(This article contains 2 sections.  Section 1 displays the mathematical absurdity of both special and general relativity.  Section 2 explores the mistakes that lead to the formulation of relativity.  The text in both sections belong to a single paper that challenges relativity with up to US$50,000.00 awards.)

ABSTRACT

      Relativity  forbids any speed accumulation from exceeding the speed of light with its speed addition equation w= where u is the speed of a moving frame, such as a train, with respect to a ground observer,  v is the speed of another moving object, such as a passenger, with respect to the train, w is the resultant speed of the passenger with respect to the ground observer, and c is the speed of light. Customarily, in explaining no speed addition can result in a speed that is higher than the speed of light, relativity would replace the passenger with a light ray, and thus uses c in place of v, to show the "inevitable" mathematical result w=c, by which relativity addresses that the light ray emitted on the train is observed to have the same speed of c by the ground observer as well.  Of course, as relativity can replace the passenger with a light ray, we can equally replace the train, instead of the passenger, with a light ray, and thus have u=c.  Subsequently, however, we are once again led to have w=c.  Needless to say, this mathematical result means that any passenger must possess the speed of light, but no other value of speed, with respect to the ground observer, as long as any light ray moving along the same line as the passenger is detected by the ground observer.  If we must consider the case that the passenger should have -c as its speed value with respect to the light ray that is in place of the train, we will have  w=0/0.    This is an invalid expression in mathematics.

      The derivation leading to the formulation of Lorentzian transformation in special relativity is actually a duplication of an  ancient  “miracle”  in  algebra: 2x-x=0, 2x=x, 2=1.  Dominated by such a mathematical confusion, relativity displays fundamental uncertainty in understanding physics.   As such, with equations, it claims to have “discovered” two speed limits in nature: (1) the speed of light in the vacuum space and (2) the speed of light at the mass center of a material body.  Needless to say, these two speed limits repel each other, not to mention that the second speed limit is even against nature.   Relativity then further extends these confusion and uncertainty in physics to make up many self-contradicting concepts.  These concepts include the so-called homogeneous gravitational field and the idea of having (circumference/diameter)>3.1415926... for a spinning circle. With the same mathematics guiding to its “success”, however, relativity presents no homogeneous gravitational field but a monster that must be called a homogeneously inhomogeneous field for its appropriation. Based on the same erroneous mathematics, relativity must force itself to have (circumference/diameter)<3.1415926... for a spinning circle.  With the idea of a homogeneous gravitational field, relativity believes that it can establish the validity of the so-called Principle of Equivalence for the legitimacy of general relativity.  However, Newtonian mechanics, supported by the close orbital movements of numerous heavenly objects, must witness the nonexistence of such a “principle” in nature.

   

1. General Failure Of  Special Relativity In Mathematical Terms

     The crucial role of x=ct and x’=ct’ in the derivation of the Lorentzian equations must evidence the indisputable presence of the following equation set in special relativity:

Given v and c as constants, either of the two Lorentzian equations in Eq. set A can be obtained by the other three equations within the set by the following simple operation:

First, with the first Lorentzian equation in set A, we have

Second, with the second Lorentzian equation, we have

     The exactness between Eq. A-1 and Eq. A-2 allows us to duplicate one Lorentzian transformation equation from the other back and forth without losing any mathematical equivalence between them.  This further allows us to decide that Eq. set A can be reduced to

     The first equation in set B means that there is always only one x that can be found matching x', no matter how time may have advanced.   This can so happen only if there exists no movement, or, v=0, between the two axes. Any nonzero value of speed found in the Lorentzian equations in Eq. set A, such as v=0.2c, for example, must now be ridiculed, unless v=0.2c=0 is accepted.

     If anyone applies the two Lorentzian transformation equations with some numerical constants assigned to two of the four variables, such as x=m and t=n, for example, where m and n can be any nonzero constants, he must have organized an equation set that reads as:

     As mentioned before, x=ct and x’=ct’ are inseparable from the two Lorentzian equations, he who establishes Eq. set B-I has actually established the following equation set:

     With six equations and four unknowns, set B-II is easily seen as containing no solution.  Any nonzero solution of Eq. set B-I turns out to be a mirage arrived at only because x=ct and x’=ct’ have been omitted from set  B-II without valid reason.

2. Failure Of Lorentzian Transformation Equations In Studying Movement.

     The constant speed v' of the x axis along the x’ axis can be expressed as dx’/dt’=v'.  Now, if we take the derivative of the first three equations of equation set A with respect to t’, we have

     Within Eq. set C, substituting the third equation into the second equation leads to.   Then, this new relationship and the third equation in set C together will lead to dx’/dt’=c , instead of dx’/dt’=v’, out of the first equation in the same set.  dx’/dt’=c is a relationship that can be displayed by the fourth equation in equation set A.  Clearly, this means that the Lorentzian transformation equations cannot distinguish c, the speed of light, from v’, the speed that is supposedly possessed by the x axis with respect to the x’ axis.  Given that v and v’ must be of equal magnitude, setting aside their directions, the Lorentzian transformation equations must obviously force such a relationship between the two axes: c=dx’/dt’= v’=v=dx/dt.

3. Invalidity Of the Length Contraction.

     Length contraction from relativity means that a segment  of length of  (x₂ – x₁) is worth

when this length is cut from the x axis but attached to x’ axis and viewed by the observer on the x axis.   Therefore,

is an  apparent length of (x₂ – x₁) on the x' axis to the observer on the x axis.  Conversely, for a moving length from the x’ axis to cover the length of (x₂ – x₁)  on x axis, it takes a length that is worth cut from the x axis and moving .   Let us call   the equivalent length of (x₂ – x₁) on the x' axis to the observer staying on the x axis.
 

     So, for the observer on the x axis, if it takes (x₂ – x₁) from his own frame to obtain speed v, it is well justified to say that it takes an equivalent length from the other but moving axis for him to obtain speed v. The equivalent length, if cut from the moving axis but made stationary to the observer who is on the x axis,  is measured as          .
 

      All these inevitably lead the observer on the x axis to conclude

v=(x₂ – x₁)_/(t₂ – t₁) = [  ]_/  (t₂ – t₁).

This relationship must lead to v=0.  Once again, any nonzero speed written in the Lorentzian equations is ridiculed.   Mathematics prohibits the promise of length contraction to the observer.

4. Invalidity About Time Dilation.

     The second equation in equation set A advocates that a single time instant registered by a clock on the x axis can identify different time instants between different clocks on the x’ axis.   Subsequently, this allows a zero time interval quoted on the x axis to match a nonzero time interval quoted on the x’ axis.

     Relativity must let the above principle conversely hold so that a zero time interval quoted on the x’ axis can correspond to a nonzero time interval quoted on the x axis.

     Now, relativity has the following options:

(1)  It allows (t’₂ - t’₁)= (t₂ - t₁), where (t’₂ - t’₁) is zero, while (t₂ - t₁) is nonzero. This option, of course, cannot be tolerated by any mathematical rule.

(2)  Corresponding to t'₂ and t'₁  that are quoted on the x’ axis, two time instants at two locations on the x axis can be  identified as:
and.
Then, a time lapse between the two axes must match each other according to the following:

This further leads to
(Eq. D)

     For a nonzero time interval shown by a clock on the x axis, we must have t₂ and t₁  not equaling each other.  With these two time instants, a clock must always identify two locations x’₂ and x’₁ on the x’ axis because of the movement by this clock with respect to the x’ axis during said time interval.  For a single time instant, or a zero time interval, on the x’ axis, we must have t’₂ = t’₁. With  t’₂ = t’₁, any pair of x’₂ and x’₁  that are identified in the above manner must obey the truth of
(x’₂ -x’₁)=v(t’₂ - t’₁)=0.

      So, if (t₂ - t₁) must be nonzero, the only choice left for relativity by the relationship shown in Eq. D is to have  ,  inevitably, v=c. With v=c, Lorentzian transformation equations must fail.   Another popular conclusion, i.e., time dilation, from relativity is then unacceptable in mathematical terms.

         Isn’t it now apparent that special relativity must encourage the v term of any value in the Lorentzian transformation equations to take this form: 0=v=c?

(1) Speed Of Light and The Limits of Speed.

Besides the speed limit asserted by special relativity, general relativity announces another speed limit in nature:

     “For measuring time...” relativity states, “...at a place which, relatively to the origin of the co-ordinates, has the gravitation potential ø, we must employ a clock which___when removed to the origin of co-ordinates___goes (1+ø/c²) times more slowly than the clock used for measuring time at the origin of co-ordinates.  If we call the velocity of light at the origin of co-ordinates c₀ , then the velocity of light c at a place with the gravitation potential ø will be given by the relation  c= c₀(1+ø/c² )” (ON THE INFLUENCE OF GRAVITATION ON THE PROPAGATION OF LIGHT, by A. Einstein, 1911)

     This statement has obviously placed the mass center of a gravity body at the origin of the co-ordinates.  With the gravitational field so arranged, ø, the gravitational potential, must be zero at the origin but negative elsewhere. So, general relativity, with an equation as shown in the above statement, must hereby assert that it has “discovered” that light can travel through the mass center of a material body with a speed and that this speed can even be concluded as being the highest in nature.

     In addition to ridiculing the speed limit advocated in special relativity, the “newly found” speed of light also expels a concept that is extremely important to general relativity: the so-called homogeneous gravitational field.  According to general relativity, no measurement made in a homogeneous gravitational field is supposed to be varied between different coordinates.  The above equation for light’s different speed at different location obviously claims that such a homogeneous field is unfound in nature.

(2) A Homogeneously Inhomogeneous Field

In the explanation of homogeneous gravitational field by relativity, one can find the following:

     In the article [ON THE INFLUENCE OF GRAVITATION ON THE PROPAGATION OF LIGHT, by A. Einstein, 1911], relativity states that “...relatively to K, as well as relatively to K’, material points which are not subjected to the action of other material points, move in keeping with the equations

...”, where K represents a coordinate system that is at rest in a gravitational field, and K’ represents a coordinate system that is (mechanically) accelerated.  Since relativity has apparently compared accelerations in all 3 dimensions between the two systems, relativity must allow X//X’, Y//Y’ and Z//Z’ as the only orientation between all the axes of the two systems.

     In another article, relativity states that “ Let K’ be a second system of reference which is moving relatively to K in uniformly accelerated translation.” [THE FOUNDATION OF THE GENERAL THEROY OF RELATIVITY, by A. Einstein, 1916]  In this statement, K is referred to as an inertial system.  In order for us not to confuse K, which represents an inertial system in this statement, with the K, which represents a system rest in the gravitational field in the previous paragraph, let us use Ko to represent the inertial system mentioned in this paragraph.  The axes of Ko will then be named Xo, Yo, and Zo.  In the movement comparison presented by this paragraph, relativity of course must have restricted the orientation of all axes between K’ and Ko in such a way that X’//Xo, Y’//Yo and Z’//Zo.

     Putting together all of the above restrictions regarding the orientation of axes, relativity must have stressed the overall relationship between all the axes in all three systems aforementioned as X//X’//Xo, Y//Y’//Yo, and Z//Z’//Zo.

     With the geometrical orientation that is specified above, we immediately recognize that relativity has forced an impossibility upon the gravitational field.

     While we agree with relativity that the mechanically accelerating field does provide in its entire field for any point the satisfaction to the mathematical relationship  , we must stress that such a satisfaction is unfound in the gravitational field.  If in the gravitational field we happen to have found a certain point where the material movement obeys this group of equations, at any other point of the same field the material movement must disagree with this group of equations.  The reason is simple: while the value and direction of acceleration in the entire mechanically accelerating field is coordinate independent, as shown by the above equations, it is absolutely not so in the gravitational field.

     We all know that, in describing the dependence of the acceleration upon the coordinates in the gravitational field, Newtonian physics has a highly accurate expression, which is,  where and M  is the mass of the body causing the gravity.

     Newton’s equation allows no homogeneity for the gravitational field, but matches the reality with such an accuracy that human beings so far can confirm no universal data for a dispute.

     Homogeneity, by definition in physics, is the quality of coordinate independence.   So, before relativity can remove the inhomogeneous characteristics from the gravitational field, i.e., the coordinate dependence of the gravitational acceleration, in naming a homogeneous gravitational field, general relativity has actually created a term that must read as a homogeneously inhomogeneous field.

     As a matter of fact, relativity even fails itself in recognizing a gravitational field that is homogeneous.  In the preceding analysis regarding the speed of light in the gravitational field, relativity apparently finds no homogeneous gravitational field for itself.   In another statement concerning a gravitational field, relativity states:

      “The unit measuring-rod thus appears a little shortened in relation to the system of co-ordinates by the presence of the gravitational field, if the rod is laid along a radius, …With the tangential position, … the gravitational field of the point of mass has no influence on the length of a rod. ” [THE FOUNDATION OF THE GENERAL THEORY OF RELATIVITY, by A Einstein, 1916]

     Relativity does hereby predict the variation of measuring results at different locations (Radii can have different orientations and lengths) in the gravitational field.  Now, is the gravitational field homogenous, as what relativity imagines, or inhomogeneous, as what relativity fits itself in?

(3) Principle Of Equivalence

     If the gravitational field and the mechanically accelerating field are really equivalent so that material movement can only follow the equations
then calculation will lead us to believe that the free movement path of all objects can only have one type of trajectory: parabolic curves in the coordinate system with respect to which acceleration is detected.  However, the movement of the abundant heavenly objects must disagree with this belief.  Many of their moving paths demonstrate as close loops. Relativity fails to provide equations to match this kind of movement.  In contrast to the inability of relativity, Newtonian physics can provide us an equation to describe those moving paths.   This can be evidenced by the following conic section equation that, as shown in Fig.1, illustrates the path of an object, named as B, flying by a massive object, named as  A, in this figure:

 where R is the instantaneous distance between the mass centers of A and B,
         ve is the abbreviation of designation for a point in space named as the  virtual equilibrium point,  where the gravitational force between A and B precisely cancels out the centrifugal force produced by the movement of object B around body A,
          Rve, a constant (proof offered in section 2), is the distance between the mass center of A and B when B’s mass center coincides with the ve point.
          v_Rve , a constant (proof offered in section 2), is the moving speed of B at the ve point but resolved along Rve,
           v_Tve , a constant (proof offered in section 2), is the moving speed of B at the ve point but resolved along a line that is tangential to Rve.

     We can always orient the coordinate system x-o-z in such a way that the ve point coincides with the x axis, then    is the angle between R and the x axis.

                    Fig. 1
     The three types of curve described by the above conic section equation will be
1. A hyperbolic path if ,
2. A parabolic path  if ,
3. An elliptical path if .  In the case of  , of course,  , the ellipse is actually a perfect circle.
The more detailed development of the aforementioned conic section equation will appear in the later text of this article. Since the free moving path of an object in the gravitational field shows three types of path,  but only parabolic curve in a mechanically accelerating field (Its equation is omitted in this paper), the equivalence drawn by relativity about two different fields can only be explained as a product of immature consideration in physics.

(4) (Circumference/Diameter)>3.1415926... For a Spinning Circle

In the third section, titled “The Space-Time Continuum... General Laws of Nature”, of the article “THE FOUNDATION OF THE GENERAL THEORY OF RELATIVITY”, by A. Einstein, one can find a paragraph that reads: “In a space which is free of gravitational fields we introduce a Galilean system of reference K (x,y,z,t), and also a system of co-ordinates K’(x’,y’,z’,t’) in uniform rotation relatively to K.   With a measuring-rod at rest relatively to K’, the quotient (circumference/diameter, noted by this author) would be greater than .  This is readily understood...that the measuring-rod applied to the periphery undergoes a Lorentzian contraction,... Hence Euclidean geometry does not apply to K’ .”

The paragraph is quite long, but can be condensed to say that, according to general relativity, an observer can conclude that  (circular circumference / diameter)> for a circle that is spinning with respect to him.   Without offering any kind of proof to support such an assertion, general relativity believes that, based purely on its imagination, it can even remove the universal validity of Euclidean geometry from anything that is moving. Very unfortunately, using the very same idea it advocates, general relativity must only lead itself to encounter a conclusion that is the exact opposite of its assertion, i.e.,  (circular circumference / diameter) <.

If we construct equilateral polygons of sides of 3, 4, 5,...n to circumscribe and inscribe a circle in the manner as shown in the following diagrams

 we must have ( n l_i < L< n l_o ), where

l_i  is the length of each side of the inscribing polygon,
l_o is the length of each side of the circumscribing polygon,
n is the total number of sides of the corresponding polygons and can approach infinity,
L  is the length of circumference of the circle.

     Now, let us rotate both the circumscribing and inscribing polygons about the center of the circle with the same angular velocity, leaving the circle stationary relative to an observer who is at the center of the circle. Disregarding the magnitude of the angular velocity, we must accept that:

1. Once defined, n, the number of sides of each polygon, remains unchanged.
2. Once n is defined, the polygon circumscribing the circle can neither expand nor shrink. That it can not expand is obvious. If it shrinks, however, it would contradict a statement that can be found in the above quotation: "...while the one (the measuring-rod) applied along the radius does not (undergo a Lorentzian contraction)."
3. The above analysis can also be applied to the polygon inscribing the circle. Further, each side of the inscribing polygon can serve as a measuring-rod along the circle’s periphery for each of them always has both of its endpoints on the circle. Please note that each of these rods is now moving.
4. Basic geometry tells us that as n increases, the total sum of the length of the circumscribing polygon decreases while that of the inscribing polygon increases, but both approach the same limit, which is the periphery of the circle.

Because of the assumed effect of the Lorentzian contraction on each side of the rotating polygons, as specified by relativity in the above statement, all of the above deductions would further lead us to have  ( n l'_i < L< n l'_o ), where the primes denote the new respective length of each of the polygon's sides that undergoes Lorentzian contraction.  Subsequently we must have

where v_i is the linear speed of the sides of the inscribing polygon and v_o is the linear speed of the sides of the circumscribing polygon. .  If the angular velocity of the rotation increases, both v_i and v_o must increase and result in smaller and smaller values out of both the left and the right terms in the above inequality, forcing the value of L to decrease and even approach zero.  Any circle that is between these inscribing and circumscribing polygons and spins with them together, according to the above mathematical deduction, must also have its circumference decrease and finally approach zero. With a decreasing value of L, we are naturally unable to have L/D> , where D is the diameter of the circle and, as predicted by relativity, will not undergo a Lorentzian contraction but stays constant.  So,  a decreasing L and a constant D must result in L/D <!

     What makes general relativity believe that it can remove the validity of Euclidean geometry before any proof can be presented?

****************************************

Please continue to  Section 2 , a discussion on the mistakes that lead to the formulation of relativity