INDEX
CONTENTS




Lorentz and AD Transformations

 

This article collects in one place several topics related to the Lorentz and Autodynamic (AD) transformations.

 

The argument that the transformation needs to be linear and that successive transformations need to be invariant applies to AD, of course.  That is, in going from T1 to T2 and from this to T3, T3 must be equal to T1.  However, the equations that satisfy that transformation in Lorentz drive SR to a contradiction because TWO observers will see different values according to each one's state of motion.
[1]  Contrarily to SR, in AD TWO observers in different Inertial System will deduce the same value independently of their state of motion. The AD equations satisfy the invariance and physical reality is equal for all observers.

 

If Lorentz’s transform from T1 to T3 forms a “group," it also forms a “group” in AD, even though in AD the concept is physically irrelevant, because the concept of groups is only a mathematical concept unrelated to any physical phenomenon.

 

The transformation needs to be linear, but we need to know what should be transformed from the physical point of view, not from a blind mathematical transformation.

Though there are many ways to effect the transformation using Algebra, Calculus, or Determinants we will use Algebra since it is the simplest.


The Lorentz Transformation.

Fig 1.

 

 

If the light source is at the origin of the frame O (O for Observer) and a spherical electromagnetic wave is emitted at t = 0, the wave front will be at a distance ct  (Point P) from the origin at time t. The distance to Point P is

 

                        (1)

 

or

                      (2)

 

If O’ is the moving reference frame, it will see the same wave front at P at a distance c t’ at time t’ and the equation is

 

                 (3)

 

or

 

           (4)

 

The transformation will be invariant if replacing x’ y’ z’ and t’ as function of x,y,z and t in equation (4) the two equations (2) and (4) give an equal value.

 

                      (5)

 

That is:

 

0 = 0

To find the equations Lorentz used:

 

x’ = a x - b t         (6)

 

t’ = e x - f t           (7)

 

a, b, e, f  [2] are values to be found.

 

If O’ is in motion with respect to O, Lorentz’s distance is

 

x’ = x - v t     (8)

 

.

Equation (6) is rearranged to

 

x’ = g (x - v t)        (9)

 

The function of t is v t because x is a distance and v t is another distance to be added or subtracted to x.

 

Here is the problem. Independent of the velocity that O’ has with respect to O, x is a constant distance to O and will stay constant independent of the O’ velocity with respect to O. Contrarily, if x is affected by a value that will be function of a variable velocity v, as really happens with the Lorentz’s coefficient , x will be not a constant value for each change of velocity of O’ with respect to O. 

As the values a, b, e, f need to be determined in equation (6) and (7), there is a possibility that a could be equal to ONE, but looking at equation (9) shows that a = b = g and, as g is not ONE, consequently a ¹ 1.

Precisely as in the Lorentz equation (12)

 

a = b = g = 1 / root (1 – b^2) = g      (10)

 

We can see this clearly, because equation (9) can be written as:

 

x’ = g x - g v t          (11)     

 

The very well known Lorentz equations are: (See Footnote 5)

 

                         (12)

 

                    (13)

 

These equations must satisfy the Lorentz transformation given by equation (5).

If in equation (5), setting y’ = y and z’ = z, x’ and t’ is replaced by its values in equation (12) and (13) we have

 

        (14)

 

               (15)

 

Simplifying and multiplying by (1 - b2)

 

          (16)

 

 

      (17)


0 = 0   


The result is 0 = 0, but this result doesn’t say anything beyond the initial condition of v = c or a different velocity such as c = v, as happens in AD.

This mathematically correct solution involves physically wrong results given by equations (12) and (13), as we will show using very simple examples.

Setting these values (using exactly the same example as in Feynman’s paper “Superfluous System” in the SAA’s Home Page):

 

x = 100 m         t = 10-7 s             v = 0.8 c               c = 3 10-8 m

 

Applying equation (12) and (13) we have

 

           (18)

 

           (19)

 

 

These two results don’t make sense.

The first, in equation (18) shows that O’ traveling to the point P's direction is far away from it and the time of travel is negative.


Distance x inserted in equation (6) or (9) or (11) or (12) will automatically increment its value because the root also divides x, this being erroneous because Point P is supposed to be “at rest.” The same happens with t. We need to point out here that v is not the phenomenon (P) velocity with respect to the Observer O’. v is the constant velocity between System O’ in relative motion with respect to O (See Fig. 1). Nevertheless, the velocity v, used in all the SR equations, is the velocity between Observer and Phenomenon and in this case the System O’ is not necessary. The contradiction is evident because in SR the phenomenon velocity will result from the x variation. (Phenomenon position through a mathematical differential that gives vx (equation (1.13) on page 16 on the AD’s book)) even though x is a constant distance.

We will see that in AD everything makes sense.

 

 


AD’s Transformation

Using Fig. 1, AD’s reasoning is exactly the same as using Fig. 2. (See footnote 3)

           

  Fig. 2.       

 

When Carezani found the physical contradiction in Lorentz’s transform, he simultaneously discovered that to achieve relativity, it is not necessary to “introduce” two systems of coordinates to observe a phenomenon, even though he respected the Lorentz mathematical transformation for ONE Observer (O’) that sees the object's (wave front) motion from position X to X’(Fig 2). As will be shown later, this is not a unique solution, because the same will happen when “two systems” are involved. (See further “Two” frames in AD).

[In Fig. 2 the Observer “at rest” is O’]

The electromagnetic wave starts from the frame origin of O’. Distance x is the distance from the Origin of the System of Coordinates, x = y = z = 0 to point X, which is given by c t, or x = c t, written in the classic notation is:

 

x2 – c2 t2 = 0            (20)

 

When the electromagnetic wave front moves to point X’ (y’’), the distance is given by x’ = c t’ to Observer O’ and

 

x’2 – c2 t’2 = 0          (21)

 

The transformation needs to be invariant, that is

 

x2 – c2 t2 = x’2 – c2 t’2         (22)

 

That is  0 = 0

 

This equation (22) is exactly the same as equation (5) after setting y’ = y and z’ = z.

Remembering that the velocity c is constant and independent of the Observer or Phenomenon velocity, t’ is the time that Observer O’ measures at the origin of his coordinate system.

[In AD, only one system is needed to build a relativity theory.]

 

 

“Two” frames in AD.

There's a misinterpretation that AD cannot use “two” systems to observe a phenomenon, or that it cannot interchange information between “two” systems observing a phenomenon. AD says that in Physics, only a single System of Coordinates or Frame and only ONE Observer is needed. AD says that it is possible to build a Physics in which the Phenomenon is directly observed by any number of observers in relative motion and to interchange information between them. AD also says that we cannot “use two tied” frames (Lorentz’s transform) to observe a phenomenon because this introduces a false physical solution even though the mathematical solution is right. This doesn’t mean that we cannot use two independent (untied) systems to observe a phenomenon.

What does this mean, practically speaking?

In Lorentz, the constant distance x is introduced in the equation that defines two systems in relative inertial motion, and x is automatically decreased or increased by the Lorentz coefficient g. In Lorentz, Fig 1, x “is tied” to the g relativistic coefficient and is simultaneously “tied” to the System O’ in motion represented by v t.

 

(Equation (23) (Lorentz increases what is constant and Lorentz and SR are equivalent to applying the Kinetic Energy equation to an object at rest).

 

x’= g (x + v t) = g x + g v t      (23)

 

In AD, the constant distance x1 (Fig. 3), which is not related to any motion in its system or frame is not introduced in the equation that defines the relative motion between two different positions or between two Observers, who are not “tied” together through the g coefficient (Equation (23)).  In AD, x1 “is not tied” to the g relativistic coefficient. (Equation (24). P, at x1 distance from the origin of O, is moving together with O in Fig. 3 and O as the inertial frame is moving with respect to O’. Consequently, the distance x1 is constant and distance x is the only distance that will change. (Fig.3). It is impossible to apply the Principle of Relativity to something that is not in motion, P with respect to O or x1.

(AD doesn’t increase what is constant and consequently AD doesn’t apply the Kinetic Energy equation to an object at rest).


x’ = x1 + g ( v t)          (24)


 Fig. 3.                                                    

Using two systems of coordinates or frames, O and O’, will not change anything under the AD concept. Starting [3] the electromagnetic wave from the origin of frame O’, an observer will measure a distance x at time t equal to x = c t, written in the classic notation as

 

x2 – c2 t2 = 0                         (25)

 

The wave front will reach P at distance x’ at time t’ and the equation is


x’2 – c2 t’2 = 0                       (26)

 

We can write equation (25) and (26) as

 

x2 – c2 t2 = x’2 – c2 t’2                 (27)

 

The transformation must also be invariant if replacing x’ and t’ by their values as a function of x and t when the velocity of one system with respect to the other is v, giving

 

0 = 0

                     

Equation (27) is equal to equation (22) and also to the Lorentz equation (5).

 

Carezani used equation (22) or equation (27) to obtain the AD Relativistic Coefficient equal to g.[4] 

 

Using the traditional notation, we have

 

x’ = g v t               (28)           

 

y’ = y

                            

z’ = z

t’ = g t                   (29)

 

Equation (28) is equivalent to Lorentz equation (6) and equation (29) to equation (7).

From this, it is straightforward to get the AD equations. This creates a Physics, not just a Mathematics. It is Physics because the Principle of Relativity is applied to what is in motion, System O (x) not to what is at rest, P (x1). The Lorentz mistake – which creates only a Mathematics – is to apply the Principle of Relativity to that which has no motion, P (x1).

 

         (30)   

 

         (31)   



If in equation (27) x’ and t’ are replaced by the values in equation (30) and (31) under the initial condition that v = c (Electromagnetic wave propagation) the equation (27) is invariant and simultaneously is valid when c = v without involving any contradiction, as will be shown in the numerical examples.

            (32)  

 

    (33) 

 

        (34)  

 

Simplifying

 

                (35)   

 

               (36)  

 

The equation's right term equals 0 if v = c, the initial condition: An electromagnetic signal traveling at c with a radial propagation makes the transformation invariant. If v = 0, x is equal zero. This is a physically correct value because v = 0 (See footnote 4) means that the body has no motion and x = 0. What happens if c is different, that is equal v? Evidently if in equation (36) c is taken equal to v, the right side term is also 0.

 

 

Numerical examples. (SR and AD)

To show the Lorentz incongruencies, nothing is clearer than numerical examples.

Using exactly the same setting as in the “Superfluous System:”

x = 100 m         t = 10^-7 s             v = 0.8 c               c = 3 10^8 m

As is shown in equations (18) and (19), the result are:

 

x’ = 126. 666 666 666 m              (37)

 

t’ = - 2.777 777 777 10-7 s           (38)   

 

 

Time is negative, which is an absurdity.

Of course, SR believers always invent some ad hoc hypothesis to explain any SR contradiction. In this particular case, it will not work because the ad hoc hypothesis is the Lorentz postulated arm contraction to explain the Michelson-Morley experimental failure to detect the Ether. But the Lorentz contraction concept applies to material objects such as the Interferometer bars used by Michelson-Morley’s experiment. And what is worse, the Lorentz concept creates a dilemma: Length contraction or Ether! If there is length contraction, the Ether exists. The SR supporters don’t believe in the Ether but accept the Lorentz contraction and the Einstein time dilation!

No serious physicist can understand how an Inertial System produces the force to provoke the Lorentz contraction on material bars, nor understand how something that doesn’t exist, the force, can contract something immaterial such as length or space.

SR applies the concept of length contraction to distance, which is space. But as the Ether doesn’t exist (Aberration) the Lorentz bar contraction or the SR length contraction are both flawed.

Nevertheless, even though the concept is erroneous, SR insists on applying it to space as an ad hoc hypothesis to correct the failure in the Lorentz-SR equations to calculate distance and time.

Nevertheless, in the present example, if the distance could be corrected, the time will continue to be flawed.

 Correcting x = 100 by the Lorentz’s contraction equal 0.6, the distance is x’’ = 60 m and

 

 

  (39)

 

Time continues to be negative, which make no sense.

 

SR. (First case). [5]

 

      (40)     (35)


 

                                                                                                                                                  (41)        

 

 v’ = 1.127  272 c          (42)

 

 

When Frame O, Fig. 2 and 3 [6] passes Frame O’, the respective observers synchronize the clocks, interchanging information.

O is telling O’ that he will travel 10-7 second at v = 0.8 c and the distance to Point P is 100 meters.

When O’ is an SR Observer he thinks: Observer O is trying to deceive me. He is telling me that his distance to P is 100 m but I know that the length is contracted. He applies the Lorentz contraction measuring 60 m because the Lorentz factor g with v = 0.8 c is equal to 0.6 but he will not apply the value to time equation (13) or (38). He will make a trick applying the value directly to the result (39) and x’’ = 206.666 666 666 – 60 = 146.666 666 666 m.

 

He will make another trick: He will not apply time dilation to t (10-7 s) and will not introduce the length construction to x in the original equation (39).

Dividing 146.666 666 666 by the “normal time” 6.111 111 111 (40) he get v’’ = 240 000 000 m or 0.8 c. The trick is working! Applying the same concept directly to time dilation with the results in equation (41) we have 6.111 111 111  10-7 / 0.6 = 10.185 185 185 10-7

Now

 

v’’’ = 146.666 666 666 / 10.185 185 185  10-7 =  144 000 000 m/s = 0.48 c This doesn’t work!

 

Applying SR’s rules, the failure is evident: v’’’ is not 0.8 c!

 

 

The technique is simple: To pass off the contraband of a solution that gives good results, even though it violates the rules of SR.

 

What happens if SR works with its own concepts and rules?

 

Lorentz’s contraction is 100 * 0.6 = 60 m and

 

               (43)

 

     (44)

 

                               v’’’’   = 1.076 923 c               (45)

 

           

Without applying any trick, the velocity of System O is not 0.8 c: It is faster than Light!

Applying the other rule Time Dilation we have

 

 

           (46)

 

          (47) 

 

                                      v’’’’’  = 0.857 142 c                (48)

 

Accomplishing all SR’s rules the velocity is not 0.8 c: It is 0.857 142 c

 

The failure is total, because the equations were obtained from the initial condition that the electromagnetic wave travels at c, not v’ = 1. 127  272 c or v’’’’ = 1.076 923. In the example, System O is traveling at 0.8 c, not at any other velocity.

 

 

AD. (First case).

               (49)

 

 

       (50)

 

 

    (51)

 

In units of light velocity:

 

 

              (52)

 

This is System O traveling at velocity v. AD finds the right solution without any tricks or ad hoc hypotheses.

 

SR. (Second case)

In this example only the value of the distance is changed.

 

x = 200  m         t = 10-7 s        v = 0.8 c  m/s             c = 3E8  m/s

 

 

     (53)

 

                       (54)


v’ = 373.333 333 333 / 10.555 555 555 10-7  = 353 684 210 m/s                   (55)

 

v’ = 1.178 947 369 c

 

A velocity larger than c.                           (56)   

 

Now, employing the contraction trick

 

 

x’’ = 200 x 0.6 = 120 m                       (57)

 

x’’’ = 373.333 333 333 – 120 = 253.333 333 333                    (58)

 

v’’’ = 253.333 333 333 / 10.555 555 555 10-7 = 240 000 000 m    (59)

                     

Taking the Light velocity as the unit

 

v’’’ = 0.8 c

 

 

Of course, applying the same trick, it is "working."

 

Now we will apply the Lorentz contraction correctly

 

 

       (60)

 

 

                  (61)

 

v’’’’ = 240 / 7.037 037 037 10-7 = 341 052 631 m               (62)

 

v’’’’ = 1.136 842 c.



Applying the time dilation rule

 

 

           (63)

 

v’’’’’ = 240 / 8.111 111 111 10-7 = 295 890 411 m                     (64)

 

v’’’’’ = 0.986 c                                  (65)

 

 

AD.(Second case).

In AD, nothing changes regarding v and t since v and t have the same values. All that changes is the distance x, now equal to 200 m. As in the AD equations the distance x is not included in the relativistic equation because it is constant in every case when v changes values, the total distance is different but the velocity v is the same.

 

 

SR.(Third case)

Here only the time changes.

 

x = 100     t = 10-6         c = 3E8               v = 0.8

 

 

            (66)

(67)

 

                                                                                                                                                   

 

         (68)

     

Taking the Light velocity as the unit

 

 

                 (69)

 

This velocity is smaller than c!

In the Lorentz equations, the velocity is never c and it varies according to the values given to the parameters!

 

 

AD. (Third case).

 

 

            (70)

 

 

                (71)

 

 

            (72)

 

Taking c as unit

 

 

               (73)

 

    

It is important to point it out here that in the SR’s third case, the velocity in equation (42), v’ after calculation, is larger than c, v’ = 1.127 272 c.

In the present example, SR’s third case, the time from 10-7 s to 10-6 is the only value changed but the final velocity v is smaller than c, v’ = 0.894 736 c. The final velocity depends on the time duration, which is an absurdity!

Of course when SR uses the double trick to discount the “contraction” directly, without introducing the “contraction” in the Lorentz equations, the trick will work.

Without any trick, as was shown earlier, SR is plagued with inconsistencies, contradictions and wrong results.

The SR equations fail to explain decay, and the Neutrino is invented.

The SR mass increase is contradicted by experimental results, and the fantasy of “the increasing mass has no physical reality" is invented.

Of course SR cannot answer the question: Why are we using the concept of “no physical reality” if we are doing Physics?

And what is worse, a second later SR applies the opposite to calculate the Momentum where the most critical factor is the increasing mass!


[1].- “It follows that mass is not a unique property of a body. Two observers, equally competent, will measure the inertial mass of a body differently if their velocities relative to the body are different.”  F. Miller Jr., College Physics, Fifth Edition page 671, Editor Harcourt Brace Jovanovich, Inc. NY, SF, London, etc..

Another practical example is the Doppler effect.

If a source emits a light beam at a steady frequency, each Observer in a different inertial frame will observe a different light frequency according to his state of motion. In SR, what each Observer observes is “reality.” In SR, if we “observe” that the Earth is flat, that is the reality: The Earth is flat because SR sees it so, but we perfectly know that the Earth is spherical, not flat!

In AD, each Observer will deduce that the frequency observed by him is only a measure of his state of motion with respect to the light beam source.

In AD, “reality” is represented by the frequency emitted by the source and the different frequencies observed by different observers under different states of motion are only the measure of their motion with respect to the source. The source’s emission frequency is the “reality.”     

[2].- We reserve c and d for light velocity and derivative, respectively.

[3].- Even though the Lorentz equations satisfy the invariance with negative ((12), (13)) or positive sign ((39), (40)) the physical facts are different. In Fig. 1, which represents the equations with negative sign, the electromagnetic wave emitted at the system’s origin reaches point P at rest with respect to this origin.

In Fig. 2, which represents the equations with positive sign (See footnote 5) the electromagnetic wave reaches point P in motion with respect to the O’ origin. In other words: In Fig. 1, P is at rest with respect to the emitter (O) and in Fig. 2 P is in motion with respect to the emitter (O’).

 

[4].- What does this mean? This means that in Relativity v and t need to be changed by a coefficient to make the transformation invariant.  It must be realized that v t could reach position X (distance x) or position X’ (distance x’) because the time could take any value, t or t’. To derive the AD equations, the standard Lorentz procedure is used but without “two tied systems,” Fig. 2. But as AD creates a Physics with only ONE system or ONE Observer, x’ could represent the System Motion to position O or, if the phenomenon is moving with respect to O, x’ could represent the phenomenon (P) in motion with respect to O, because AD needs only ONE Observer in the same System of Coordinates or Frame. It is a subtle difference, but it must be understood to appreciate the AD difference regarding Lorentz. In a few words: x’ = v t is universal, that is, it could represent the distance between system O’ and O or between O and P if P is moving with respect to O. (Relativity between Observer and Phenomenon).

 

[5].-  Equation (34.15), page 34-8, Volume 1 on “The Feynman Lectures on Physics” by Feynman, Leighton and Sands, Addison-Wesley Publishing Company 1975.

.  

[6].- When System O, Fig. 3, passes origin of system O’ this observer sees first Point P passing him and later the origin of System O at the same velocity, knowing O’, consequently, that P is at rest with respect to O. After synchronizing their respective clocks, System O, traveling at v, will reach a distance v t at time t.

Observer at O’ doesn’t worry about the distance to P with respect to O because he knows that the distance is constant and he also knows the value because at synchronization time O told him (He can also measure it) the distance between P and his coordinate origin and the time value of his trip.Observer O’ wants to know what distance and time he will measure regarding the trip of System O.