_{1}

^{*}

Assuming the light as composed of
*longitudinal-extended elastic (and massive) particles* emitted during an emission time
*T* at speed
*c* (=
*u*) (with u the escape speed from all the masses, toward the infinity), it is shown that
*c* is invariant (under the Newtonian mechanics laws), for an Observer fixed to the initial emission point E
_{p} (the point where the emission starts), in spite of any motion of the source (of light) with respect to E
_{p}. On the contrary, an Observer, in motion from E
_{p} during the emission, will state (indirectly) a
*Galilean* variation of
*c* which can be proved and evaluated by an appropriate
*feasible* experiment described here.

This paper, in accordance with the Newtonian laws, is based on following assumptions:

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

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

III. Light composed of longitudinal-extended elastic particles (as defined in Section 3) moving at speed c = u. This equality is supported by a cosmological reason.

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

1) The relation u = (−2U)^{1/2}, where u is total escape speed, from all the masses, toward the infinity and U (total gravitational potential, due to all the masses in space), while the assumption c = u (which implies the massiveness of the light and also c_{∞} ® 0), gives to the speed of light a cosmological/anthropic reason to its value, as highlighted in Section 2.

2) The constancy of the speed of light, under the Newtonian laws, for an Observer fixed to the (initial) emission point, E_{p}, (the point where the emission of light starts), in spite of any motion of the source with respect to E_{p}.

3) A variation of c (in accordance with the Galileo’s velocities composition law), if the Observer is in motion with respect to the said emission point (taken as reference frame): for instance, on Earth, an Observer in motion from the source (of light) fixed to the ground during the emission. This variation of c (contrary to the 2^{nd} postulate of Special Relativity) can be proved by a feasible experiment described here.

Here we get the escape speed due to one, two and n masses; then we assume u = c (= −2U)^{1/2}, with u the total escape speed (from all the masses), and U the total gravitational potential .

As known, considering in space one only mass M (regarded as a point-like), the gravitational potential U acting on a particle having mass_{¥} = 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 (= 1/2u^{2}) represents the unitary (for unit of mass) kinetic energy of our particle arriving from the infinity (where u_{¥} = 0), for E = 0, gives U = −K, leading to

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; thus u (escape velocity) has to be referred to M; if M is a real mass, u has to be referred to the point of M where _{p}. Therefore we have to write

meaning that the escape velocity of our particle m has to be referred to C_{p}.

Considering now two masses M_{1} and M_{2}, having, at a given time, distances s_{1} and s_{2} from a considered point, the potential U_{1,2} in this point becomes

Without the assumption I, the Equation (3) is still valid, but U_{1,2} could have a different value, depending on the assigned values of s_{1} and s_{2}, at the considered time.

Now, the escape speed from two masses can be written

representing the value, in the considered point, of the (escape) velocity u_{1,2} which has to be referred to the Centre of potential C_{p}, (where

Then, as

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

with _{p} (centre of potential of all the masses).

Now, considering as origin of our Reference frame, the initial position of m, we may call it Emission point E_{p}, (the point where m will be given the speed u), the following relation

may be called as relative escape velocity of m from E_{p}, with

The particle m, at the time of its emission, has, in general, a velocity _{p}, hence, from C_{p},

it has a velocity

in particular, if the velocity _{p}, more than half of them will surely tend to the infinity.

We assume now the equality c = u, hereafter supported by the estimated mass of the universe and also by this cosmological/anthropic reason: in fact, 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, going toward the infinity in an unlimited time, tends to avoid the said collapse.

Here we also point out that the energy of light (mc^{2}) seems to be not in accordance with the said relation E = U + K = 0, where K = 1/2mu^{2}, but, as shown on [^{2}, complying the said relation (plus an internal energy of the same value).

This following part of Section 2, is only a support to our assumption c = u.

The mass of universe, by some authors, is estimated [_{¥} = 0, we may assume their density as decreasing, from the Local Group (LG) toward the infinity, like a function ρ = ρ_{c}e^{−as} with ^{−}^{27} kg/m^{3} the critical density [

yielding

From the LG, the variation of U, 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 Equation (6), on Earth it is

Therefore, on Earth, u_{0} = c_{0}, so that

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

Here it is shown that under the assumption III, should a source of photons, during their emission, move from its initial Emission point (E_{p}), their length will vary, and since their transit time, (time to cross an Observer), will also vary, their speed becomes constant for any Observer fixed to E_{p}, in spite of any motion source-E_{p}.

The Galileo’s velocities composition law, which is related to point-particles, cannot be, apparently, applied to a particle defined as follows:

“longitudinally-extended, elastic non divisible particle emitted at speed u by a source during an emission time T, and moving along rays, (continuous succession of photons)”.

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, every tail, (along a ray free path), corresponds to the front of the next photon.

Now, on _{p} (the point where the emission starts), be the origin of our reference frame; then let S

move, during the emission time T, from E_{p} to S_{T}. For instance, E_{p} may be a generic point on the Earth’s surface, with S coincident to E_{p} at t = 0.

Now, the relative escape velocity of the front A (being a point-particle), from Equation (7), can be written

which gives, as hereafter shown, to the (whole) photon, for an Observer in E_{p}, (or fixed to it), a constant speed (u), in spite of any motion of S from E_{p}. At this purpose, still referring to _{p}; hence the velocity of the front A, with respect to S, that is, v_{SA}, from Equation (13), becomes

Therefore, at t = T, when S has reached the point S_{T}, we get

where λ' is the photon AB emitted with the source in motion from E_{p}, while λ (º uT) should be the photon (AE_{p}) if S, during the emission, should be fixed to E_{p}.

Thus, after the emission time T, as for a source receding from the front A, as in

where_{p}-S_{T} covered by S during T, with

Now, the speed of a point-particle is defined through two Observers, while the speed u' of a photon, (since its length could vary), does not correspond to the speed of any point of it, hence to define the speed of a photon we must consider its length referred to the time T' (transit time) the photon (front to tail) needs to cross one Observer, leading to

which is the average speed of the photon along the path λ'.

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, that is why we need the Equation (16) to define the speed of the whole photon.

Still returning to _{p}, the transit time T' of the photon AB is given by the time the front A needs 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} =

Thus, see Equation (17), the speed of the photon AB, referred to our Observer fixed to E_{p}, becomes

showing that, on our bases, an Observer, fixed to the Emission point E_{p} (point where the light starts to be emitted), states that the speed of light is invariant (and equal to the total escape speed), in spite of any speed Source-E_{p}.

Now, what about for an Observer in motion from E_{p}? Well, the length λ' of an emitted photon does not depend, see Equation (16) on the Observer motion, while the photon transit time T', depends, for an Observer fixed to E_{p}, see Equation (18), on _{ }and it also depends, for an Observer in motion from E_{p}, on_{SO} = 0 (Observer co-moving with S, both moving from E_{p} to S_{T} during the emission), the photon length, see Equation (16), is still λ', while its transit time (for the Observer and for S too) is T' = T, hence u' = λ'/T ≠ u, that is c' ≠ c.

Hereafter, to detect this variation, we describe a feasible experiment where the Observer (measuring the times), has to be in motion from the Emission point (taken as reference frame): usually, the measurements of c (through the method d/t), to be accurate, requires the Observer(s) to be fixed with the photon emission point), but, on our proposed experiment, the distance d, (variable during the experiment), is not involved into the evaluation of Δc.

To evaluate our predicted variation of c stated by an Observer O in motion from the initial Emission point E_{p}, we may assume, for simplicity, the source S fixed to E_{p}.

Thus, see _{p} (our reference frame) and let _{p}. So, at t = 0, it is: O and A_{1} (front of the first emitted photon) both in E_{p}. At the end of the emission, lasting t_{e} (= nT), where n is the number of consecutive photons (A_{1}B_{1}-A_{n}B_{n}), the path covered by the front A_{1} becomes L = nλ, the Observer has covered the path Δd = v_{O}t_{e}, while the tail B_{n} is still in E_{p}. Then, when the tail B_{n}, at a certain time t_{O}, reaches O, the path covered by O is ΔL = v_{O}t_{O}, where t_{O} corresponds to the (transit) time the n photons need to cross the Observer (that is from t = 0 to the time the tail B_{n} reaches O), hence we can write t_{O}c = L + ΔL, so we get

giving

which, for

(valid for) (22)

and then

which is the variation of time stated by the Observer O between the transit time t_{O} and the emission time t_{e} where t_{e} is also the transit time for the Observer if this would be fixed to S. For instance, if v_{O} = 10 m/s and t_{e} = 3 s, one gets Δt = 10^{−7} s, well measurable.

Therefore the speed of light as stated by the Observer O (moving along the same direction as the light) becomes

while the speed stated by S (or by an Observer fixed to E_{p}) is:

Referring to

t = 0 (start of times, when the first photon, A_{1}-B_{1}, reaches O);

t = t_{O} (end of time, when the last photon, A_{n}-B_{n}, reaches O).

As for the clock of the source, not represented on figure (and not necessarily synchronized with the Observer clock), we have:

t_{S} = 0 (start of emission from S);

t_{S} = t_{e} (end of emission).

We point out that the value Δt given by Equation (23) does not depend univocally on the speed of the Observer, hence the predicted (by the Relativity) time dilation due to the Observer motion, cannot be claimed.

Now for an Observer, the frequency of photons of the same ray has to be defined as

・ if S is moving from E_{p}, (as in _{p}, he will state: c' = c and

・ if the Observer is moving from E_{p} along the same direction as the photons (like in _{p}, the Observer will state:

We have seen that, on our bases, the speed of light (corresponding to the total escape speed) follows the Newtonian laws, and we have described a feasible experiment showing that the speed of light also follows the Galileo’s velocities composition law, if applied to these photons.

AlfredoBacchieri, (2015) A Feasible Experiment Contrary to the 2^{nd} Postulate of SR. Journal of Modern Physics,06,1342-1349. doi: 10.4236/jmp.2015.69139