as for the compensating speed source-Observer (in order to restore the resonance between them), predicts opposite directions with respect to the ones we have obtained: we hope that now (after 50 years from the related experiment) an appropriate (similar) experiment will give a sure answer.

References

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- Pound, R.V. and Snider, J.L. (1964) Physical Review Letters, 13, 539. http://dx.doi.org/10.1103/PhysRevLett.13.539
- Pound, R.V. and Snider, J.L. (1965) Physical Review, 140, B788. http://dx.doi.org/10.1103/PhysRev.140.B788
- Kragh, H. (1996) Cosmology and Controversy. Princeton University Press, Princeton, 212.
- Gogberashvili, M., et al. (2012) Cosmological Parameter. ArXiv:physics.gen-ph, 2.
- Immerman, N. (2001) Nat’l Solar Observatory: The Universe. University of Massachusetts, Sunspot.
- Gott III, J.R., et al. (2005) The Astrophysics Journal, 624, 463-484.
- Van Dokkum, P. (2010) Nature, 468, 940-942. http://dx.doi.org/10.1038/nature09578
- Schutz, B.F. (2003) Gravity from the Ground Up. Cambridge University Press, Cambridge, 361.http://dx.doi.org/10.1017/CBO9780511807800
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Appendix A (Compton Effect)

Here, see Figure A1, an incident photon (length
λ, frequency n), ejects a circling electron (m_{e}) but there is also a
reflected photon (length λ' frequency) so the electron, while emitting
a photon λ' toward the Observer A, represents a source in motion from A along the
direction w, thus (considering the component w_{A} of w) there is an indubitable
Doppler effect.

Now, on the basis that the scattered photon starts to be reflected at the same time
when the incident photon starts to hit the electron, and since
^{ }(=1/) is the emission time of the
photon, it turns out that
is also the whole interaction time, meaning that there is not a complete absorption
of the incident photon followed by an emission: this means that the internal energy
of the photon is not involved in this action, hence the momentum transferred from
the incident light to the electron is p = mc (=γc/T) as per (38), and the same value
p = mc is the momentum transferred from the scattered photon to the electron.

Therefore, the Conservation of Momentum (CoM) along the direction normal to w, becomes

giving.

Moreover, the length of the reflected photon, for the Observer, according to (17) is

(52)

where and where is the component of the electron speed along the direction of the Observer A and (=1/) is, for A, the photon transit time, so we get

(53)

Now the CoM along w is giving

(54)

and plugging this value into (53) we get

(55)

Now, , hence and therefore

(56)

and since, we get the Compton equation:

(57)

which cannot be obtained through the relativistic Doppler effect equation which, as for a source receding from the Observer, is

Figure A1. Compton effect.

Appendix B

Regarding the Conservation of Energy (CoE), E = U + K = 0, leading to
the term K is the unitary (per unit of mass) kinetic energy of a massive particle,
hence as for the light (having total energy, see (34),
with K_{i} its internal energy), we
have to consider, to comply the CoE, its kinetic energy only, giving, like any other
massive particle. Anyhow, one can observe that the internal energy of light () of photons going toward the infinity could
be lost, but this is not the case: indeed, we have seen on §6 (see also Figure 4(a))
that the frequency of a photon is constant along its path, and since, see (34),
, it turns out that toward the infinity, where
c_{∞} → 0, we get, yielding K_{i∞}
→ 0 (In other words, the internal energy of light only depends on the length of
their photons).