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Di hua research Fellow (ret.) of Stanford University



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Fundamental Revision of Einstein’s Theory of Relativity

Galilean Relativistic Mechanics with Variable Speed of Light

Di HUA

Research Fellow (ret.) of Stanford University

Academician of Russian Academy of Astronautics

huadi1936@gmail.com


Abstract

Many mistakes in Einstein’s theory of relativity are revealed. Both Einstein’s postulate of constant speed of light and Lorentz’s postulate of length-contraction are invalid. Einstein’s Lorentz transformation cannot provide invariance for Maxwell’s electromagnetic field equations, nor for the electromagnetic wave equations. His theory cannot explain the Doppler effect and the aberration. His famous principle of mass-energy equivalence fails for moving bodies. He commited many mathematical errors in the development of his relativistic mechanics. His formulas of moving mass and kinetic energy are inconsistent with each other. His explanation of the perihelion motion of Mercury was a fabrication. Based solely on the Galilean principle of relativity, without any additional postulate, the author develops a new theory of relativity with a new relativistic mechanics. The new theory proves: Time-synchronism is universal; Speed of light is variable and is not the limit of speed; The variable speed of light causes the Doppler effect and the aberration; The principle of mass-energy equivalence can be precisely extended to moving bodies; Photon has static mass; etc. The new theory can precisely and unconditionally prove the equivalence between gravitational mass and inertial mass, demonstrate a comprehensive correspondence between gravitational and inertial fields, and make the special and the general relativity theories consistent with each other. The new relativistic mechanics can precisely calculate the angular deviation of light caused by the sun’s gravity, the red shift of the sun’s light, the perihelion motion of Mercury and, thus, give the gravity a mechanical interpretation instead of Einstein’s geometrical one.


§1 Einstein’s Theory of Relativity Was Wrong at Its Birth

Einstein published his paper On the Electrodynamics of Moving Bodiesin 1905. That year is popularly known as the relativity theory’s birth-year. In that paper Einstein tried for the first time to deduce his Lorentz Transformation. We will show Einstein’s errors in his study.
For easier reference to Einstein’s original text, we will follow his notations, for example, the speed of light is , not .

Einstein introduces two systems of reference and with their axes parallel to each other respectively. The -system moves translationally and uniformly at velocity in the positive direction of the -system’s -axis. Einstein also sets up a mathematical model with a ray traveling forth and back between a source of light and a reflector. The source is fixed at the -system’s origin and the reflector is moving with the -system. With this model, which was incorrectly used by him (see below), he gets where “ is a function ”. So, there must be:

(1)

He claims that “if we put ” and “if we substitute for its value, we obtain”:

, where .

Then he proves “ must equal 1” and he obtains: .

Actually, however, by placing into (1), he should have obtained:

.

Because of , there ought to be: . (2)

It is , not. Einstein has made an algebraic error!

More sadly, Einstein misused his “Source-Ray-Reflector” model. Let’s analyze his study carefully. At the beginning time , the source emits a ray in the positive direction of the -axis. At the time , the ray catches up with the moving away reflector and is instantly reflected back toward the source. At the time , the reflected ray arrives back at the source. With this model, Einstein writes:

. (3)

Einstein applies his postulate of the constant speed of light to this model. At the beginning time , when the ray is at the -system’s origin (), he assigns for . At the time , when the ray reaches the reflector, he writes for , which means the reflector is at the -system’s and the ray departing from with velocity spends time to reach the reflector at . He violates his postulate of the constant speed of light, according to which the light must have spent time to reach the reflector at so that he ought to have . Next, he writes for , which means the reflected ray with velocity spends time to return from to the source of light at the-system’s origin (). He again violates his postulate of constant , according to which there must be . His assumes the velocity of the reflected ray to be , which not only violates his postulate of constant but also means that the reflector (the source of the reflected ray) is moving toward (not away from) the original source of light!

Einstein’s misuse of his model can be ascertained by placing his expressions of into his equation (3):

,

^ They do not satisfy his equation (3) because his . With this violation of his equation (3), however, Einstein insists on placing his erroneous expressions into (3) and obtains:

.

Next, he writes, “if is chosen infinitesimally small”, then

or .

From this erroneous partial differential equation, he obtains his .

Strictly adhering to his constant postulate, however, there ought to be , and . Indeed, they satisfy (3):

.

Correspondingly, there must be: .

Now, if is chosen infinitesimally small, we obtain:

or or constant.

The constant regardless of variable signifies a universal time-synchronism!

However, the time-synchronism and the postulate of constant together would lead to the following kinematic equations of a spherical wave of light in the two reference systems:

-system: and -system:

which in turn lead to a trivial solution: or , without any relative motion between the two reference systems. Thus, Einstein’s =constant postulate becomes a senseless triviality. Without the relative motion (), of course, the speed of light always remains constant.

If we reject his constant postulate, then there must be: , and . Because, the -system’s reflector is moving away from the -system’s source of light at velocity , so the ray from the source has speed with regard to the moving reflector. ^ The moving reflector is the source of the reflected ray, so the reflected ray’s speed is with regard to the original source of light which is fixed in the -system.

Our expressions satisfy the equation (3):

.

Correspondingly, we have: .

Now, if is chosen infinitesimally small, we obtain:

or or constant.

Again, we come to the universal time-synchronism. Therefore, the conclusions can only be: “The speed of light is not constant due to the relative motion between a source of light and a reflector. The time-synchronism is universal”.

§2 Lorentz Transformation vs Galilean Transformation

In his 1907 paper On the Relativity Principle and the Conclusions Drawn from It, Einstein had given up his “Source-Ray-Reflector” model. (^ Had he realized his errors in 1905?) He turned to deduce his Lorentz Transformation by maintaining invariance of spherical wave’s kinematic equation between two relatively moving reference systems. For easier reference to Einstein’s original text, we will follow his notations, for example, the speed of light is , not .

Suppose two reference systems and with their axses parallel to each other respectively. The -system moves translationally and uniformly at velocity in the positive direction of the -system’s -axis. A spherical wave of light is radiated at velocity from a source fixed to the -system’s origin. In the -system, the sperical wave’s kinematic equation is:

(4)

According to the relativity principle’s requirement of invariance, the spherical wave’s kinematic equation in the -system must be: . (4’)

§2.1. Einstein’s Lorentz Transformation.

Einstein tries to deduce a group of transformation equations capable of maintaining invariance between (4) and (4’). This can be done as follows:

From (4)

or

or

or

or , where . (5)

Comparing (5) with (4’) and taking into consideration and for a relative motion along the -axis, the following group of transformation equations can be obtained:





(6)





The above group of transformation equations is commonly known as the Lorentz Transformation. But, let’s call it “Einstein-Lorentz Transformation” (E-L Transformation) in order to stress that it contains Einstein’s postulate and Lorentz’s postulate.

Einstein claims that he has succeded in proving his E-L Transformation by use of only two principles: the relativity principle and the principle. He upgrades his from a “postulate” to a “principle”. In fact, his violates the relativity principle (See^ Appendix A§1) and cannot be entitled as a “principle” just because his E-L Transformation can mathematically provide the invariance for the spherical wave equation. Moreover, Einstein avoids mentioning that Lorentz’s postulate is indispensable for him to deduce his E-L Transformation (See Appendix A§2 and A§3).

§2.2. Galilean Transformation.

Based on the classical Galilean and , we can obtain from (4’):

. (7)

(7)-(4) gives: or . (8)

So, the transformation for the speed of light is: . (9)

The following group of transformation equations can invariantly transform (4’) into (4):





(10)





This indicates that the classical Galilean Transformation can also transform the spherical wave’s equation invariantly between the two relatively moving systems of reference. Einstein has no right to arbitrarily claim his E-L Transformation only legitimate one. Contrary to it, we will prove that, in the physics domain, his based E-L Transformation (6) is misleading while our based Galilean Transformation (10) is correct (See3,4,5,6,7). The principle of relativity in physics, which governs physical laws, involves more than just the mathematical invariance.

§2.3. The Time-Synchronism Comes Directly from the Principle of Relativity.

We can prove the time-synchronism directly from the principle of relativity.




Suppose the two systems’ origins and are initially overlapping. During a period of time (by the -clock), an observer in the -system sees the -system’s origin moving through a distance of in the positive direction of the -axis. The -observer calculates the -system’s velocity as . With regard to the same event, another obserevr in the -system finds the -system’s origin moving through a distance of in the negative direction of the -axis during a period of time (by the -clock). The -observer calculates the -system’s velocity as . According to the principle of relativity, there must be and so that .The universal time-sychronism is not a postulate but a principle based on the principle of relativity.

§2.4. Questioning Einstein’s Time-Transformation Equation.

Generally speaking, in case of a relative motion with velocity in an arbitrary direction etween the and the systems, Einstein’s time-transformation equation can be written as:

, where .

The -system’s relative motion in the -direction can be resolved into three independent relative motions along the -system’s axes respectively. To make it simple, let’s analyze a two-dimensional case in the plane, where and .




For a motion along the -system’s -axis, Einstein’s time-transformation is:

, where .

For an additional motion along the -system’s -axis, the time-transformation is:

, where .

Since a superposition of these two motions is equivalent to a single relative motion in the -system’s -direction, so the resultant time must be the same time directly transformed from the -system’s time . Hence, there must be .

Let’s analyze a special case with 45? so that ,, and . The resultant time is:

.

Obviously, Einstein’s time-transformation is wrong. Only when and , which is a meaningless trivial case without relative motion, then .

In contrast, the velocity-transformation in our based Galilean Transformation (10) is true. Indeed, for a motion in the -direction, our velocity-transformation equation is:

or .

For a motion along the -system’s -axis, we have . For an additional motion along the -system’s -axis, we have . Since,, , , so and . Therefore, we have:

.

The resultant velocity from a superposition of the two motions is exactly the velocity directly transformed from the -system’s velocity . Obviously, the and based Galilean Transformation (10) is correct.

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