#### Introduction

Part 1 of this series explained why relativity is so popular and well–defended. As leading scientists proudly proclaim: “it works.” So it should come as no surprise that challenging relativity – a theory that “works” – is a difficult endeavor.

Despite an overwhelming level of experimental support and an intense bias in its favor, relativity contains a serious mistake that renders it completely invalid. While this last sentence may invoke a visceral reaction in the most ardent supporters of Einstein’s work, science requires that we remain open to the possible that relativity is wrong – no matter how remote the possibility. The idea that a theory is fallible is what separates science from dogma.

Changing the belief in relativity’s validity is *difficult*, especially when the evidence shows that it works. Fortunately, *difficult* is not the same as *impossible.* As discussed in Part 1 of this series, there are three ways to show that a theory – in this case, relativity – is wrong:

**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

The first argument that scientists have agreed can disprove Einstein’s theory is to **show that relativity contains a significant and incurable mathematical mistake**. Here we will reexamine the failed spherical wave proof to satisfy the first requirement. The material in this post will feel like a review for readers already familiar with my analysis of the proof.

#### Evaluation Criteria

To show that relativity contains a significant and incurable mathematical mistake, a successful challenge must satisfy two conditions:

- Clearly show a significant mistake that invalidates the theory.
- The challenge cannot introduce ideas, concepts, or equations.

The first condition is important, because easy to correct mistakes and typos do not automatically invalidate the theory. The mistake has to be big. The second condition is important, because the scientific community will summarily dismiss new and novel ideas as reflecting the challengers lack of understanding of theory. This means that a challenge can only use Einstein’s original work, which in this case is his 1905 paper; and material that is part of the accepted bodies of knowledge in physics and mathematics.

Any challenge that does not satisfy both conditions is unlikely to sway the scientific community at–large. Fortunately, an analysis of the spherical wave proof meets both criteria. In addition, this mathematical analysis can be objectively evaluated by a broad community of stakeholders, including those without expertise in relativity.

#### Einstein’s Spherical Wave Proof

Many modern–day scientists believe that deriving Einstein’s transformations equations is alone sufficient to prove that relativity is valid. This is incorrect and contradicts Einstein’s position. After deriving his equations, Einstein says that that he must prove that the *principle of relativity* and the *principle of the constancy of the velocity of light* are compatible.

To prove his principles were compatible, Einstein developed a proof: called the spherical wave proof. This proof is found in Section 3 of his paper that established relativity: “Zur Electrodynamik bewegter Köerper” (*On the Electrodynamics of Moving Bodies).* [1] An English language translation of the proof is:

The proof is quite ingenious and is based on the idea that when a spherical wave exists in the stationary system, the application of Einstein’s translation equations on that wave will produce a spherical wave in the moving system. The proof is clever because it demonstrates the compatibility of both postulates. If the proof passes, then the use of *c* in both equations means that light has the same velocity in both systems, which validates *the principle of the constancy of the velocity of light*. This relationship is illustrated in Figure 1 with an orange circle below *c* in both equations.

If the shape in stationary system is the same as the transformed shape in the moving system, Einstein will show that *the principle of relativity is valid*. This is why the shape – a spherical wave – is so important. The idea that a spherical wave exists in both systems is based on the mathematical equation for a sphere and is illustrated in Figure 1 by the blue boxes.

#### Analysis of the Spherical Wave Proof

The failure of the spherical wave proof has been presented in Chapter 1 of * DISRUPTIVE*, in the paper entitled the

*Failure of the Einstein–Lorentz Spherical Wave Proof*, and in a

*Tutorial 13*in the tutorial series on this site. [2,3,4] Rather than repeat that material and their accompanying mathematical treatments, we will summarize and highlight the key findings.

The proof begins with Einstein saying that he has not shown that his two principles are compatible, acknowledging that the equations alone are insufficient the prove relativity’s validity (Sentence 1). Einstein’s statement leads to the question: *Where exactly does relativity come into existence? *The answer is: relativity exists only after Einstein concludes “*This proves that both principles are compatible.*” That statement (Sentence 6) is the critical place in Einstein’s work where we can say relativity exists.

The proof is critically important, because Einstein clearly identifies the condition that must be satisfied for relativity to be valid: He must show both principles are compatible. If the proof, fails then Einstein cannot say his principles are compatible, which invalidates relativity.

#### Why Did Einstein Conclude The Proof Worked?

Einstein believes that the second shape is a spherical wave (Sentence 5). This belief allows him to conclude that relativity is valid (Sentence 6). Modern–day scientists who review his work also believe that the transformed shape is a spherical wave. Despite their review, there is an extremely subtle but critically important mistake that makes the proof fail.

#### The Proof Failed Because The Transformed Wave is Not Spherical

To understand why the proof fails, we have to review the mathematical definition of a sphere; which is *a three dimensional shape where all points on the surface are a constant distance from a common center*. Previous reviews of Einstein’s proof have focused on the equality of the equations alone to determine if the shapes are spherical, illustrated with triangles beneath the equals signs in Figure 1.

In addition to maintaining the second equation’s equality, a spherical wave requires that *each segment (which should be a radius) must be the same length and originate from the shape’s center*. Upon examination, neither of these requirements are met in the transformed shape.

The mistake was not found earlier, because it is not obvious. The proof appears to pass when all that is considered is the equality of the math statements. To find the problem, we must go beyond simply assessing the statements equality. We have to ask if a spherical wave shape is formed. Specifically, we have to determine if the “radius” to each point meets two criteria: each segment is the same length and each segment originates at the shape’s center.

Fortunately, the problem is revealed when the spherical wave is transformed using Einstein’s equations. To visually illustrate the problem, we use a large velocity, but the findings are mathematically valid at any velocity. A spherical wave at time *t = 1/c* in the stationary system results in the following:

Collectively, the the original and transformed points products the following shapes:

The validity of relativity is solely determine using the criteria that Einstein established in his paper, which comes down to one question: *Is Einstein’s conclusion in Sentence 5 of the spherical wave proof – that “the transformed wave when viewed from the moving system is a spherical wave” – true or false?*

If true, then any mathematical mistake in Einstein’s work must be viewed as insignificant and relativity is a valid theory. The belief that the proof passes is one of the key reasons why leading scientists argue that relativity will not be shown wrong, but might be improved upon. However this analysis shows the answer is false, because the transformed shape is not a spherical wave. The failed proof not only invalidates relativity, it means it was never correct to begin with.

There are three main defenses to this analysis. The first is **denial** of the results. The second is to focus on the equality of the statements and **ignore** Einstein’s statements that the shapes must be spherical waves. The third is to use a relativistic term to explain the transformed shape, which fails because ** a priori knowledge** cannot be used until after the proof is complete. As a result, these defenses fail and the analysis of the failed proof prevails.

The analysis of the failed spherical wave proof meets the criteria established above: It is a material failure that invalidates the theory and there is no way to correct the proof so that the second shape is a spherical wave.

#### Conclusion

The spherical wave proof is the most important aspect of Einstein’s 1905 paper, because without it Einstein is unable to show that his principles are compatible. This compatibility is the cornerstone of relativity. While the proof fails, this mistake has gone uncovered for over a century. That’s because the mistake is not obvious. The problem is that the proof is incomplete and the steps Einstein shows are insufficient to prove that the second shape is a spherical wave. A spherical surface requires a constant radius from a common center, characteristics Einstein does not consider. Upon examination, the proof fails because all of the necessary characteristics are not met.

The proof’s failure is **evidence that relativity contains a significant and incurable mathematical mistake**, which satisfies the first requirement**. **As mentioned in Part 1 of this series, satisfying the first requirement alone will not convince the scientific community at–large that relativity is wrong. They believe that relativity has been proven. Specifically, they believe that relativity is the only theory that can explain certain experiments and observations. This belief will be challenged in Part 3 of this series, where we examine why relativity is an approximation of Modern Mechanics.

#### References

[1] Einstein,“Zur Electrodynamik bewegter Köerper” (*On the Electrodynamics of Moving Bodies), *pages 900–901

[2] Bryant, * DISRUPTIVE*, Chapter 1, pages 14–30

[3] Bryant, *The Failure of the Einstein–Lorentz Spherical Wave Proof*

[4] Bryant, *Tutorial Thirteen – Einstein’s Failed Relativity Proof*

––––

**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*

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Photo of Steven B. Bryant ©2015 Steven B. Bryant, Photo by Amy Slutak*