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

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.


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 ν 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.

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 = 2 1 2 u represents the unitary, i.e. for unit of mass, kinetic energy of our particle arriving from the infinity, where 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 ( ) ( ) Now, the escape speed from two masses can be written 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 therefore the escape speed due to all the n masses in space becomes 2 2 2 ∑ 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 = 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 53 u 10 M ≅ 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 27 c 9.2 10 kg/m 3 the critical density [9].So the mass of universe can be written ) 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 ( ) Now, according to (5), on Earth it is 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 2 , 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].

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 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) 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 ) = +0.10m⋅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 -4 -1 r 2 10 m s u ∆ ≅ × ⋅ ; so, on Earth, u o is practically constant during one year, as it is for the measurements of the speed of light.

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 [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 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 where ′ λ is the photon AB emitted with the source in motion from E p .

( ) ( )
where ′ λ 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 ( ) 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 [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 ( ) Now, according to (18), the speed of the photon AB, referred to O, becomes ) 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 For any Observer, the frequency of photons of the same ray has to be defined as ν' = 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 ν' = 1/T', so from (19) we get ( ) showing that for v = 0, that is β = 0, we have ν' = ν, 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 ν s = ν 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 ( ) ( ) 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  ( ) 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 ( ) Then ( ) 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 ′ λ 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 ′ λ (emitted while S is moving along the orbit r) will be ( ) 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).

Physical Characterization of These Photons
Now, similarly to a fluid flowing in a pipe (whose kinetic energy is K = 1 2 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 = 1 2 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 = 1 2 mc 2 + 1 2 mc 2 we may infer that each of these particles is provided with an internal energy (K i = 1 2 mc 2 ) equal to its kinetic energy.Now, equating mc 2 to hν we get ( ) where m written as ( ) is the mass of light, with frequency ν , passing along one ray in 1 s, while the constant ( ) is the mass of light passing along one ray during T, we may call it "mass of one photon"; so one finds 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 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 (≡n r m) by a source of light becomes 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 r n ν = P/h.The number of photons emitted in 1 s becomes: ( ) 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

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), where ( ) is the variation of the total gravitational potential U, due to Earth, from the tower base to its top.As Eo where M E is the Earth's mass and r its radius, we get U ∆ = M E Gh/r 2 where h (=r h -r o ) is the tower height, yielding showing that, on the top, where  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 ν h = ν 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 ν h = ν o , it seems to be E h /E o = hν h /hν 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 ν h = ν 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 [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 ν h = ν o , which, on our bases, is wrong: with S on the top, see  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 ν 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 ν h < ν o , and in particu [Still referring to Figure 5, taking S to the tower top, we have c h < c o and ν h < ν o implying, contrary to GR, a decrease of the energy of light to be emitted by S, ( )

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 = kν h yielding Δν/ν o = Δt/t o where Δt (=t h -t o ) is the ticking time variation from ground to height h, with Δν (=ν h -ν 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 ν, so that the (40) can be written as 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 ( ) 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 ) 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 day 1 1

s day
as observed.This equality also confirms that λ h = λ o as for sources in altitude.

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 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) ν = ν o , we get , hence a blueshift (contrary to a red shift of the same value predicted by the Relativity).
As for 40 s <≅ 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 40 s >≅ 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 0.01 0.20 z ≅ < <≅ (where z follows the Hubble's law), the (47), written as which shows that, for z  1, U s depends linearly on z; in particular, Table 1 shows that, in the said range

2 1 2 u
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].

Figure 1 (Figure 1 .
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 .
) 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 1 O A v different from u, if he could measure such a speed)].

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

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

Figure 3 .
Figure 3. (a) Transverse Doppler effect; (b) Source circling around the Observer O.where λ′ is the length corresponding to S T O, while λ corresponds to E p O. On the contrary, see Figure3(a)right part (where the source is receding from the Observer), it will be ( ) 2 2 2 1 vT λ λ λ β ′ = + = + (valid for S receding from O) (27) 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 according to Newtonian mechanics should be written as their kinetic and their internal energy, that is E = mc 2 we obtain

Figure 4 .
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.

Figure 5 .
Figure 5. Harvard tower experiment scheme, with the source on the top.(a) S on top, S and A at rest: c h , ν 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.
, and therefore when these photons reach the base (where c h-o = c o ) their energy becomes 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, give a cosmological reason to the high redshift of sources where |U S |  |U o |.

3 ≅
Bly) the red shifts (here in the range 0.01 -0.20 ≅ ), 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,

,
where ν 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 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 d the distance Earth-Sun; so on the Sun,