12 – Absolute Dynamics.

1- Momentum and Kinetic energy.

Let S_A be the reference frame associated with the aether. All the distances, velocities and accelerations mentioned in this subsection 1 will be referred to S_A and should be considered magnitudes measured by the Official Observer OO that uses ordinary clocks.

Suppose that a body of mass m is acted by a constant material force F (called material because it is originated by matter and to distinguish it from the aether drag force).

Hypothesis: The acceleration of the body produced by a constant material force F is:

[12-1]                              

where

u     is the absolute speed (i.e. relative to the aether) of the body
c     is the speed of light in vacuum
m     is the frame invariant property of bodies called mass.

The vector equation relating the force F, the absolute velocity u and the acceleration a is:

[12-1c]                         

In other words the acceleration produced by the force is always along the direction of the force, and as will be seen in the coming new version of Section 5, the force that a body A exerts on a body B that moves with a velocity u relative to A is always along the direction AB at the epoch of observation of the force (independently of the angle that u is making with AB).

The expression [12-1c] will be defended below (in subsection 3 of this article) arguing about the nature of the aetherinical forces originated by matter.

NOTE: In earlier versions of this work it was instead believed that the needed relativistic corrections to Newton’s 2nd law a = F/m could be explained by microscopic discrete aetherinical considerations affecting the macroscopic acceleration  acquired by a particle (suffering the collisions of the aetherinos that implement the force).  It was believed that it could be shown that the macroscopic acceleration suffering a constant force decreased when the speed of the body relative to the aether increased. I.e. that  mass should increase with absolute speed due to microscopic discrete aetherinical considerations. But the computer simulations of the effect of aetherinical forces on bodies moving through the aether, do not manifest such mass increase. The theoretical analysis made on Annex M pretending to describe such effect is neither convincing.

It is of interest to deduce the speed u[t] of the body as a function of time supposing that u[0] = u₀ and supposing that the body is accelerated by an external constant force F according to [12-1]. Solving the following differential equation:

gives

[12-2]                   

Momentum.

It will be assumed, in similarity with Newtonian Physics that a constant force F applied on a free body during a time Dt increases a dynamic property of the body called its momentum by an amount F Dt. But now, although in this scenario of an aether it is controversial to consider that the body is free, it will nevertheless be defined that, in the proposed "aether dynamics" there is an intrinsic vector property p_A called momentum of the body in the absolute frame or absolute momentum that is zero when the body is at rest in the aether and such that when a force F[t] is applied to the body between the epochs {t₁, t₂} it increases (according to the classic law) by an amount:

[12-3]                              

Notice also that a given absolute momentum always implies the same speed of the body independently of the history of impulses that it has suffered to reach such speed. For example suppose that to a body of mass m, initially at rest in S_A, is applied (always in the same direction) first a force F during a time DT and next a force f during a time Dt:

The first impulse F DT endows the body (according to [12-2] with u₀ = 0) with a speed:

[12-4]                         

and the second impulse increases such speed (according again to [12-2] with f instead of F, Dt instead of t, and u₀ = u[DT] ) to the final value:

[12-5]                    

in which the symmetry of  (f Dt + F DT) reflects the commutative feature of the impulses.

Due to the commutative character of the impulses applied to a body, an easy way to calculate its absolute momentum as a function of its absolute speed  u   is to imagine that, starting from rest, the body is accelerated by a constant force F during a time interval DT and therefore its absolute momentum is p_A = F DT.   Replacing the F DT of Eq[12-4] by the symbol p_A and solving for p_A:

[12-6]                           

that can be considered a scalar equation (valid for u > 0), but that can be converted in the following vector definition of absolute momentum:

[12-6b]                      

where |u| is the modulus of the absolute velocity u of the particle.

Kinetic energy.

Again it will be assumed, in similarity with Newtonian Physics and the above treatment of momentum that a constant force F applied on a free body along a path Ds increases a dynamic property of the body called its kinetic energy by an amount F.Ds (dot product of 2 vectors) . And again it will be defined that, in the proposed "aether mechanics" there is an intrinsic scalar property K_A called kinetic energy of the body in the absolute frame or absolute kinetic energy that is zero when the body is at rest in the aether and such that when a force F[t] is applied to the body between the positions {s₁, s₂} it increases (according to the classic law) by an amount:

[12-8]                           

that, as usual, when the velocity u[t] of the body is known as a function of time and so is the force F[t], the above integration can be performed using time as the integration variable considering that ds = u[t] dt

[12-8b]                          

Note: Similarly to the commutative character of the impulses shown above it can also be proven that independently of the order of the works (F_i.Ds_i) applied to the body it always reaches the same final speed. Therefore:

The absolute kinetic energy K_A of a body of mass m in the absolute reference frame can deduced evaluating for example [12-8b] for the case in which the body is accelerated by a constant material force F starting from rest. The above Eq[12-4] (substituting DT by t) gives the speed u[t] of the body at any epoch t in such circumstances :

[12-9]                              

that solving for the elapsed time t as a function of the speed u acquired by the body gives:

[12-9b]                             

According to equations [12-8b] and [12-9], the absolute kinetic energy of the body at the epoch t can be expressed by:

[12-10]                             

that can be expressed as a function of the absolute speed u of the body substituting t by its function given in [12-9b]:

[12-11]                             

The newly defined absolute momentum and absolute kinetic energy are now compared with the momentum and kinetic energy of Einstein’s Special Relativity given by:

[12-12]                     

[12-14]                     

the following graphic compares the speed dependence of the absolute momentum [12-6] with that of the Relativity momentum [12-12] :

                    Fig[12-1]    Absolute momentum (red curve) compared with the momentum of Relativity.
                                 

The following graphic compares the speed dependence of the absolute kinetic energy [12-10] with that of the kinetic energy of special Relativity [12-14] :

         Fig[12-2]  Absolute kinetic energy (red curve) compared with that of Relativity.

 

Conservation: It will be assumed that in closed systems the net absolute momentum is always conserved and the net absolute kinetic energy is conserved in elastic collisions of particles.

A quantitative comparison with Special Relativity in a high energy collision.

Consider the following elastic collision between two equal particles:

Particle 1 (the projectile) of rest mass m has a high initial lab speed v_i1 (close to c).
Particle 2 (the target) of rest mass also m, is initially at rest in the lab.
Suppose that the collision is such that both particles emerge from the collision with the same lab speed (v_1f = v_2f)

It is often invoked in support of Special Relativity (SR) that in such kind of high energy collisions the particles emerge with trajectories whose angle in the lab reference frame decreases when the speed/energy of the projectile is increased. Such experimental result can not be explained with classical mechanics that predicts instead that the particles should emerge in all cases (whatever the initial speed of the projectile) with trajectories whose angle is p/2.

The conservation of the quantities "absolute momentum" and "absolute kinetic energy" proposed in this section do also predict that the particles should emerge from the collision with trajectories whose angle in the lab reference frame decreases when the speed of the projectile is increased. Although here, for a given projectile speed, the predicted "emergence angle" differs quantitatively with the one predicted by SR it can be seen that (the same as with SR) the predicted angle tends to zero when the projectile is given a speed increasingly closer to c. It can be argued in defence of the new proposed expressions of the "absolute momentum" and "absolute kinetic energy" that to decide which dynamics fits better the experiments the speed of the projectile should be measured directly with good precision. Inferring the projectile speed from indirect theoretically related quantities (like energy) is not considered valid because it is the theory itself that is being tested. But it seems difficult to distinguish with direct measurements for example a speed .99c from a speed .9999c and it is nevertheless necessary to dispose of such resolution of speeds to evaluate the results of the comparison.

2- Non invariance.

According to the proposed expressions it should be possible in theory (adding enough energy) to accelerate any massive particle to an absolute speed smaller than c but as close to it as wished. Therefore, according to the Galileo transformation of velocities (the one adopted in this model), in relation to an inertial reference frame S moving at a velocity -v_A relative to the aether, a particle could have a speed c+v_A (bigger than c). That implies that the above expressions of the absolute momentum and kinetic energy can only be valid for the reference frame of the aether. Of course, trivial expressions for the absolute momentum and absolute kinetic energy of a particle can also be written using as a variable the velocity v of the particle relative to any frame S. For example if the reference frame S of description moves at a velocity +v_A relative to the aether (i.e. to the absolute frame S_A) the absolute momentum of the particle would adopt the form:

[12-17]                 

in which the specific velocity v_A of the frame S relative to S_A appears in the expression, but such momentum would still be the absolute momentum.

Note:  The | v +v_A|  appearing in [12-17] is the modulus of the vector sum  v +v_A

Nevertheless, for description purposes in some context, it might be useful to define also two frame dependant (non absolute) magnitudes that will be called "frame momentum" and "frame kinetic energy":

In an inertial reference frame S (different in general from the "absolute" reference frame S_A associated with the aether at rest), the "frame momentum" of a body of mass m and velocity v relative to S will be defined as the net impulse given to the body to move it from the condition of zero velocity (in S) to its final velocity v.

[12-18]                               

but since to any velocity v in S corresponds an absolute velocity u in S_A given by u = v +v_A and since it has been shown that the net impulse needed to change the absolute speed of a body from u_i to u_f is independent of the history of impulses given to the body (commutativity), the frame momentum dependence on the velocity v can be computed as:

[12-19]                            =

where a sub index _A indicates that the quantity is referred to the Absolute (or Aether) frame. Therefore:

[12-19b]         

becoming evident that now the modulus of the frame impulse depends in general of the final velocity (including its direction in space) of the body relative to S and not only on its speed.

Noting that the first term of [12-19b] is the "absolute momentum" of the particle and that the second term is independent of its velocity in S, then since the net absolute momentum of an isolated system of particles is conserved in collisions, it can also be asserted that the net frame-momentum is conserved.

A similar definition and properties can be put forward for the frame-kinetic energy. (Omitted here).

Although the laws of the dynamics proposed so far are of similar form in different inertial frames of reference, they all include a constant v_A (velocity of the reference frame in relation to the aether) that is specific of each frame S and therefore those laws are not invariant in the strict sense that is mainstream in Physics.

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

It must be realized that when a body changes its direction in the Earth's lab without changing its lab speed (e.g. when a particle moves in the lab in a circle at constant speed or when the moon orbits the Earth) it may happen that it suffers fast changes in its speed relative to the aether (if the lab is moving in the aether). But according to the proposals of this section it must be assumed that when a fast circling body turns into the semi direction of the aether wind its absolute kinetic energy decreases and when it turns into the opposite semi direction its absolute kinetic energy increases. That does not imply that, to conserve its energy, the body must suffer compensating speed changes in the lab reference frame if it can also explained that the potential energy of the system (source of force + target of force) changes simultaneously in an opposite amount so that the total energy (i.e. absolute kinetic energy of the interacting bodies plus potential energy of the system) remains constant. This would be consequence of the fact (unevaluated yet but consistent with the model) that, when the material forces are analysed at the aetherinical level it should happen that, if the redistributing matter source of the force moves relative to the aether, it produces an anisotropous disturbance in the aether.

3- Aetherinical interpretation and plausibility of the hypothesis [12-1].

An aetherinical force f acting on a material body gives it (according to the model, see Section 3) an IO acceleration equal to f/m where m (mass of the body) is a constant of the body related with its number n of Simple particles and with some constant Q relating dimensionally an aetherino’s elementary impulse on a Simple Particle with the speed increase produced on it. The net aetherinical force on a body that moves relative to the aether and that is also subject to an external material force F_M is just the sum of both forces f = F + F_D . If the body has a mass m and a velocity u_I relative to the aether it was found in Section 2 that it suffers an aether drag force that can be written as

[12-25]                       F_D   =  - m m u_I

therefore, considering to simplify just the 1-D case in which F and u_I are parallel (and working with scalars instead of vectors), the IO acceleration suffered by the body is:

[12-26]            

It has also been shown in the first sections of this work that the bodies accelerate instead for the Official Observer OO as if the aether drag force did not exist (whatever their speed relative to the aether) and that the OO acceleration that the body suffers is given simply by  F_M/m  (where F_M is the "material" force originated at some matter that exists in presence of the body suffering the force). Such matter redistributes the speeds of the aetherinos that collide with it and therefore the body target of the force detects a flow of aetherinos different than if it was alone in the aether.

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

where u_I and t are respectively speeds and times evaluated by the Ideal Observer (for which the aetherinos travel at constant velocity in some referential frames called rectilinear).

If the IO acceleration is due to an aetherinical material force F_M (plus the aether drag force) then substituting a_I by its value given in [12-26]:

When the interest is in comparing the forces or the accelerations at only one instant of time it can be supposed that the IO clocks are synchronised with the OO clocks at such instant making dt = dt and therefore setting in such instant both kinds of clocks to read t = t = 0.

Note: [12-31 is actually valid at all epochs because it has also been found that the models of redistribution predict that the aetherinical material forces decrease with time according to Exp[- 2 m t] cancelling therefore at any epoch the exponential factor of [12-30]

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

Assuming then that the acceleration that the Official Observer measures in a body (acted by an aetherinical force F_M) is given by a = F_M/m, it must also be assumed that if the acceleration that he observes is of the form [12-1] a = F/m (1-u²/c²) it must be due to the fact that the aetherinical force acting on the body is of the form

[12-32]                              F_M = F (1- u²/c²)

although the mainstream physicist believes that according to his theories the body is acted by a constant force F that does not depend on the absolute speed of the body.

But the expression F_M(u) = F_M(0) . (1- u²/c²) has the behaviour predicted by the model for the aetherinical material forces based on redistribution of aetherinos. More precisely it has been calculated in Section 5 that the force exerted by a body A, that is at rest in the aether, on a body B that moves at speed u relative to the aether is of that type. For the calculus it has been supposed that the body A creates a redistribution of aetherino speeds A according to the suppositions presented in Annex D.

NOTE: The prediction of the model of the force that a body A moving relative to the aether at a speed u produces on a body B that is at rest in the aether is not available yet. (Neither is of course available yet the prediction of the force in the more general case of a body A that moves at velocity u_A on a body B that moves at velocity u_B). To predict the value that the model gives to such forces it must be first calculated the exact form of the redistribution produced by a body that moves relative to the aether and the task seems consuming.

It will probably be considered controversial to attribute the increasing "difficulty" in accelerating the bodies (see the basic law [12-1] ) not to an increase with absolute speed of the mass of the target body but rather to a general aether speed dependence of the material forces. This raises the question: if the force acting on the body is (1- u²/c²) F, instead of F, how is it that the impulse and the work "done" in accelerating the target body has been assumed above to be given respectively by F Dt and F Ds instead of by F (1- u²/c²) Dt and F (1- u²/c²) Ds ? (For example, notice that if the "work done" on a particle is assumed to be based on the integration of F (1- u²/c²) Ds instead of F Ds then it is evident that the result would not differ from that of Newton’s dynamics and the kinetic energy of a particle of speed u would be computed to be 1/2 m u²). The reason is believed to be the following: When two bodies (or particles) A and B interact (e.g. due to the repulsion of their electric charges) the aetherinical force of the body A on the body B is in general of different magnitude than the aetherinical force of the body B on the body A. In other words, the interpretation of Newton’s 3^rd Law as F_AB = -F_BA is not correct. It is the reaction force F acting on the body A "source" of the force that must be accounted for in the evaluation of the impulse and of the work "spent" on the "target" body B. The terminology "work done to" or "impulse given to" often used in mainstream mechanics is therefore confusing and should be substituted for example by "work spent by" the acting body to accelerate the target body or "retro-impulse suffered by" the acting body to accelerate the target body. It is important to notice that when it is instead the body B that is considered the subject of the force acting on A the situation is not in general symmetric to the former because in the proposed "absolute dynamics" the relative velocity between A and B is not the only factor that affects the forces because now the intensity of the aetherinical forces depends also on the absolute velocity of the redistributing body being considered "source" of the force and specifically on the direction in which is placed the target body relative to the source body.

Some diagrams have been drawn to give an idea of how the aetherinical forces are expected to behave in some example situations. The particles A and B are supposed to be in all the examples at the same given distance apart at the moment of the evaluation of their respective forces. The forces of the examples correspond to repulsive forces between equal-sign electric charges. (It should also be remembered that, as said above, the acceleration suffered by a body acted by an aetherinical force f is just equal to f/m).

   It has been said that expressions of the type [12-33] are the "natural" predictions (i.e. consistent with other features of the model) for the force produced by a redistributing matter on a particle that moves at speed u relative to the aether. But the examples of Annex A have actually been based on the specific supposition that the redistributing matter Q, source of the force, was at rest in the aether.  It therefore remains to be shown that other sources of force (originated at matter that is not at rest in the aether) are also consistent with the dynamics proposed.  (See in this respect the example diagrams discussed).  However the form of a plausible redistribution produced by a body Q moving in the aether has not been computed yet. The reader is invited to do so. Actually the reader is encouraged to develop, test and correct the model within its basic paradigms.

 Home               Example diagrams