where v_r is the velocity of the aetherino relative to the Simple Particle (SP).

---------------------------------------------------

NOTE 3-1:

A velocity increment (i.e. change) is an invariant in a Galileo transformation between any 2 reference frames moving at constant velocity relative to each other. It is a vector that points in the same direction and has the same modulus in all reference frames (that share the same space standards and the same clock). It is a classical, simple, defined concept that is nevertheless known to produce some discomfort when one has the habit to think in terms of the theory of Relativity thus conceiving space time as a physical reality instead of as mathematically defined concepts.

By the moment the model assumes the existence of only one kind of material Simple Particle, with a specific geometrical cross section s. (The possibility  remains open to postulate in the future the existence of different kinds of Simple particles if  such need appears in other contexts of application of the model.   But, for the sake of simplicity and fidelity to its geometric nature, the expression  "different kinds of Simple particles" should in a first attempt be equivalent to "different sizes of Simple particles". Only if this point of view proves insufficient, an attempt should be made to adopt different [3-1] laws for the different kinds of Simple particles. The constant Q of the hypothesis must therefore not be associated with a specific type of Simple particle neither with a mass).  The hypothesis B is  a microscopic law that must not be confused  with the laws of collisions between material particles in mainstream mechanics.  In today's physics, the concept of mass is an enigmatic attribute of a material body.  A geometrical model must instead face the challenge to represent such concept in a comprehensive way.

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Imagine first a single Simple material particle (SP) in a given "macroscopically smooth" aetherino distribution.   At  the microscopic level, according to hypothesis B, it can be inferred that the particle suffers abrupt short living velocity changes at each collision (i.e. it jiggles).

To calculate the macroscopic movement of the SP (e.g. its average velocity in a time interval in which it suffers a big number of aetherino collisions) it will seem reasonable to postulate that between two consecutive collisions with aetherinos, the Simple particle maintains a constant velocity in any given rectilinear frame (i.e. in a reference frame where the aetherinos travel, by definition, at constant velocity). (See hypothesis B-2 below and Annex M).

-------------------------------------------------------

NOTE 3-2:

Let r(v) be, in a reference frame S,  the aetherino's velocity distribution in the neighborhood of a material particle at the epoch of observation.  To be able to mathematically predict a law of movement of the material particle subject to the aetherinical force produced by its local aetherino’s distribution this force must be unambiguously defined. But, the way it has been defined, an aetherinical force is only well defined if the material particle has a constant velocity in S during a time interval big enough for many aetherino collisions to occur. This is so because such force is dependent on the velocity of the particle relative to the distribution and because an aetherinical force is  a  macroscopic statistical concept that  must  be understood  as the "average" net impulse per unit time.  But if the material particle keeps jiggling about (due to the aetherino collisions implementing the force) it can no longer be strictly said that it has a constant velocity in S. (Only if the constants q.Q of Eq [3-2]  happened to be so small that all aetherino collisions produced a speed change of the material particle much smaller than the average aetherino speeds relative to the particle it could be said that the aetherinical force is a strictly defined concept. But nothing can be said a priori of the size of the constant Q.q ). There nevertheless seems to be a semi classical way out of this description problem without the need of a quantum mechanics formalism. The instantaneous aetherinical force will by definition be understood to evaluate the impulses (by unit time) of the aetherinos on an hypothetic material particle that doesn’t jiggle but has a constant velocity equal to the average velocity of the real jiggling particle that it represents. It is when calculating the "effect" of this aetherinical force on the average movement (velocity change) of the material particle that the description will take into account the consequences of the microscopic jiggling (as will be explained below: Hypothesis B-2 and Annex M). Therefore the aetherinical force will be supposed to admit a classical  treatment  and there is then  no contradiction in imagining for it a time interval big enough for the averaging and at the same time small enough to allow its treatment as an infinitesimal time interval from the   macroscopic point of view and  hence to allow the aetherinical force to be treated as an instantaneous force. This force can then be assumed to be expressible by a continuous function of continuous time derivatives without conflicting with its microscopic roughness.

-------------------------------------------------------

NOTE 3-3:

(This note is just a first glance at some epistemological questions related with the model. It tries to give a feeling of how the model is understood, but most of what is said here might need to be revised or simply ignored).

From some point of view the laws of physics can be classified either as "classical laws" or as "statistical (or probabilistic) laws". Let incognita be the generic name of the physical magnitude predicted by a (mathematical) law when a given set of other magnitudes is known.

A given theory might define exactly all the magnitudes involved in some phenomena, relate them in a deterministic way and enounce some laws predicting single exact values for the corresponding incognitas. These laws will be called classical. The experimental physicist will assert the validity of  this classical law when each experimentally determined error segment of  the incognita contains the exact mathematical value given by the function defining the law.

Another possibility is that a theory makes use, together with non fluctuating magnitudes, of some other magnitudes defined in a probabilistic (but unambiguous) way (e.g. leaning on some well defined mathematical distribution), and relates them enouncing some laws that will therefore make probabilistic predictions of the corresponding incognitas as well as single valued predictions of their averages. These laws will be called statistical. The experimental physicist will check the validity of  this statistical law when, experimentally measuring the incognita an enough number of times (through a repetition of the experiment under the same conditions), he obtains a distribution of values that fits the probability distribution predicted by the law. The measured average value of such incognita must differ from the predicted average value given by the law less than the experimental error. But the individual measured values differ in general from the predicted average more than the experimental error.

In the aetherino model, a law predicting the movement of a material body cannot be classical from a strict theoretical point of view although it may happen, from the experimental point of view, that if the body is made of many particles, the  averaging between  these may hide the fluctuations  in  the position of the global body to the experimenter.  But from a theoretical point of view, the time and space resolving powers of the observer could be increased at will and hence even the fluctuations of a big body, caused  by the statistical behavior of the aetherinos could "in principle" be evaluated with the consequence that any previously called classical law should from then on be treated as a statistical one.

From the point of view adopted in this work, a quantum mechanics law is considered simply a statistical law whose statistical character is caused by the aetherino's semi-random behavior. The main difference between quantum mechanics and mainstream statistical mechanics would be that in the former there is interest in knowing (as precisely as possible) the "partially" random positions and velocities of some microscopic constituents (the material particles although not the aetherinos) while in the latter there is no interest and no point in describing the "entirely" random positions and velocities of its microscopic constituents (atoms, molecules, …).

From another point of view the model is a deterministic description of the physical world. This does not contradict the assertion that the laws deduced from the model are intrinsically statistical: The aetherinos are admitted to exist and behave in a deterministic precise way. But on the other hand they can not be seen (by a real observer making physics) because they are the vehicles of light and touch... , and therefore only statistical predictions can be made from a strict point of view.

Nevertheless the concept of determinism is considered a matter of convention, because, (even supposing that the model proposed could explain all physical experimental facts), since the aetherinos can not be seen, no claim can be made to identify the model with the "reality" of Physics. Because it could happen that a different non deterministic theory (e.g. quantum mechanics) could also explain all experimental facts.

---------------------------------------------------

This section seeks to deduce a "classical" law (like Newton's 2nd law) for the movement of a material body in the aether of the model. A classical law will only fit the real microscopic movement of a single Simple particle if Q.q << 1. For a composite body made of n Simple particles, a classical law will fit the microscopic movement of the body (represented by the average position of its component particles) when Q.q/n << 1. All these considerations seem to be directly related with the conventions of mainstream physics in setting the boundaries between classical and quantum phenomena (i.e. Heisenberg's principle of uncertainty, etc). It is believed that the particles of small mass of the real world (like the electron) move in a typical jiggling way, being the reason why a quantum probabilistic description is needed to explain their experimental facts. Hence, the Simple particles of the model, that are believed to be as "light" as (if not lighter than) the electron will also exhibit such non-classical behavior and therefore Newton's second law is more appropriate for the description of material bodies made of a "big" number of Simple particles (i.e. it doesn't seem likely that Q.q<<1, but bodies can always be found for which Q.q/n <<1).

    The position of a composite body will be defined as the mean position of its Simple particles:

[3-3]                                   

It is then natural to define the velocity of a composite body as:

[3-4]                          

therefore:

[3-5]                       

    Similarly for higher derivatives. E.g. for the acceleration:

[3-6]                    

    The goal of this section is to deduce a law for the macroscopic movement of a material body that suffers an aetherinical  force. The deduction will lean on the application of hypothesis B to the aetherinical collisions suffered by the body.

    The "macroscopic movement" is understood to be described by the time evolution of the macroscopic or average position of the body over many identical experiments . If the body is made of many SPs and hence its Q.q/n << 1 this average position will be in practice relatively very close to the real position (see [3-3]) of the body in any single experiment.

    The "macroscopic (or average) velocity" of a particle or body at time t can be defined as the time derivative of its average position at time t. Again, if the body has a small Q.q/n this velocity  v(t) will in practice be relatively very close to the real velocity (see [3-4]) of the body in any single experiment.

    Suppose that a material body made of n Simple particles is subject to an aetherinical net force F(t) .

    F(t) is the net aetherinical impulse given to all the SPs of the body in unit time, at the epoch t, by all the colliding aetherinos whatever their origin (i.e. whatever ascription to a specific type of influence might be made for other purposes). Therefore, by definition,   the aetherinical force on a composite particle (or body) is the sum of the aetherinical forces acting at a given epoch on its Simple Particles:

[3-7]                            

    As a direct consequence of hypothesis B, Eq [3-1], adding on one side the velocity increments and on the other its corresponding impulses (suffered by a SP in unit time), it follows that due to an aetherinical force F_i(t) the SP labeled i increases its velocity in unit time by an amount:

[3-8]                              a_i = Q F_i(t)                 (for a SP)        

[3-8b]                      (for a CP)

Consider first the case where the net force F(t) reduces to the aether drag  force calculated above. (i.e. the body can be imagined to be moving at the epoch t with a velocity u in an "undisturbed" canonical aether and therefore suffering the corresponding aether drag force.  No other external forces are supposed to be present. Standard Physics would say that the body does not experiment any force).

    Using the linear approximation deduced in Section 2 for the aetherinical drag force acting on a composite particle then Eq [3-8b] becomes:

                                                                                [3-9]

therefore

                                                                                     [3-10]

    The integration of differential equation [3-10] gives:

                                                            [3-11]

where m = k Q is a constant and u(0) is the aether speed of the body at the epoch t = 0.

    Expression [3-11] will be called the "aether Slow Down Law of free bodies" in a canonical aether.

-------------------------------------------------------

NOTE 3-5

-------------------------------------------------------

NOTE 3-6

    The "law" Eq [3-11] for the slow down of bodies moving in the aether has statistical validity and limitations. It is a  macroscopic law in contrast with hypothesis B , Eq [3-1], that can be considered a microscopic law.  The speed u(t) of the body must be considered an average speed along a time interval "big enough"  to allow for many aetherino collisions to occur on the body but at the same time "small enough" compared with  the resolution power of the clocks  used  in  the observation of the law so that it can be considered  as an instantaneous speed.  In other words such speed u is "average" in relation  with microscopic time and "instantaneous" in  relation with macroscopic time (i.e. the time read in normal clocks).

    Insisting again, although  the model does not allow a strictly classical law for the movement  of a particle the possibility of a (non  strict) classical law can happen in the following sense: Imagine a composite particle made of many Simple particles. Its position has been defined in Eq [3-3].  If the  number n of components is big enough it may happen that the positions of the composite particle in a set of identical experiments never fluctuate from the average more than a given quantity Dx.  If  Dx is smaller than the space resolution of the experimenter then this observer will not detect such fluctuations and he will conclude that the composite  particle  behaves classically.

--------------------------------

    Macroscopic time (the reading of real clocks) could in some way be treated  as a mean statistical magnitude conditioned by the microscopic postulated time. First it can be postulated that microscopic time advances uniformly  (in relation with a description postulate of constant velocity of all the aetherinos), being for example given by the x component of the position vector (in a rectilinear  reference frame) of a given aetherino. Then, some cumulative process can be imagined to be fed by microscopic statistical events so that  the quantity accumulated in a given system determines the macroscopic reading of the time elapsed in a given real clock. It  will then happen that not all real clocks advance in  exact synchrony  when compared at their maximum resolution due to the hazards of the statistical cumulative process in each clock.

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Official time and Ideal time.

    It  seems reasonable to state that the advance of the clocks used in official physics  to measure "time" is conditioned  by  the movement  (oscillatory  in many cases) of some specific matter. (The aetherinos are not considered matter).  Examples of such official clocks are the atomic ones,  the pendulum  ones,  those based  on the positions of celestial bodies,  etc..  With all of them it can  be verified that a material body subject (according to official physics) to a constant force acquires a uniformly accelerated movement (at least for speeds u of the body such that u << c ). According to the model it then also seems reasonable to admit that, being all matter bathed by the aether, the continued collisions with the aetherinos will cause a  general and gradual slow down of all the internal and external movements of such matter and in particular of the matter that sustains the clock's mechanisms.  Being the "slowing down" a relative  concept  it must be understood that "matter" suffers  the general slow down when measured by an "Ideal clock" (for which the aetherinos move by definition at constant speed).   But then it can be expected that matter will not suffer any general slow down in relation with official clocks since these are themselves made of  matter which from the point of view of the ideal clocks is slowing down at the same rate.

    Hereafter 2 different observers will be invoked distinguished by different time standards but sharing the same space standards:

-  Ideal  Observer   IO  :  Uses Ideal  clocks  for  which the aetherinos  move  at constant speed (in virtue of a description postulate) in the so called rectilinear reference frames.

- Official Observer OO :   Uses Official clocks (the material ones commonly  used  in experimental  physics). For this observer due to the aether drag slow down of all matter, the aetherinos move at ever increasing speeds.

    Likewise subindex I (from Ideal) will be used for the speeds and accelerations observed by IO and subindex F (from oFficial) for those measured by OO.

    Although it is not a priori evident at what rate can the official clocks be expected to slow down relative to the ideal ones the following choice seems reasonable in this context:

                                                                                         [3-15]

where m is the same constant of Eq [3-11].

    It would be unfair in this case to blame the "Tempo rate law" given in Eq [3-15] of being an ad hoc hypothesis of the unwanted type, since, considering that the official clocks are immersed in the  "same" aether it can be expected that their mechanism slows down relative to IO at the "same" rate at which a free body does (i.e. according with the law given in Eq [3-11] with the same constant m).

Inertia.

Therefore from Eq [3-15]:

                                                                                [3-17]

    If  the body observed is the "free" body (subject only  to  the aether drag) described above (see Eq [3-11] ) then, the official observer assigns to it a speed:

                                                 [3-18]

    Since u(0) (initial speed of the body at an arbitrarily defined epoch t = 0 ) is a constant, Eq [3-18] expresses the  property called "inertia" of free bodies.

    Notice also that the Official Observer sees the aetherinos increase their speed according to  Eq[3-17] where u_I stands in this case for the constant speed IO speed of an aetherino.

    Integration of Eq [3-15] gives:

                                                                                [3-19]

                                                                                 [3-20]

the description will correspond to the case in which both observers assign the epoch zero to the same given physical event.

    An "event" is understood as a distinct position's distribution of a set of elementary objects (aetherinos, Simple particles,...) of the physical world (or model).

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NOTE 3-7

    The fact that the time readings (epochs) assigned by each of the two observers to a later (or earlier) event will differ does not alter the fact that the description continues to assume the existence of  "Absolute Time"  as a valid concept for both observers. I.e.:

    If two events are simultaneous for one of the observers (IO or OO) they also are simultaneous for the other.  The time readings of each observer for any given event are related by the function of  the type of Eq [3-20] assumed.

    The word  "instant" will be reserved for the set of all the events of the physical world that are simultaneous to any given event. Any given event is therefore assigned to the same set of events by both observers.

    The concept "instant" is considered an abstraction of "epoch" for all observers. Epoch is the number (i.e. name) assigned to an instant by a given observer.

---------------------------

    The description postulates the existence of a classification of events  in  instants (absolute time) such that any  given event belongs to only one instant (set of events).  The name of a given instant, epoch, may differ for different observers.

    This description (in the sense of organized information from which it is possible to make predictions on the behavior of  the physical  world)  adopts the classic approach of causality:  the existence of an event can in principle be deduced, with the use of simple rules (laws), from the information of a complete set of "earlier" events (i.e. from events corresponding to an instant of smaller associated number).

    In practice,  the physicist, who can not see the aetherinos, can not strictly (but only statistically) determine a complete set of earlier events and therefore he may incorrectly infer violations of causality.

    Of course,  the description must be consistent with the classification of events and the rules relating them. It must not happen that two contradictory predictions are made by  two different observers using complete sets of earlier events.

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Newton's second law.

    Suppose now that the body is not free but that in addition  to the aether drag force there is an  external "material"  force acting  on  it.  The adjective "material" is used to point  out that the force has its origin in some known matter thus distinguishing it from the aether drag force (due to the movement relative to the reference frame in which the aether is considered at rest). The official description of physics does only treat as forces those that have here been called material forces since in that description there is no recognition of the aether drag.

    The generic net force F(t) that appears in Eq [3-8b] can,  in this case, from a theoretical point of view, be  "separated" in the two kinds  of  forces assumed to be acting on the body:

Therefore using Eq [3-8b] and dropping the subindex C meaning "composite" in that equation and introducing instead the subindex I meaning "as observed by the Ideal observer" :

                                                                      [3-23]

                                                                             [3-24]

                                                                                     [3-25]

the derivation of both members with respect to t gives:

                                                                             [3-26]

therefore the acceleration for the official observer OO is given by:

                                                                      [3-27]

                                                        [3-28]

                                                                                    [3-29]

and therefore:

                                                                                 [3-30]

    It must now be noticed that an "aetherinical force" is a defined concept whose numerical value depends on the type of observer (IO or OO) since it depends on the values that these observers respectively assign to the aetherino's (relative) speeds in their collisions.

    Nevertheless,  in this work,  when referring to aetherinical forces (either  material or drag) it must always be understood that their values refer to the Ideal Observer since the model takes advantage calculating such forces for this observer for which the aetherinos move at constant speed.

    But the example of redistribution studied in Annex D (see Note D2 in that Annex) suggests that the aetherinical material force should be considered proportional to the square of the mean internal IO speed w of the component  Simple particles of the bodies producing the material force. Hence  if the "model of material force" expressed in Eq [D-11] is assumed, when adding the consistent supposition  that  the internal IO speeds w of  the responsible matter decrease (for IO) at the same rate of slow down of matter in the dragging aether (and of the slow down of the (material) official clocks), the IO time evolution of any OO constant material force is given by:

     [3-31]

                        [3-32]

which is the sketch of Newton's second law.

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NOTE 3-7b

- The preceding assertions implicitly assume that the epoch t = t = 0 is a well defined epoch (related to some physical event) in which both observers synchronized their clocks. But this synchronization could have been done at an earlier or later epoch (simultaneous to a different physical event) and it can be asked if this earlier or later synchronization would affect the value of the aetherinical force assigned by IO at the new epoch t = t = 0 (and hence assigned by OO at any epoch). The answer is no considering that the natural way for the model to define a synchronization of the IO and OO clocks is the following:

1) in simultaneity with some given event of the physical world , IO and OO set their clocks to read t = 0 and t = 0 respectively.

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NOTE 3-8

    A common subject of discussion and confusion in Physics is whether to consider Newton's second law "F=m.a" as the primary definition of force, as an experimental law or as the primary definition of (inertial) mass.

    The "F=m.a <=> definition of force" point of view must assume that the mass m of the body is known by independent ways. Then, for example, pulling the body with an elastic spring at different accelerations and "drawing marks" at the elongations acquired by the spring, this spring can be considered as a calibrated dynamometer with which to measure other forces (for instance magnetic or electric ones). But this "definition" point of view seems a very risky one. It would be unacceptable to find a posteriori that two parallel dynamometers indicating each a force F=1 are unable to equilibrate the pulling of a third dynamometer indicating a force F=2. Of course, it could be argued "the dynamometers are not well calibrated " (i.e. our definition of force is inadequate) because we have not taken into account that the mass m of the body used in the definition varies with its speed. But unless the concept of mass was universally agreed and independently understood, no one would agree how to relate the mass of a body with its speed (or why not with its acceleration, temperature, density, chemical composition, etc). Saying, "well, let's look how the mass varies recalibrating ad hoc the dynamometers until they are able to pass the later equilibrium of forces tests" would just be cheating and introducing confusion in Physics. (A possible independent definition of mass that at first sight looks promising to be used in a later F=m.a definition of force is based on predefining the momentum of a particle as its mass times its velocity and then postulating the conservation of momentum in particle collisions. The study of the emerging velocities of the particles in elastic collisions could then be used to determine their masses and their speed dependence. But high speed mass determinations using this method can only be done in practice with microscopic particles. When later wanting to evaluate a force using the definition F=m.a one should be aware that the m used in the equation has really been independently determined in a direct or indirect way with (for example) the momentum paradigm and not using m=F/a in another experiment because this would add to a circular reasoning and the force determination would be unjustified).

    The "F=m.a <=> experimental law" point of view, that will consist in measuring accelerations, must assume that there exist independent methods to know at all times both the value of the force and of the mass involved in the experiments . The determination of the force can depend for example on some defining independent law (for example related with electricity or magnetism or with elasticity, etc). The determination of the mass can be based on gravitation (comparing weights with a balance at the Earth's surface) but since these type of measurements do not study the dependence of mass with velocity the values of m deduced in the weighing are the only ones allowed to be introduced in the equation a=F/m to check its validity. Then, according to the high energy accelerators F=m.a would not be a valid law for high speeds of the particles if m has been determined by gravity methods. If instead the masses used in the experiment have been determined with the momentum conservation method then it would turn out that F=m.a is a valid experimental law. This seems to be the point of view of mainstream relativistic Physics. But it is believed that mass determinations based on momentum determinations with high energy collision experiments are not as reliable as they should considering the complications introduced by the appearance of photons or other particles transporting a part of the momentum and even worst the ad hoc invocation of neutrinos done by some authors.

    A third point of view is to consider F=m.a as the definition of (inertial) mass. Applying an independently known force F to a free body it will (in an inertial frame) always suffer an acceleration (this acceleration could happen to be zero or, why not, negative but those will still be values of an acceleration). Once accepted m=F/a as a definition of mass then F=m.a could also be called (as is general practice) "an experimental law" but of an axiomatic and therefore not very illuminating kind. The high energy accelerators indicate that if the masses of the particles are defined by m=F/a then these masses increase with speed. Assuming that the equations of the electromagnetic forces exerted by the ground based apparatus on charged particles moving at very high speed relative to them should be considered correct, then the masses increase according to the well known relativistic relation (in which the masses tend to Infinity as the particle's speeds tend to c). The problem is that official physics lacks a defined concept of force of general application in all its fields. The expression of the electric (or the magnetic) force is not a really independent definition of force as this third point of view requires but is somehow a law dependent in a not explicit way on a previous assumption of F=m.a as a definition of force, falling therefore in ambiguity. 

    If, for example, the aetherinical definition of force could be transported and applied in a satisfactory way to most areas of physics, then m=F/a could provide a more secure definition of inertial mass (see [4-32] below). Even more, if it could be considered that the model is able to describe the concept of mass and its speed dependence in a convincing way, then it could be said that Newton's second law is a "relation" explained by the model.

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Shrinking distribution.

    Up to now it has been implicitly supposed that the aetherino's velocity distribution (e.g. the canonical distribution) is the same at all epochs for the Ideal Observer.  This implies that such distribution varies in time for the Official Observer.  (Remember that the aetherinos constantly increase their speed  for this observer and that it has been assumed that there exists neither sinks nor sources of aetherinos. Notice also that the total number of aetherinos contained at any given  "instant"  in a given domain of "vacuum" (aether)  is  the same for both observers ).

    But a distribution that varies in time for OO seems at first sight unable to account for the experimental evidence concerning the time invariance of  the universal constants of  physics. Therefore it is considered necessary to question the above supposition and study a description based on the new supposition that the aetherinos velocity distribution of any given region does not change significantly in time for the Official Observer OO. Therefore when viewed by IO the distribution must "shrink" (or cool) in time.

    It continues to be assumed that the speed of each  traveling aetherino is constant in time for the Ideal Observer IO but not for OO.

    The new supposition can perhaps be justified admitting that when the aetherinos collide with matter they reemerge with speeds smaller,  on the average,  than the incident ones.  This redistribution process should occur at  an  adequate rate to compensate the ever increasing speed of the aetherinos (between collisions) observed by OO.

    It is not yet clear if it must be supposed that our area of the Universe, where the time invariance of the physical constants has been  determined,  must  be surrounded  by  matter in a very restrictive way to maintain a rate of  "cooling" redistributing collisions capable to explain the stability of the aetherino's distribution observed by OO.  On the other hand the rate of the redistribution process depends on the rhythm of matter in a feedback process. Furthermore, the distribution observed in our vicinity has only in a very small proportion been affected by collisions with local matter since other facts force to suppose  that  most aetherinos can traverse  massive blocks of matter with very small chance of suffering collisions. Hence most of the aetherinos in our neighborhood have traveled long distances, from deep space (/time) without being redistributed. The distribution of the aetherino speeds observed by IO at the Earth's position must depend on the cosmological model of  the Universe which is far from being agreed.

    Alternately,   an expanding aether,  where the so called "local" inertial reference frames fly apart from each other, could perhaps account by itself for the cooling (i.e. shrinking) of the distribution that IO must observe.

    The aetherino's speed canonical distribution that is being used in this work was expressed in Eq [1-47] as:

                                                              [3-33]

    Therefore [3-33] seems an appropriate expression to describe a shrinking distribution in a model with no sources nor sinks of aetherinos. It apparently suffices to replace in [3-33]

                                                                             [3-34]

to account for a distribution that does not appear to change for OO (whose clocks slow down at the same rate).

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NOTE:  The considerations of Section 6 about a possible interpretation of the constancy of the speed of light suggest that the canonical distribution to account for most facts should be of the simple exponential type:

                                 [3-33a]

rather than of the Maxwell-Boltzmann type.

A more convenient form to write this distribution is:

                                 [3-33c]

where V_M  is the speed for which the distribution reaches its maximum. Again [3-33c] has the property that the total number of aetherinos in unit volume is N/4 and therefore not dependent on V_M .

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NOTE 3-9

    If a shrinking aetherino distribution needs to be introduced to explain other facts, then, first, to be consistent with other ideas proposed in this model (like the justification of material forces by the aetherino speeds redistribution at the sources) the elementary impulse defined in Eq [1-1c] must be redefined as:

                                                                      [3-36]

and interpreted like a transfer of an impulse q.v_r   occurring with a ("shrinking coupled") probability given by:

                                                             [3-37]

Supposing for example that the distribution shrinks according to [3-34], and applying Eq[3-8b] for the acceleration produced by an aetherinical force (with n=1 for a single SP):

     ( u<< V_M )     [3-39]

Calling  0.29 Q q s N V_M(0) = k m and integrating the differential equation  [3-39]:

                                                                       [3-40]

would be the slow down law for a "free" particle in an aether shrinking according to [3-34] if it where the case that Eq [3-38] was the exact expression of the drag force and not only an approximation for u << V_{M .}

It can be seen in this example, that as advanced above, u(t) is no longer of the type u(0).e^- m t and therefore it decreases in time at a different rate than the arbitrarily postulated shrinking given in [3-34].

It is nevertheless interesting to remark that if there is no shrinking (i.e. the aetherino distribution does not change in time for IO) the only drag slow down law that is able to explain without contradiction the inertia observed by OO in all "free" material bodies is u(t)=u(0).Exp[-mt] but if a shrinking is admitted (and therefore an epoch dependent aether) any law of the type u(t)=u(0).f(t) for the IO drag slow down of material bodies can in principle be imagined to take place and also show able to predict the inertia of all speed bodies observed by OO using the same reasoning as above. ( f(t) is a generic function of t like for example that of Eq[3-40] ).

    (The reason why, in the case of a constant aether, u(t)=u(0).Exp[-mt] has been said to be the only valid slow down law is that in this case the slow down rate of all bodies must be constant at all epochs and the exponential function is the only one with such a property. But if the aether is admitted to change with time then this changing aether can be blamed for the epoch dependence of the slow down rate).

--------

should instead be the natural election. 

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    In the case of a non shrinking aether the model has yet to calculate and predict which of the now called "universal" constants would change with time and at what rate and check the predictions with the experimental facts. It seems premature to attack this problem now. Nevertheless some descriptive considerations suggest that the model will better find its way admitting a shrinking aetherino distribution:

- First, a slow down law of the non shrinking type like u(t)=u(0).Exp[-mt] predicts that a body slowing down with that law can only travel a limited distance:

which is likely to present some difficulties even if m happens to be very small. (A slow down according for example to Eq [3-40] predicts instead an infinite advance as t tends to Infinity).

Newton's third law.

    The same is true in the aetherino model for the forces between two bodies since here the interaction is also transmitted at a finite speed, that of the aetherinos traveling from one body to the other.

4 - MASS AND REVISION OF NEWTON'S LAWS.

Section 3 can be considered just a simplified approach to introduce the main features of the model related to Newton’s laws. But a deeper description of the concept of mass of a body requires to take into consideration (1) the speed of the internal constituents of the body (relative to the body as a whole) and (2) the absolute speed of the body relative to the aether. The effects of these will be better analyzed using the expressions of the new dynamics shown in Section 12.

The above "sketch" of Newton's 2^nd law (Eq[3-32]) suggests that the concept of mass is represented in the model simply by n/Q. I.e. except for the constant 1/Q the mass of a body would be given simply by the number n of its Simple Particles. Pre relativistic 19^th century Physics would pose no objections to such simple interpretation of the concept of mass. But modern physics has established that:

1- Newton's 2^nd law is only an approximation for small (non relativistic) speeds of the body suffering the force. More precisely: a constant force does not produce a constant acceleration on a body but as the body increases its speed its acceleration starts to decrease. For high speeds of the body suffering the force, it can still be written F=M.a but keeping in mind that M, the so called "relativistic mass" (in the terminology of the early relativists) is a magnitude that increases with the speed of the body.

NOTE 4-3:

If the inertial-mass concept can be considered "explained" by the model, then, in what respects to its dimensional analysis, the situation can be considered similar to that of the kinetic theory of gases explaining the concept of temperature as a mean kinetic energy. No new basic dimension is needed for temperature since the dimension of this magnitude is derived from its theory and therefore from the other basic dimensions (L,M,T). This idea could be applied to the "explained" concept of mass with the consequence that M should no longer be considered a fundamental magnitude of physics (leaving thus only L and T).

--------------------------------

NOTE 4-4:

Once again it must be said that the calculations done in this work only pretend to give a hint on the descriptive possibilities of the model but do not claim to make exact quantitative predictions. As will be shown in Section 12, the inertial mass of a body varies with its absolute speed and that implies that a revision must be made of the magnitudes (energy, momentum,...) whose conservation is predicted by the model (in the interactions between particles of matter or in the interactions between matter and radiation). The theory of Relativity had to introduce a similar revision when it stated that the laws of Newton where no longer valid for high speeds.

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