The Origin And Nature Of AutoDynamics

1-2 The differences.

The difference between SR and AD is simple yet profound: their derivation. SR is derived using two frames of reference for describing a moving object: a reference frame and observer frame. AD eliminates the reference frame and uses only that of the observer. Carezani showed that the use of two frames for relativity was physically unjustifiable and mathematically superfluous.

All differences stem from this simple fact. The AD derivations follow two different paths. The first one is a systematic discussion of Galilean systems in relative motion (1-3). Here the AD concept of systems reduction is clearly espoused. The second one (1-7-2) is a small mathematical (algebraic) demonstration following the classic relativistic step of two observers, one at rest and the other in motion with respect to the light emitter.

1-3 A first derivation of Autodynamics

Frames in Relative Motion

Let us examine two parallel frames of coordinates x,y,z,t and x',y',z',t'. Frame F is moving with relative velocity v with respect to frame F'.

Fig. 1-1

Using the Galilean transformation, the abscissa x of the point P in F is given in F' (Fig. 1.1) by

x'= x + v t y'= y z'= z t'= t ....................(1.1)

The velocity of P observed by frame F' is,

v'= v .......................(1.2)

Introducing a third frame of reference F'',

Fig. 1-2, with respect to which frame F' is moving with relative velocity v1, the abscissa of P in F'' is

x''= x' + v1 t
x''= x+v t+v1 t = x+(v + v1) t ..............(1.3)

We are using three coordinate reference frames, F, F', F'', and we shall try to demonstrate that two of them are useless, inoperative, and superfluous, by doing the following analysis:

Fig. 1-2

1-3-1.- First: It is impossible to measure the abscissa of P in F from F', independently of the relative velocity v, supposing that P is moving with velocity -v with respect to F, because P will be at rest with respect to F', and, in this particular case, F' is useless, because it cannot observe a phenomenon at P.

1-3-2.- Second: We can also suppose that P is moving together with F with relative velocity vx, or with a relative velocity u with respect to F', because nobody in F' can distinguish the velocity vx from v, even though F' receives signals from F. The relative velocity u is the sum of vx and v, and the unique velocity measurable by F' is u. If F receives signals from P and it is transmitted to F', the observer in F' will measure the relative velocity u, because if frame F first transmits v and then v + vx, the observer in F' will think that the relative velocity v increases to u. It will be easier for the observer in F' to receive directly the signals from P. So, the reference frame F is unnecessary, and useless.

Presented as follows it is formally possible to demonstrate that either frame F'' or frame F is useless, and inoperative.

The velocity of the point P moving in F with velocity vx, measured from F' is

v' = vx + v .................(1.4)

Being x the coordinate of P in F when t = 0, its coordinate at time t will be x1 = x + vx t (1.5) Its coordinate in F' is

x'= x1 + v t
x'= x+vx t + v t = x +(vx + v) t .............(1.6)

Equation (1.6) and (1.3) are totally equivalent. Is it not the same to say that P moves with respect to F with relative velocity vx, which gives equation (1.6), as to say that P moves with respect to F' with relative velocity v, which gives equation (1.3)?

Equation (1.6) results from applying only "two systems of reference", F and F', and so F'' is not operative, and is useless. Equation (1.3) is obtained by applying "three systems of reference" but point P is at rest in F, v'=v, and consequently frame F is useless, and inoperative.

If point P is moving together with F an observer in F will never observe a phenomenon at P, that is to say, the self-observation of a point P is impossible.

Even so, we may insist upon the following: Can velocity vx and v be distinguished from each other while we are observing the physical phenomenon?. No. For each phenomenon there exist the phenomenon itself and the observer. The relativity of space and time is given by light speed, which is constant or, at least, has a finite value. Can we divide space in such a manner as to have an arbitrary quantity of reference frame?. A physical phenomenon naturally presupposes two objects: the observer and the object under observation. If the process is relative - and it truly is - then it should be relative to both, due to their mutual interaction.

Consequently, we shall define a " physical system " as one that consists of a moving point P -"the observed"- and a " reference frame of coordinates " -"the observer". A brief, formal and complete exposition follows1:

: Let us consider three parallel frames F, F', F'', Fig. 1.2, their origin in coincidence at the instant t = 0, and a point P of abscissa x in F. If F moves relative to F' with velocity v, and F' with respect to F'' with velocity v1, the coordinate of P in F'' is

x''= x' + v1 t x''= x + (v1 + v) t .......................... (1.7)

During this exposition we have presumed a relation of P with F even though P is at rest with respect to F. This, in our point of view, is not logical due to the fact that self-observation of a point is absolutely impossible. In other words, with a point P and a geometrical system F linked to it, we have no physics whatsoever. We need another system, one not linked to P, (and which we may individualize by its origin) in which we place an observer describing the physical changes, which P undergoes. In this case frame F is superfluous.

Equation (1.7) maintains their values if we suppose that at time t=0, the point passes the origin of F'' coincident at that instant. To make it easier we shall run up the prime values in the following manner: F' shall be F and F'' will be F', v is the velocity of P with respect to F, v1 the velocity of F with respect to F'; therefore (1.7) shall read

x'= x + v1 t x'= (v1 + v) t ................. (1.8)

This is equal to (1.7) when we only consider two systems. The foregoing analysis of systems in relative motion leads to the following definition: a physical system consists of two points in relative movement, P and F, or the origin of F. Up to now this has been known as a system F on one side and a point P at rest in the other. This new concept or conclusion will be used to analyze what happens in special relativity.

This analysis drives AD to simplify the Lorentz's equations in the form

.....................(19)

Lorentz's simplified equations.

These equations found conceptually through the above analysis, have a simple mathematical derivation without any new definition, axiom or hypothesis.

1-5 The law of relativity

AD accepts the two known postulates of relativity:

1-5-1 The laws of Physics are the same in all inertial frames of reference.

1-5-2 Measurements of the light velocity always give the same numerical value, regardless of the relative velocity of the observer and the light emitter.

The first postulate is basically the Galilean relativity postulate with an important change. Instead of mentioning only the mechanics, all laws of physics are included, including the electromagnetism.

The second postulate abandons the idea of absolute time and space. This is an issue that has occupied physicists to this day spurred on by Einstein's mention of a Stationary frame3 in one of his papers.

AD accepts these two postulates yielding new solution to many different problems that SR and GR cannot.

1-6 SR sum velocity

A clear example of the law of relativity is the SR sum velocity derivation. Another is the SR force derivation.

Lorentz's equations are:

.....................(1.11)

Writing Lorentz's equations in derivative form we have

...................(1.12)

Therefore

..................(1.13)

dx/dt = vx is the velocity of P in frame F that was supposed at rest as an initial condition. P is at rest with respect to F, but as a result of a mathematical operation, the point P starts to move. The differential of a distance, x, or the variation of x with respect to time is a velocity, and this presupposes an acceleration. That is to say, the spontaneous introduction of energy inside the system. The question is very simple: What is the origin of the energy that sets point P into motion? A popular answer by SR supporters is that we are doing kinematics and as a consequence we do not need energy. Is this what really happens in the real world when a physical phenomenon is studied? No. In the real world where a phenomenon can be repeated, energy plays a principal role. SR equations routinely are applied to dynamics phenomena, to mechanical phenomena. This apparently irrelevant observation acquires extraordinary importance when considering the phenomena and its relationship with acceleration and the law of energy conservation. As was mentioned above, this extra energy spontaneously introduced in the system needs to be carried away by the Neutrino to restore the loss of energy and momentum conservation because the SR equations yield values larger than the experimental ones. This equation also involves two mistaken solutions that will be mentioned here and developed in another section: Namely, that this equation does not maintain energy and momentum conservation if we compare its results with the SR kinetic energy and momentum equations. (See Appendix 7 and footnote in section 1-4). As will be show later in the derivation of force, the simplification of vx = v to obtain SR mass variation makes the equation (1.13) inoperative, because the two velocities are always equal which does not reflect reality (The combinations are infinite) as will be explained later on.

1-7 Autodynamics derivation.

There are two ways to arrive at kinematics. The first applies the concept (which has already been described) as systems or frames reduction (1-3). The second is formally following Einstein's classical derivation of the theory of relativity.

1-7-1 Derivation through concept.

F' being a relative reference frame with three coordinates x', y', z', Fig. 1-3, and a moving object P, the observer in F' will only measure different positions on the axis belonging to F' at time t and t' with the same criterion.

Fig. 1-3

If F is placed in the center of P, the coordinate x of point P in F will be zero but its velocity with respect to F' is still v, and if at time = 0 the origins coincide, Lorentz' equations will take an abbreviated form

........................(1.14)

This means that space and time are already relative to the observer in his own system and only depends on the relative velocity v of point P. (See A19)

Taking the derivation of equations in (1.14)

..............................(1.15)

Dividing dx' by dt' of equations (1.15), we have

..............................(1.16)

In words, the resulting velocity is the same as the original one, no matter the position of the point P. It could not be otherwise since no motive, force, or phenomenon exists to change the relative velocity v.

1-7-2 The mathematical derivation.
(See also A40.-)

Equations (1.14) may be obtained following a purely formal algebraic method. SR describes the phenomenon using two relative frames of reference. In AD we only have one reference frame.

Fig. 1-4

One of the observers A, is at rest with respect to the light emitter E, Fig. 1-4, and the other observer, A', is in motion with relative velocity v with respect to the first observer. This does not mean that observer A and emitter E are at rest. They also are in motion in the same direction, because in Nature, nothing is "at rest," even though both at the same velocity. The preceding descriptions - 1-7-1 and 1-7-2 are concretely expressed by the AD thesis:

The coordinates x', y', z' and the time t' are only a function of t.

(x', y', z', t') = f (t)............. (1.17)

This is demonstrated as follow:

Each observer will deduce his own coordinates in such a way that

x2 + y2 + z2 - c2 t2 = 0 ..............(1.18)

x'2 + y'2 + z'2 - c2 t'2 = 0............... (1.19)

Relating these measurements we shall always have for parallel axes

x' = v t
y' = y ...	...(1.20)
z' = z t' = t

............

Replacing and evolving

2 v2 t2 + y2 + z2 - 2 c2 t2 = 0

y2 + z2 = ( 2 c2 - 2 v2 ) t2............... (1.21)

Equalizing coefficients of t^2 in equation (1.18) and (1.21) we have:

2 c2 - 2 v2 = c2............ (1.22)

And is given by

............................(1.23)

Replacing in the equations (1.20) we have

...............(1.24)

These are the equations as (1.14): The Lorentz's simplified same equations. For historical and clarification purpose we will call those equations the "Carezani's equations" from now on.

1-7-3 Derivation's detailed analysis.

To clearly understand the preceding derivation, a detailed analysis follows:

If emitter E is moved successively closer to A', nothing changes. When the emitter coincides with position A', the Observer at Xo (A) will no longer see the other Observer, but only the Phenomenon at A' (the emitter). (System or frame reduction).

Now we have only the Observer at A, equation (1.18), and the phenomenon at A'. Nothing changes for the Observer at A, because for him, only the Observer and Phenomenon exist. If we exchange the role of each, nothing changes: A' is now the Observer, equation (1.19), and A is the phenomenon that contains the emitter that transmits the information at velocity c to the Observer at A', receding from A.

We need to realize that X' is only the position A' - that is, the distance (x') between the two positions A (X) and A' (X'). Then

X' - X = x'

This value is given by the first equation in (1.20).

Now we know that a relationship exists between the Observer at A' and the Phenomenon at A, but for the moment we don't know what this relationship is.

If v is known, the only variable is t, and the factor that leads to RELATIVITY ( , alpha) needs to be the same, because t is the same in both equations - Equations (1.20).

Of course, t' will be the time that Observer A' (X') observes on his clock, which is larger than the time at A (X), and equal to t, because he (A') is receding from A. Relativity is conserved, because c is constant for the Observer and Emitter, independent of their own velocity, and the time measured by t' is a real time. (See A19).

Of course, Fig 1-4 in 1-7-2 is "symbolic." The "complete" Figure would be the following:

Fig. 1-5

1-8 Remark on vx = v.

In section 1-7-1, equation (1.16), AD demonstrated that v' = v. It is interesting to point out that if we set the velocity vx = 0 in the Lorentz equation (1.13) in section 1-6 (known as the SR sum velocity equation), we also obtain v' = v. This will be repeated later in a similar situation with respect to the SR derivation of force.

A new, easier and complete explanation is in "Superfluous System," which follows.

[A1]
[A2]