Without a doubt, relativity is one of the most well–reviewed, widely–accepted theories in the history of modern science. Despite the level of scrutiny it has undergone since its introduction and its apparent level of experimental support, relativity is a mathematically invalid theory.

As explained in DISRUPTIVE and in the Tutorial Series, Einstein makes four incurable mathematical mistakes during his derivation. Each of these mistakes cannot be detected by examining his final equations alone. Each must be identified and assessed when it happens. Whether taken individually or collectively, they represent mathematical errors that invalidate Einstein’s work.

These mathematical mistakes, which are found in §3 of Einstein’s 1905 derivation, are not dependent on the meaning, acceptance, or rejection of any term or concept commonly attributed to relativity. This is a tremendous benefit, because it means that anyone with a firm understanding of mathematics can evaluate the validity of these findings. Expertise in the meaning of relativity or what Einstein intended is not necessary. All that is required is an understanding of commonly accepted mathematical rules and concepts.

1. Einstein believes the spherical wave proof passes when it actually fails

Einstein’s theory is based on the idea that two assumptions, called postulates, are related to one another by the equations he derived. Einstein says that relativity does not exist unless he can show that his two principles are compatible. He recognizes that he cannot simply say that the postulates are related to one another by the equations; he knows that he needs to prove it. To accomplish this goal, he develops a short proof called the spherical wave proof. Einstein must show that when he starts with a spherical wave, his equations will produces a second spherical wave. This would prove that relativity was theoretically and mathematically sound. But if he cannot show that a second spherical wave is formed, then the proof fails. A failed proof not only means that his postulates are not compatible, it means that relativity is a failed theory.

A spherical wave, like any spherical shape, requires all points of the shape to be the same distance from the center of that shape. What’s interesting is that Einstein believes the spherical wave proof passes, when it actually fails. A spherical wave requires that, in addition to maintaining the equation’s equality (which is the only characteristic most people consider), each segment must be the same length and originate at the transformed shape’s center. Neither of those later conditions are met. This mistake in Einstein’s analysis is hard to detect, because if you follow the same steps Einstein performs then you will reach the same conclusion. The steps are not wrong; the proof is incomplete. And the steps Einstein performs are insufficient to prove that a second spherical wave was formed.

The acceptance of the proof when it should have been rejected is called a Type I error. The failure of the spherical wave proof doesn’t simply prove that relativity is wrong; it means that relativity was never right to begin with!

2. Einstein makes a math error when he mistreats the τ function definition like an equation

When Einstein developed his theory he created the time function τ, or Tau. Tau is a function, not an equation. Although similar, equations and functions differ in several important ways. Equations are defined and used simultaneously. An example of an equation is:

y = x²

In this equation, you know that the value of x will be squared. While similar to equations, functions require two steps: function definition and function invocation. Unlike an equation, a function cannot be used until both steps are complete. For example, in the function definition:

f(x) = x²

we do not know what value will be squared until after its invocation.

Although Einstein tells us in §3 of his 1905 derivation that “τ is a linear function,” he mistreats it as if it were an equation. This is very hard to detect, because he incorrectly writes the τ function definition so that it looks like an equation:

τ = t – vx’ / (c² – v²)

It is easy to mistake this function definition for an equation the way Einstein writes it in his paper. The only problem is that it isn’t an equation; it’s a function definition. This helps to explain how this mistake has gone unnoticed for so long. We can make this mistake visible by properly writing the function definition using modern nomenclature as:

τ( x’, y, z, t ) = t – vx’ / (c² – v²)

When properly written, it is apparent that τ is a function definition. A function definition cannot be simplified before the function is invoked. Mathematically, τ is the function’s name, not a variable to which an expression is assigned! This means that Einstein’s simplification of the τ function definition to arrive at his final τ transformation equation is a math error. 

3. Einstein fails to recognize that his equations are averages

Not only did Einstein fail to properly treat τ as a function definition, he did not properly recognize the function’s purpose. To understand its purpose, we have to look at the equations for finding an arithmetic average. Most readers will recognize the addition mean equation:

A = ½( M + N )

For example, we use this equation to find the average of six and ten as eight. Less recognizable is the subtraction mean equation, which is:

A = M – ½(M – N)

It is a straightforward proof to show that the addition mean equation and the subtraction mean equation are equivalent. The subtraction mean equation is simply an alternate way of finding an arithmetic average! We can use this equation to illuminate several aspects of Einstein’s derivation that would otherwise go undetected.

In the subtraction mean equation, the absolute value of the expression:

½(M – N)

is called the half–difference, which when added to the smaller operand or subtracted from the larger operand produces the arithmetic average.

Assume for a moment that you have to find the average of two expressions, both of which Einstein uses in his derivation. The first is:

x’ / (c – v)

and the second is:

x’ / (c + v)

As shown in Tutorial five, the half–difference of these expressions is:

vx’ / (c² – v²)

As discussed in Tutorial fifteen, when this half–difference is subtracted from x’/(c – v), the larger of the two expressions, it produces the arithmetic average. Einstein performs this step in his derivation, but fails to recognize the subtraction mean equation or the resulting average equation. Einstein multiples this average by c to arrive ξ, which is called the average intercept length. It is only through an understanding of the subtraction mean equation and reverse engineering of the τ function that we can clearly state the function’s purpose:

“In a general sense, Tau uses the subtraction mean equation to return the average intercept time when provided the coordinates of a point in the moving system as the first three arguments and the forward intercept time as the fourth argument.” – DISRUPTIVE

This mistake leads to the mistreatment of lengths as positional coordinates. Specifically Einstein fails to recognize that he has found three different average lengths associated with three different points on the moving system; he never transforms the point (x’, y, z) as he intended.

4. Einstein makes a math error when he drops a β term in his final equations

This mistake is one of the easiest to independently confirm and one of the hardest to accept. Einstein begins with:

ξ = x’ / (1 – v²/c² )

When he says that he is “substituting x’ with its value,” he should replace x’ with x–vt to produce:

ξ = (x – vt) / (1 – v²/c² )

Notice that completing this substitution does not require the introduction of a β term.

To see how Einstein’s introduction of the β term facilitates his mathematical mistake, we must replace 1/(1–v²/c²) with β² resulting in:

ξ = β²(x – vt)

Here, without any mathematical support for the operation, Einstein drops one of the β terms to produce:

ξ = β(x – vt)

which is generally rewritten as:

ξ = (x – vt) / Sqrt(1 – v²/c²)

This mistake cannot be made without the introduction of the β term. Interestingly this mistake is often ignored or overlooked, because it is what makes Einstein’s equations perform better than the unadjusted classical mechanics equations. It is also why they perform worse than the Modern Mechanics equations. Mathematically, you can’t simply drop a term because it is convenient. Einstein makes a math error by dropping this term out of convenience.

Summary

It is hard to accept that relativity is invalid when it seems to work well and appears to be experimentally supported. As discussed in Chapter 7 of DISRUPTIVE and in Tutorial twelve, the acceptable performance of the relativity equations is explained through the finding that, in practice, Einstein’s equations are often good approximations for the Modern Mechanics equations. However, providing good results cannot overcome the mathematical mistakes that invalidate Einstein’s work. In the end, these mistakes cannot be corrected in a way that retains relativity theory.

Science advances not because someone finds something wrong with the existing theory alone, but because we find something that works better. Einstein’s equations perform well, but not as well as the Modern Mechanics equations. Science requires us to explore those equations and ideas that provide the best quantitative results. To do otherwise is not scientific.

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