(1) Mistakes That Lead To The Formulation Of Special Relativity

     In order to keep things simple, let us examine the following equation set:

     The task of set 1 is to pursue the numerical relationship between the variable elements of (x, t, x’, t’).  Obviously, this task cannot be realized unless (a₁₁, a₁₂ , a₂₁, a₂₂) are defined with numerical values.  However, if we must apply set 1 in an effort of defining (a₁₁, a₁₂ , a₂₁, a₂₂) , we would have naturally placed ourselves in a position of solving equation set in which (a₁₁, a₁₂ , a₂₁, a₂₂)  are regarded as variables.   Subsequently, the other elements, i.e.,  (x, t, x’, t’), must be regarded as playing the role of known constant coefficients in the same set.  In doing so, beginning with set 1, the derivation of Lorentzian transformation equations normally go through the following steps:

      Step one: By supplying the conditions x’=0 and x=vt to set 1, the following set is reached:

       Step two:  By substituting set 2 into x’²=(ct’)² and, after some term rearrangement, by having the new equation equate the equation  x²-(ct)² =0, a solution set for the a’s is reached:

     Step three:  Finally, equation set 1 is rewritten as

     Solution 2 is then the so-called Lorentzian transformation equations.

     Apparently, upon the performance of step one, the following equations must have been bundled together to form a set:

     Upon the performance of step two, relativity has actually bundled set 2 and x’²=(ct’)² and x² – (ct)² =0 together to form the following set:

     Given how set 2 is achieved, set 4 is actually a collection of all of the following equations together before any derivation work for the Lorentzian transformation equations begins:

     Now, special relativity's mistakes are apparent:
     (1)  It has shown no consistence in whether taking (a₁₁, a₁₂, a₂₁, a₂₂) or (x, t, x’, t’) as variables in solving an equation set.  Its derivation of the Lorentzian transformation equation is well known aiming at (x, t, x’, t’) as variables, but during its process, it has actually pursued for a solution of (a₁₁, a₁₂, a₂₁, a₂₂).
     (2)  Most seriously, by requiring x=vt and x= ± ct in the same set, while designating v=/=c relativity actually has no legitimate equation set to begin its derivation.
     (3)  By designating x'=0 in the set, relativity, before its derivation begins, already rejected the legitimacy of x' to serve as a variable in the set upon which the Lorentzian transformation equations are developed.

     Now it is only fair to say that the "success" of special relativity depends on an accumulation of mathematical mistakes.

     In the original paper of special relativity concerning the derivation of Lorentzian transformation equations, one can easily find that the “success” of the derivation of special relativity also depends on the derivative taken with respect to a constant instead of a variable; a critical rule in calculus is not respected.  But enough is enough, this author do not intend to pursue further on this matter, until some readers feel interested and request proof.

(2) Mistakes That Lead To The Formulation Of General Relativity

What serves as the fundamental support to the general relativity is:

(1) Special relativity,
(2) The Principle of Equivalence.

That special relativity is unable to offer support to anything is obvious, because, as what has been shown so far, special relativity itself is a product of fallacious derivation in mathematical terms.  As to the Principle of Equivalence, its invalidity is evidenced in two folds:

(1) A gravitational field that needs to be homogeneous is indispensable for the existence of this “principle”.  However, as we learn previously, even general relativity itself presents enough ideas to have the concept of homogeneous gravitational field rejected.
(2) More detailed derivation guided by Newtonian mechanics will show that an object must demonstrate a significant difference, instead of  equivalence, by its free moving path between a gravitational field and a mechanically accelerating field.

     Let us refer to Fig. 1 again.

                      Fig. 1

In the gravitational field, abbreviated as GF, produced by the massive object A, we attach a coordinate system X-O-Z to A, with the origin O of the system located at the mass center of object A.

     Assume that at a certain time instant, in the vicinity of A, of mass M_A  , we found a projectile B, of mass M_B , having a distance of R  from the mass center of A.  With respect to the inertial frame X-O-Z, this projectile is found moving with velocity v_B , which forms angle  ß  with R.  This velocity can be resolved into two components: a tangential component  v_B/T , and a radial component  v_B/R .

     We can always find the gravitational force F  between object A and object B according to the following formula:

The total mechanical energy of  B with respect to A is

If there is no foreign interference, E  is a constant.  We can divide both sides of Eq.3-2 by M_B  to get

where e represents the total mechanical energy per unit mass of projectile B;  e  is also obviously a conserved quantity.

     It can be shown that in the course of B’s movement there always exists one point where the centrifugal force, Fc , developed by B’s sideways movement with respect to A, precisely cancels out the gravitational force, Fg , between A and B, leaving us with Fc + Fg =0.   We will call this point the virtual equilibrium point, abbreviated as VEP.  Naturally, at the VEP, we should also have an equation similar to Eq.3-3 for B’s mechanical energy per unit mass with respect to A:

where the subscript ve stands for the quantity at the virtual equilibrium point.

     At VEP, Fc + Fg =0 will lead us to have

where  is the angular momentum of B with respect to A, a quantity that must always be conserved. (In the upcoming text, we will replace M_B with m for simplicity of notification.)  If  J is a constant, must then also be a constant by Eq. 3-5 once all the initial moving status are defined, including the speed with which B was found.

    At a point other than VEP, we can set , with f  being any positive number.  Then Eq.3-3 can be rewritten as

where v_R  is the component of B’s speed along R and v_T  is the tangential component of B’s speed.
With  , Eq. 3-6 becomes

Eq.3-4 and Eq.3-7 together lead us to have

 

Because at VEP, Fc + Fg =0 , we have:

Substituting Eq.3-9 into Eq.3-8, we have

Eq.3-10 further leads to

     The positive direction of  R  is assumed to be pointing away from A. is along the radial line and in the direction of decreasing  R , so we take the negative sign for, i.e.,

      If we express B’s sideways movement with angular speed  , then, because of conservation of angular momentum, we would have

Dividing both sides of Eq.3-13 by Eq.3-12, we have

Since  Eq.3-14 becomes

     At VEP, if   takes the negative sign, the ratio of   is a negative value.  Besides, at VEP,  , thus f=1.   The continuous movement of B serves to further decrease R , or, to further increase f .  These conditions enable us to assume an initial condition of  for C .   So, Eq.3-15 gives us

          Eq.3-16 is a conic section formula.  In this conic section formula,  is a constant according to Eq.3-5.  Onceis obtained,   can be calculated from Eq.3-9. Then,   can be obtained from Eq.3-7 by setting f=1 at VEP.

      This conic section equation can predict three types of paths for a free moving object in the gravitational field.  Can we find any compatible equation in general relativity?  The answer is: NO!  If the Principle of Equivalence is ever valid, parabolic curve is the only curve to be shown by relativity for the path of a free moving object.

(1) What Is Speed?

Before the debut of special relativity, speed was always a simple and clear concept: completion of linear displacement during a unit time interval. However, by introducing the concept of length contraction and time dilation, relativity has confused the concept of speed.   Let us refer to Fig. 2 and Fig. 3.

     Assuming the position of frame AB and frame A’B’ in both figures are recorded by an observer riding with frame AB with a clock that is stationary to him.   Which of the following expressions will relativity accept as a speed expression that the observer can use for the speed term v in the Lorentzian transformation equation?

(1) v=length CB/(t₂-t₁)
(2) v=length C’B’/(t₂-t₁)

     Given that relativity can also instruct the observer that there is always a dilated time interval, say (t’₂-t’₁) , recorded by clocks on the A’B’ frame to match the time interval (t₂-t₁), the observer must also have the following expressions

(3)  v=length CB/(t’₂-t’₁)
(4)  v=length C’B’/(t’₂-t’₁).

Can relativity answer whether to accept any one of these four expressions but reject the others, or to accept them all, or reject them all in determining a v value for the Lorentzian transformation equations?
 

(2) Why Are The Methods Different?

     If it is said that the measurements concluded by special relativity such as length contraction, time dilation, and mass escalation are observation dependent, the same measurements concluded by general relativity are no longer observation dependent, but absolute. In other words, for example, special relativity asserts that, in an environment free of gravity, the determination of the length of an object depends on how fast an observer is moving with respect to the object that he is measuring.   The speed no longer plays any crucial role in general relativity when the same measurement is made.  Once the mass quantity of the object that causes the gravitational field is determined, the other factor that governs the result of the measurement would be the location with respect to the massive object.  (Not to mention that this is another violation brought about by general relativity against its own concept of homogeneous gravitational field, with which relativity draws the Principle of Equivalence.  The Principle of Equivalence, however, can tolerate no difference between locations when measurements between these locations are made.)

     Why is the method in determining certain measurements in general relativity absolute, but, once acceleration is absent, as stressed by special relativity, must the method of making the same measurements be observation dependent?
How can this fit into what Einstein said: “The laws of physics must be of such a nature that they apply to systems of reference in any kind of motion”? [THE FOUNDATION OF THE GENERAL THEORY OF RELATIVITY,  by A Einstein] .

     What more is needed from relativity to convince people about its fallacious nature, both in mathematics and physics?

REFERENCES

1. THE PRINCIPLE OF RELATIVITY, a collection of original memories on the special and general theory of relativity, by Dover Publications, Inc. New York, U.S.A. (Library of Congress Catalog Card Number: A52-9845)

2. Robert Resnick. 1968. INTRODUCTION TO SPECIAL RELATIVITY,  John Wiley & Sons, Inc. (Library of Congress Catalog: 67-3111). New York, U.S.A.

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