#### Introduction

Part 1 of this series summarized how relativity is well–supported and defended. It also serves as a reminder that any theory, specifically relativity theory can be proved wrong if any one of three requirements is met:

- Show that relativity contains a significant and incurable mathematical mistake
**Show that relativity is an approximation of the new theory**- Show an experimental result that contradicts relativity

Part 2 of this series revealed that Einstein’s spherical wave proof fails. This failure means that Einstein did not prove that his two principles – *the* *principle of the constant velocity of light* and *the* *principle of relativity* – are compatible. Because Einstein says the proof is required to validate his theory, the failed proof alone shows that **relativity contains a significant and incurable mathematical mistake **that invalidates his work.

While this failure should invalidate Einstein’s theory entirely, it will not be enough to overcome the bias that his theory is right. Supporters use two counterarguments to discount the importance of the failed spherical wave proof. First, they argue that *relativity works;* a finding based on experimental evidence that is not disputed. In fact, any challenge that does not acknowledge where relativity works cannot be taken seriously.

Second, supporters argue that *relativity is the only theory that works.* This belief is based on the idea that there are only two theoretical explanations for certain experiments: classical mechanics and relativity. Because relativity outperforms classical mechanics in explaining experiments associated with the electromagnetic force, supporters assert that it is the only theory capable of explaining those experiments. This leads to the question: *How can relativity, the one and only theory that seems to work, be wrong*?

To overcome these objections and answer the question just raised, we must show that relativity is not the only – or even the best – predictor of specific experiments. To accomplish this, we must **show that relativity is an approximation of a new theory called Modern Mechanics****. **Modern Mechanics is built upon the mathematical foundation call geometric transformations. It is introduced and derived in chapters 2 thorough 4 of * DISRUPTIVE*. Here we simply use one the equations from Modern Mechanics to compare it with one of Einstein’s relativity equations.

#### Evaluation Criteria

To overcome the bias that relativity is the only theory that is capable of explaining certain experiments and show it as an approximation for Modern Mechanics, we must:

- Show that Modern Mechanics and relativity produce similar results
- Show that Modern Mechanics and relativity both lead to
*E=mc*^{2} - Show that Modern Mechanics outperforms relativity in at least one experiment

If all three conditions are met, we conclude that **relativity is an approximation of a new theory called Modern Mechanics**, satisfying the second requirement for invalidating Einstein’s theory.

#### Relativity as an Approximation

Many scientific experiments are based on the idea of relative measurement, where a researcher might compare the original value with the observed (or transformed) value. In a general sense, relative measurements for this kind of experiment are written in the form:

*∆x = x’ – x
Equation 1*

where *∆x* is the difference of the observed value *x’* and the original value *x*. Many experiments that confirm relativity, like the Ives–Stilwell experiment that will be discussed shortly, use relative measurements. In other words, their actual results and their expected results are both expressed in terms of *∆x*.

Einstein’s relativistic equation expressed in the form of Equation 1 is:

*Equation 2*

While Equation 2 is independently developed in Chapter 7 of * DISRUPTIVE*, Einstein derives it over a century ago. In fact, this equation first appears in Einstein’s 1905 paper:

*Does the Inertia of a Body Depend on its Energy Content?*[1] Equation 2 is not only used to evaluate experiments like Ives–Stilwell, it is also critical in developing Einstein’s energy equation:

*E = mc ^{2}*

*Equation 3*

We’ll come back to how Equation 2 leads to the energy equation shortly. First, we have to show that Modern Mechanics and relativity will produce nearly identical answers.

The Modern Mechanics equation, when expressed in the form of Equation 1, is:

*Equation 4*

Equations 2 and 4 are not the same equation and a visual inspection of both equations does not immediately suggest that they would produce similar results. Yet, what is interesting is that equations 2 and 4 will produce nearly identical results! To understand why this is the case, we have to examine both equations using a mathematical technique called a Taylor series. The relativistic equation given in Equation 2 when expressed as a Taylor series is:

*Equation 5*

The Modern Mechanics equation given in Equation 4 when expressed as a Taylor series is:

*Equation 6*

Even if you are unfamiliar with a Taylor series, you can make two observations about equations 5 and 6. First, both equations have the same first expression:

*Equation 7*

The accuracy obtained from this first expression alone explains why Modern Mechanics and relativity produce nearly identical answers. In fact, as you will soon read, Einstein felt that the level of accuracy of this expression alone was sufficient enough to discount the remaining expressions when he produced his energy equation.

Second, equations 5 and 6 differ beginning with their second expression. An examination of both Taylor series reveals that the maximum difference in the second expressions is just under a factor of 0.125*λ*, a value that is only produced when a moving system is traveling just shy of the speed of light. When a moving system is traveling at slower speeds, the difference is so small that it may not be measurable or if it is measurable might be misinterpreted as *experimental error*.

Even the most ardent supporters of relativity must acknowledge that **Modern Mechanics and relativity produce nearly identical result**, satisfying the first item in the evaluation criteria (above). This means that many experiments that have been used to confirm – or “prove” – relativity will also confirm – or “prove” – Modern Mechanics.

#### The Making of *E=mc*^{2}

^{2}

A Taylor series is a very important mathematical tool. It can convert an equation that might be hard to solve (especially in 1905, before calculators and computers) and transform it into a highly accurate approximation equation that can be easily solved. Because Taylor series are approximations, technically they should be written using the approximation symbol ‘≈’ instead of the equals sign ‘=’. This correct use of the approximation symbol is important when both series are truncated to the first expression, resulting in:

*Equation 8*

Notice that it would be technically incorrect to use the equals sign because we have already shown that equations 2 and 4, while close approximations, are not equal.

To align with Einstein’s choice of variables in his 1905 paper [1], we rederive equations 2 and 4 using the variable *L*, instead of *λ*. Einstein noticed the similarity between the first expression in the series and the classical mechanics energy equation. Although not explicitly shown in his paper, he introduces mass into the equation by setting both equal to one another (which we show using the approximation symbol):

*Equation 9*

Following algebraic simplification and the replacement of *L* with *E* to align with modern nomenclature, Einstein arrives at:

*Equation 10*

Equation 10 is significant for two reasons. First it shows that Einstein’s energy equation is properly written using the approximation symbol, not the equals symbol. Second, it shows that **relativity is not unique in its ability to produce the energy equation. This equation can also be derived from the Modern Mechanics equations**, using the first expression of its Taylor series and by following the same steps Einstein performs in his paper: *Does the Inertia of a Body Depend on its Energy Content.* This satisfies the second item listed in the evaluation criteria (above).

**Analysis of the Ives–Stilwell Experiment**

As mentioned above, the scientific–community at–large believes that relativity is the **only** theory capable of explaining certain experiments or worldly observations. For example, Adrian Sfarti in a paper entitled “*Experimental Test Theories for STR: Part 1 The Ives-Stilwell Experiment*,” [2] writes: “*it is much more difficult and sometimes impossible [for an alternative theory] to produce data coincident with the experiments. One [example] is the well-known Ives-Stilwell experiment*.” In other words, Sfarti argues that relativity is the only theory capable of explaining the Ives–Stilwell experiment. There’s only one thing wrong with this belief: It is completely wrong.

The Ives–Stilwell experiment is evaluated using an equation related to Equation 2. [3,4] Table 1 illustrates the experiment’s expected and actual results. A review of the table reveals that the relativistic equation is a good predictor of the Ives–Stilwell experiment. The observed error, or the difference between to expected and actual results, is extremely small; so small that it is attributed to *experimental error*. The relativistic equations outperform the classical mechanics equations and if that is the only alternative theoretical explanation then level of performance helps to explain why scientists believe relativity is unique in its predictive ability.

However, we have explained that the Modern Mechanics equations will produce similar, but not identical answers. Therefore, the Ives–Stilwell experiment can be reevaluated using an equation related to Equation 4. [3,4] Table 2 illustrates the expected and actual results when the experiment is analyzed using the Modern Mechanics equation. Interestingly, there is no measurable error to two significant digits. In other words, what was attributed to experimental error when analyzed using the relativistic equation is actually the result of *approximation error* inherent with the use of Einstein’s relativistic equation. This is an important finding, one that could not have been discovered without the introduction of the Modern Mechanics equations.

This analysis shows that the Modern Mechanics equation outperforms the relativistic equation, revealing that** relativity is an approximation of Modern Mechanics**. This satisfies the third item in the evaluation criteria (above).

#### Conclusion

In the introduction, we asked: *How can relativity, the one and only theory that seems to work, be wrong*? The answer is: *It isn’t the only theory that works: Modern Mechanics also provides answers that work*. To show this is the case, we identified three specific criteria that we had to meet. First we discussed why Modern Mechanics and relativity produce similar, but not identical answers. This discovery alone is significant, because it means that relativity is no longer the only theory capable of explaining experiments and observations. Second, we shattered the myth that relativity is the only theory that yields Einstein’s energy equation. More importantly, we showed that Einstein’s derivation of the energy equation is an approximation, not an absolute equality as it is often presented. Third, we discussed how the Modern Mechanics outperforms relativity in predicting the results of the Ives–Stilwell experiment. This is important, because it establishes relativity as an approximation for Modern Mechanics.

Contrary to popular belief, relativity is not unique in its ability to explain certain experiments and observations. The finding that **relativity is an approximation of Modern Mechanics**, when combined with the finding from Part 2 of this series of the **failed spherical wave proof**, is further evidence that relativity is a failed theory.

#### References

[1] Einstein, *“Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?“* (*Does the Inertia of a Body Depend on it Energy Content?*)

[2] Sfarti, “*Experimental Test Theories for STR: Part 1 The Ives-Stilwell Experiment*,”

[3] Bryant, * DISRUPTIVE*, chapters 6 and 7

––––

**Steven B. Bryant** is a futurist, researcher, and author who investigates the innovative application and strategic implications of science and technology on society and business. He is the author of * DISRUPTIVE: Rewriting the rules of physics*, which is a thought–provoking book that shows where relativity fails and introduces Modern Mechanics, a unified model of motion that fundamentally changes how we view modern physics.

**is available at Amazon.com, BarnesAndNoble.com, and other booksellers!**

*DISRUPTIVE*

*Images courtesy of Pixabay.com
Photo of Steven B. Bryant ©2015 Steven B. Bryant, Photo by Amy Slutak*