GPT-4 is this incredible AI model that can understand and generate human-like text. Think of it like a super-smart chatbot that you can have a conversation with, and it can help you solve complex problems. I wanted to see if it could help me dig deep into the world of physics and analyze some equations from Einstein’s work.

Now, you might be wondering, “Why would you use an AI to do that?” Well, the truth is that there aren’t always enough human experts available – time or interest – to work on these kinds of complex problems. Plus, it’s fun to explore new ways to collaborate with AI!

To test GPT-4’s potential, I carefully crafted a series of questions to guide its analysis of Einstein’s equations. At first, GPT-4 was a bit biased and tried to preserve the theory, but with some tweaking of the questions, I managed to get it to focus on pure math.

Guess what? GPT-4 actually confirmed that there’s an inconsistency in the equations derived from the Special Theory of Relativity! How cool is that? This finding shows that AI can play a huge role in scientific research, even when it comes to complex and controversial subjects.

But, of course, using AI in research isn’t all sunshine and rainbows. We need to be aware of its biases and limitations, especially when dealing with complex domains. I had to put in a lot of effort to fine-tune the questions I asked GPT-4, but it was worth it.

In the end, this research has confirmed the incredible potential of AI as a co-collaborator in scientific research. It also underlines the importance of understanding AI’s biases and working to improve their objectivity and accuracy.

So, next time you’re tackling a complex problem, don’t forget that AI tools like GPT-4 might be able to lend a helping hand! Just remember to be aware of their limitations and work together to find the best solutions. Who knows? You might just make a groundbreaking discovery!

If you’re interested in reading more about my research, you can find the full paper here.

]]>**Notes**:

- One commenter has suggested that the statement
*ξ=cτ*is only valid when*x=ct*. This defense dismisses the mathematical equivalence of the statement that**creates**the*ξ*equation presented in Einstein’s work which is unconstrained by any independent variables associated with finding*τ*. Mathematically, when*ξ*is created from c*τ*using the equation*ξ*= c*τ*, the truth of this equation must always be maintained. The proof is fairly straightforward since c*τ*is equivalent to*ξ*= the sum from i = 1 to c of*τ*_i. Said another way,*ξ*= c*τ*is equivalent to using addition to find*ξ*, as in*ξ*=*τ*+*τ*+ … +*τ*(where*τ*appears c times). Since*ξ*is created from c*τ*, the requirement for the equation to always be True is not negated by the way in which*τ*is found.

**Corrections**:

- This was a live discussion and I’ve found a few typos to math equations (which are corrected in the presentation attached above). Some of my verbals mistakes, (e.g., using “terms” where I should say “expression”, using “B” where I should have said “beta”) are easily recognized and the proper wording can be inferred via context or by the text on the slides.
- While deriving the energy equation I talk about orders of 4 and higher. The slide should show c^4 instead of c^2. My comments should be related to c^4 and higher in the denominator, not v^4 or higher in the numerator. This does not change the truncation, but for clarity, this must be called out. This is corrected in the revised presentation attached above.

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**Steven B. Bryant** is a researcher and author who investigates the innovative application and strategic implications of science and technology on society and business. He holds a Master of Science in Computer Science from the Georgia Institute of Technology where he specialized in machine learning and interactive intelligence. He also holds an MBA from the University of San Diego. Steven 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.

This post walks through the proof, explains why scientists believe it’s valid, and explains why it fails.

Einstein’s 1905 paper, *On the Electrodynamics of Moving Bodies*, consists of three parts: pre-proof, proof, and post-proof. The pre-proof contains the assumptions and derivation, and the post-proof further develops the work in the context of relativity. Many people believe that relativity exists once the equations are derived. This is incorrect. Einstein realizes that assumptions and equations alone do not make the theory. He develops a proof that begins by saying he must show that his assumptions and equations *are compatible. *He uses the Spherical Wave Proof to show this compatibility*. *Relativity exists once the proof is successfully completed.

Any challenge to the assumptions or derivation alone (pre-proof) can be defended by asking the question: *If Einstein’s work contains a significant mistake, then why does the proof establishing relativity work?* Not only does a failure to answer this question serve as prima facie acceptance of the theory’s validity, it also allows for a relativistic context to be used in the defense. Thus, a challenger is dismissed by saying that s/he *doesn’t understand the theory*. Similarly, any challenge to the paradoxes (post-proof), or to relativity in a general sense (post-proof), is defended in the same way.

A challenge against the proof, however, cannot be defended in the same way. This is an interesting area for a challenge because the rules of inference eliminates and prevents the use of the thing being proved – *relativity (and a relativistic context)* – as a defense. Not only would such use violate the rules of inference, its use in defending a challenge made against the proof is equivalent to saying: *Relativity is right because it’s relativity.*

Before I can explain what’s wrong with the proof, I must explain why people believe it’s right. So, let’s look at the Spherical Wave Proof.

Consider the following **Statements**:

- Einstein’s theory of Special Relativity is derived in his 1905 paper entitled, On the Electrodynamics of Moving Bodies
- The rules of inference (eg, deduction) are used in mathematical proofs
- A sphere is defined as the collection of all points (in three dimensions) that are the same distance from a common center
- Einstein’s proof (found in Section 3 of his paper from Statement 1) begins with “
*We now have to prove …*” and ends with “*This shows [proves] that our two fundamental principles are compatible*.” - Within the proof, Einstein’s first equation represents the equation for a spherical wave at a specific time
- When evaluated, the equality of the first equation is
**always**maintained - When the shape is evaluated with the first equation, it satisfies Statement 3
- Einstein uses his transformation equations to produce a transformed shape
- Within the proof, Einstein’s second equation represents the equation for a spherical wave at a specific time
- Einstein uses the second equation to evaluate the transformed shape
- When evaluated, the equality of the second equation is
**always**maintained - When the transformed shape is evaluated with the second equation, it satisfies Statement 3
- The rules of inference for mathematical proofs in Statement 2 have not been violated
- The proof that formally establishes relativity is complete. Relativity (and a relativistic context) is valid from this point forward

Einstein’s proof stands alone. The use of the transformation equations to associate a spherical wave in one frame with a transformed spherical wave in the second frame demonstrates compatibility with the *principle of relativity*. The use of the constant * c *as the velocity of light in both spherical wave equations demonstrates compatibility with the

Now I must show where the proof fails.

The best way to reveal the mistake is simply for **you** to *draw* the transformed shape. One way to accomplish this is to begin with a unit sphere (a sphere that exists at time 𝑡=1/𝑐 seconds with a radius of 1 meter). Convert the first shape (eg, the sphere) into the transformed shape using his transformation equations. Draw the transformed shape. (*Hint: It will not be a sphere.*) While using a large velocity makes this visually identifiable, the result is mathematically true for any non-zero velocity. Surprisingly, **you cannot discover the mistake by performing the steps (in the Statements section above) alone**. This helps to explain why this problem has not been previously uncovered: Remember, the proof looks right.

**Statement 12 fails because the requirement of a constant radius from a common center (required by Statement 3) is not satisfied**. It’s also important to recognize that *Statement 11 alone does not confirm that the shape is a sphere when the radius is not held constant*. As a result, Statements 13 and 14 cannot be completed, compatibility is not established, and the proof fails.

For some readers, any challenge against their deeply-held convictions is difficult to accept. While disagreement is expected and scientific, any defense must adhere to scientific practices and mathematical rules. So, defenders should be prepared to:

- Provide a picture of the transformed shape, and
- Recognize that a defense that presumes a relativistic context, through which the shape is “interpreted” as a sphere, will violate Statement 13 resulting in the proof’s failure

This discussion does not:

- Address the experimental evidence regarding relativity theory
- Explain how relativity can be wrong and still provide useful answers, and
- Provide an alternative that matches or exceeds the performance of relativity

I discuss these areas throughout the site. They are also summarized in a short 5–minute poster presentation that I delivered in Feb 2019 at the Georgia Institute of Technology.

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**Steven B. Bryant** is a researcher and author who investigates the innovative application and strategic implications of science and technology on society and business. He holds a Master of Science in Computer Science from the Georgia Institute of Technology where he specialized in machine learning and interactive intelligence. He also holds an MBA from the University of San Diego. Steven 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.

**Panelists (seated left to right):**

William Ford, PhD, Analytics/AI/ML Consultant

Emmanuel Matthews, Google

Ayori “Selfpreneur” Sellasie, Salesforce

Keita Broadwater, PhD, Oxygen-AI

**Moderator:** Steven B. Bryant

The video is broken into four parts:

- Introduction to Artificial Intelligence (AI) and Machine Learning (ML)
- Introduction to AI/ML continued, machine learning job roles, and solution bias
- Solution Bias continued, Diversity, How To Get Involved in AL and ML, and Q&A
- Q&A continued

I hope that you find the discussion informative, thought-provoking and motivating.

**Part 1 – AI Panel Discussion: Introduction to Artificial Intelligence and Machine Learning **(Deb Watson, NBMBAA SF Chapter President, gives a nice moderator introduction, but the discussion doesn’t begin until the 5:30 mark if you want to skip ahead.)

**Part 2 – AI Panel Discussion: Introduction to AI/ML continued, machine learning job roles, and solution bias**

**Part 3 – AI Panel Discussion: Solution Bias continued, Diversity, How To Get Involved in AI and ML, and Q&A**

**Part 4 – AI Panel Discussion: Q&A continued**

I want to thank the San Francisco chapter of the National Black MBA Association for sponsoring the event and Wells Fargo for providing the space for the meeting.

––––

**Steven B. Bryant** is a researcher and author who investigates the innovative application and strategic implications of science and technology on society and business. He holds a Master of Science in Computer Science from the Georgia Institute of Technology where he specialized in machine learning and interactive intelligence. He also holds an MBA from the University of San Diego. Steven 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.

Let’s begin with statements with which everyone should agree:

- The
**average (or arithmetic mean),**of two expressions s and*ξ*,*t*can be found using the equation:*ξ = 0.5*(s + t)*. It can also be found using an equivalent equation:*ξ = t – 0.5 *(t – s)*. If you use the second equation but fail to recognize it as an average, this does not enable it to take on new magical properties. - Mathematically, a circle (2D) or sphere (3D) is axiomatically defined as,
*the set of all points in a Euclidean plane (2D) or space (3D) that are a*. If you find at least two points that belong to the same set and those points are not the same distance from a common center, then the shape is not a circle or a sphere.**constant distance**from a common center - If given the distance equation,
*distance=time*velocity*, you can solve for any variable if the other two are known. However, you**cannot**use this equation to determine a*velocity*if you**replace**.*distance*with*grams*,*volume*,*cycles*, or*shoe size*

Now, let’s create some statements with which few people should agree. I’ll call these statements elements of a **crackpot test:**

- On a sheet of paper.
**Draw**a circle, an oval, a straight line, and a squiggle. Convince yourself that each of the shapes is a circle. - Convince yourself that each of the following equations are equivalent and will properly find the
*velocity*of a moving object:*velocity = grams/time*;*velocity = cycles/time*;*velocity = volume/time*, and*velocity = shoe size / time*. - Imagine a train approaches you with a bright light on top of the locomotive. You know the wavelength,
*x’,*of the light. You measure the light’s wavelength as the train approaches and again as it moves away from you as,*s = x’c/(c + v)*and t*= x’c/(*c – v*)*. Find the average Doppler equation,*ξ.*Convince yourself that the average Doppler shift is the train’s spatial position.

Now for the test question: *If someone builds a “cockamamie theory” (Ms. Miller’s words, not mine) based on at least one of the above statements, would you label them as a crackpot and dismiss their theory?*

Before you answer the question, recognize that a key theme of the scientific process is independent validation. To this end, review Einstein’s 1905 paper, *On the Electrodynamics of Moving Systems,* and Michelson and Morley’s paper discussing their interferometer experiment and see if you can find each of the anomalies (above). Why do I ask that you find it yourself? Because when you do, it’s no longer about someone telling you what they’ve found. Instead, finding them independently allows us to come to the table as peers and engage in a scientific conversation rather than an emotional argument. Even if we disagree on whether a finding is “right” or “wrong”, we’re discussing the same finding.

Ideally, you’ve independently found each anomaly mentioned above. But, if you’re struggling to see the problems in the original works, you can (optionally) review an academic poster presentation that I delivered in February 2019 (see: https://goo.gl/8kaF3N ). However, I still encourage you to review the original works and confirm each finding yourself.

Returning to the test question: If you answered yes (and you’ve done the research mentioned above), not only have you dismissed Einstein’s theory of relativity as a “*cockamamie theory*”, you’ve labeled Einstein as a “*crackpot*”. This is why name calling is so dangerous. While I believe relativity is invalid, I would never use such terms to describe Einstein or his work. It is this type of labeling and name calling that turns a scientific conversation into an emotional argument; at which point serious discourse no longer occurs.

So, Ms. Miller, please join me in changing the tone of the conversation. Let’s agree to stop the grade school name calling because labeling someone as a crackpot does nothing but perpetuate a culture of bias and discrimination. Let’s also agree to stop hiding behind the excuse of peer reviews when editors, many of whom share your biases, have no intention of publishing works that disagree or challenge their beliefs – no matter how well-argued and researched that work might be. Name calling and exclusion were (ineffective) tools we used as kids on a playground when we didn’t know any better. But as mature scientists, we owe it to ourselves and to the broader community to find more effective ways of handling crucial conversations. Fortunately, I’ve met many scientists who are quite open to exploring material that challenges their beliefs. We can use them as role models.

Scientific innovation and discovery advance best when we examine what others have to say and remain open to reexamining our most deeply held beliefs. I think it would be a huge benefit to the scientific community if more journals were open to publishing well-researched, critical submissions that challenge our understanding. Ms. Miller, will you please join me in moving the conversation from the playground to places more appropriate for serious scientific discussions?

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**Steven B. Bryant** is a researcher and author who investigates the innovative application and strategic implications of science and technology on society and business. He holds a Master of Science in Computer Science from the Georgia Institute of Technology where he specialized in machine learning and interactive intelligence. He also holds an MBA from the University of San Diego. 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.

The 48″x36″ poster walks through the key findings discussed on this site; beginning with the spherical wave proof finding and ending with a performance comparison of relativity and Modern Mechanics.

I’m happy to share the abstract, poster presentation, and short 5-minute audio walkthrough of the presentation.

––––

**Steven B. Bryant** is a researcher and author who investigates the innovative application and strategic implications of science and technology on society and business. He holds a Master of Science in Computer Science from the Georgia Institute of Technology where he specialized in machine learning and interactive intelligence. He also holds an MBA from the University of San Diego. Steven 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.

Images courtesy of Pixabay.

]]>In Part 3 of the Unleash Your Curiosity series, we’re going to look at circles and spheres, shapes many of us learned about in primary school. We’ll begin with a definition:

*A circle (two–dimensional) or a sphere (three–dimensional) is defined as the collection of all points that are equidistant from a common center point*.

Mathematically, this definition is represented by the equations:

*x ^{2} + y^{2} = r^{2}*

and

*x ^{2} + y^{2} + z^{2} = r^{2}*

for a circle and sphere, respectively.

You are now asked to answer two questions. First, consider Figure 1 and determine which shape(s) satisfies the *textual definition* of a circle. Of course, the only answer that satisfies the textual definition is the one in the upper right corner, labeled “Circle”.

**Figure 1**

Second, reconsider Figure 1 and answer the question: Which shape(s) satisfy the *mathematical definition* of a circle. Before you answer that question, consider how well the mathematical definition of a circle aligns with the textual definition by examining the collection of points for the line in Figure 2.

**Figure 2**

Interestingly, the collection of points that represent a straight line also satisfy the mathematical equation for a circle. Figure 2 reveals an important and extremely subtle distinction between the mathematical and textual definitions of a circle. In the textual definition, the requirement for a constant radius is *explicit*, while in the mathematical definition, it is *implicit*. This subtle but important difference means that when you have not explicitly confirmed that the radius for each point is the same, every collection of points when represented as *<x, y, r> *or *<x, y, z, r>* tuples will satisfy the mathematical equation for a circle or sphere, regardless of the actual shape that the points represent.

Answering the question from above: a failure to explicitly check for a constant radius means that each shape in Figure 2 will satisfy the mathematical equation for a circle or sphere, as long as the radius *r* is included in the tuple. Simply said, without confirming a constant radius, one can incorrectly conclude that a collection of points is a circle or sphere when in actuality it is a straight line, a triangle, or squiggle, or any other shape for that matter.

Perhaps at this point, you’re crying foul, asserting: “Of course a circle has a common radius! Everyone knows that!” While I hope “everyone knows that”, this discussion has raised two important points.

- First, adherence to the mathematical equations alone cannot confirm that a shape is a circle or a sphere if you have not also explicitly confirmed that the radius for each point is the same.
- Second, when the constant distance of the radius is not examined and it is included in each point’s tuple, one can incorrectly conclude that a collection of points is a circle or sphere when in actuality it is not.

Please keep this discussion in mind as you (re)read section 3 of Einstein’s 1905 paper: *On the Electrodynamics of Moving Bodies*. Specifically, examine Einstein’s spherical wave proof found in that section, which is included below as Figure 3 for your convenience. As you work through the proof, draw the original and transformed shapes. Pick a sufficiently large velocity so that you can visually see the problem. While the finding is mathematically true at all non–zero velocities, you will most easily see it when using larger velocities.

**Figure 3**

When you work through the proof using real numbers and draw the corresponding shapes, you’ll find that Einstein’s transformed shape is not a spherical shape because each point does not share the same radius. Notice that because no one knew to check for a constant radius, this mistake has gone undiscovered for more than a century. What are the implications of Einstein incorrectly concluding that his “principles are compatible”?

Unsurprisingly, some will reject the assertion that anything is wrong with Einstein’s work. They will ignore or dismiss the constant radius requirement and insist that the second shape is a spherical shape due to the relativistic perspective of the observer in the moving system. This defense fails to recognize the importance and significance of Einstein’s mathematical proof, where relativity does not exist prior to the completion of the proof’s final sentence. Mathematically and logically, one cannot use a conclusion resulting from the proof as evidence that the proof is correct. Otherwise, we simply would conclude *relativity is right because it’s relativity*.

Once again, I hope that this series has piqued your curiosity and that you’ve read something that made you say “Hmmm?”

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**Steven B. Bryant** is a researcher and author who investigates the innovative application and strategic implications of science and technology on society and business. He holds a Master of Science in Computer Science from the Georgia Institute of Technology where he specialized in machine learning and interactive intelligence. He also holds an MBA from the University of San Diego. 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.

Images courtesy of Pixabay.

]]>Scheduling a hair appointment:

Making a dinner reservation:

Making a dinner reservation #2:

Now that you’ve given some thought to how each person feels in each of the conversation, you might be surprised to learn that three of the six people you just heard aren’t people at all. Three of the “people” are actually computer generated AI agents. Can you guess which three? The answer is, the three callers.

This is an exciting development because it means that computer–based AI agents are able to interact with real people, in the real world, in a completely real and natural way. This is an important evolutionary step that takes state–of–the–are AI agent solutions like x.ai’s Amy that could schedule meetings on your behalf via email, or Georgia Tech’s Jill Watson which serves as a teaching assistant in its Knowledge–Based AI course, to the next level. Notice that the AI agent initiated the conversation. Also, notice in the conversations above how the human (recipient of the call) was able to ask the AI agent questions and the AI agent responded appropriately. This type of seamless, natural experience opens the door for AI agents to do a lot more on behalf of their principals.

Google Duplex is important because it means that someone can have an experience with a computer without first having to “learn” how to interact with that computer. Contrast the conversations you just heard with one of your recent phone calls into a customer support line where you were prompted with: “Say 1 for billing, 2 for sales, 3 for directions, or stay on the line for the next available operator.” I’m sure I’m not the only one who would repeatedly press zero so that I could speak to a real person.

There are clear benefits to consumers and businesses from leveraging this technology. For one, it cuts training time to zero. The people who answered the call didn’t need to learn how to talk with the agent or be prompted with a set of responses that the agent would understand. Second, it means that agents can do more. They are no longer limited to interacting with people via a computer. They can interact via phone. I can already imagine a future where you are interacting with a visual agent using augmented or mixed reality. Third, it frees up valuable time and allows you to deploy your people on more value–added tasks and activities.

Of course, this technology advance must also consider the broader implications. What happens if an AI agent incorrectly makes an expensive purchase on your behalf? Should an AI agent be required to disclose itself as one when it interacts with people? Are there implications for believing we are dealing with a person when we aren’t? These are questions we’ll need to answer as we start to build solutions that leverage this new approach.

I’m excited about this technology. In fact, while recently talking to a colleague about AI Agent–Oriented Operating Systems, I cited the movie HER as a futuristic example of what might be possible. While Artificial General Intelligence (AGI) solutions with capabilities like Samantha are still futuristic, Google’s solution represents a significant step forward in some very exciting areas.

https://youtu.be/n1AjtIAje3o?t=3m10s

For more information about Google Duplex, please visit their blog post.

––––

**Steven B. Bryant** is a researcher and author who investigates the innovative application and strategic implications of science and technology on society and business. He holds a Master of Science in Computer Science from the Georgia Institute of Technology where he specialized in machine learning and interactive intelligence. He also holds an MBA from the University of San Diego. 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.

Images courtesy of Pixabay.

Audio clips linked from Google’s blog post.

An extremely important concept in mathematics and science is called **Units of Measurement**. It helps us to distinguish a unitless numeric value from a specific measurement. Notice the difference between the number 100 and units of measurements like 100 *inches*, 100 *pounds*, or 100 *shoes*. For example, imagine you are asked to sweep a 7–foot by 12–foot patio. How big is the patio? An answer of 84 is incorrect. The correct answer is 84 *square feet. *One cannot overemphasize the importance of recognizing that a numeric answer alone is incorrect if it is not accompanied by the appropriate units of measurement.

Mistreating or ignoring units of measurement can lead to catastrophic failures. For example, in 1999 the Mars Climate Observer crashed into the Red Planet. The failure was traced to a mismatch in units of measurements, where one component calculated a numeric value in *pounds–fource seconds* and another in metric units called *Newton–seconds*. As a second example, the Institute for Safe Medication Practices reported a medication mixup where the dosage of phenobarbital (a sedative) was mistaken as 0.5 grams instead of 0.5 grains. Because the conversion is 1 grain = 0.0065 grams, the error could have been disastrous. Fortunately, the patient recovered from medication–induced breathing problems after physicians discontinued the medication.

Clearly, we agree that errors in units of measurements can lead to adverse events or catastrophic failures. However, what happens when the incorrect use of units of measurement does not lead to a catastrophic failure? Can we assume that if there isn’t an adverse event or catastrophic failure that the answer is right? You should agree that the answer is no. Any answer that uses incorrect units of measurement is wrong. To put this idea to the test, consider the equation:

**distance = time * velocity**

If you travel at 100 km/h (*velocity*) for 2 hours (*time*) you will have traveled a total 200 kilometers (*distance*). When you know two of the three variables in this equation, you can always find the third. Consider the following examples and assume you are asked to find the corresponding velocity:

**A:**10 hours, 1000 miles**B:**10 hours, 1000 pounds**C:**10 hours, 1000 cycles (or loops) of string**D:**10 hours, 1000 chickens

Ideally, you agree that only A (above) can be used with the distance equation to produce the velocity 100 miles per hour. You should also agree that pounds, cycles, and chickens are not measurements of distance and cannot be used with the distance equation to produce a velocity (even though in each case one can divide the numeric value 1000 by 10 to arrive at the numeric value 100). Just because one can compute a numeric answer does not make that answer right. Unfortunately, mistakes involving units of measurement can be extremely subtle. This means that, unless there is a catastrophic failure or you know to look for it, mistakes can go undetected for decades.

What is the relationship between units of measurement and theoretical physics? To answer this question, please keep answer C (above) in mind as you (re)read the Michelson and Morley’s Interferometer paper. Hint: Look for the units of measurement each time Michelson and Morley *use* their equation. Specifically, notice how they mistreat the *number of waves* (or *cycles*) for a *distance. *

The *number of cycles* (or *waves*) is not a *distance*, which means that they cannot use the distance equation to properly calculate the Earth’s orbital velocity.

Some readers might object by claiming that Michelson and Morley intended to use “cycles per seconds” (or Hertz). However, to produce the correct velocity, two adjustments are needed. First, a new expected result equation would need to be recomputed using the base equation:

**velocity = wavelength * frequency**.

Second, they would need to use *frequency* and *wavelength* in their computation instead of *time* and *distance* to avoid a similar error in their units of measurement with the revised equation. Imagine the impact on modern physics should the proper treatment of units of measurements in their paper, and the accompanying computation of the Earth’s orbital velocity, show that their experiment was a success! Regardless of your belief in the validity of the Michelson–Morley experiment, I hope this insight leads you to once again say “Hmmm?”

––––

**Steven B. Bryant** is a researcher and author who investigates the innovative application and strategic implications of science and technology on society and business. He holds a Master of Science in Computer Science from the Georgia Institute of Technology where he specialized in machine learning and interactive intelligence. He also holds an MBA from the University of San Diego. 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.

Images courtesy of Pixabay.

]]>In Part 1 we are simply going to look at how to calculate an average (arithmetic mean), *ξ*, of two numbers, *s* and *t,* where their average is found using the **addition mean equation**:

*ξ = (t + s) / 2*.

Consider a specific example where a laser of wavelength *x’* is emitted from a stationary location and a moving vehicle is approaching or receding from this location at velocity *v*. The corresponding Doppler equations are:

*c * x’ / (*c – v*)* and *c * x’ / (c + v)*.

Using the addition mean equation, their average is:

*ξ = c^2 * x’ / (c^2 – v^2)*.

Does this average define the position or a location coordinate of the moving vehicle? You should agree that the answer is no.

Alternatively, the average Doppler equation can be found using the mathematically equivalent, but not widely–used, **subtraction mean equation**:

*ξ = t – ½ * (*t – s*)*.

It is important to understand that the absolute value of the expression *½ * (*t – s*)* is called the **half–difference**.

Now I will ask you to use the subtraction mean equation to find the average using the following steps:

- Factor out the “
*c*” variable from the numerator in the original Doppler equation so that you begin with*x’ / (*c – v*)*, which we’ll use at “*t*”, and*x’ / (c + v)*, which we’ll use as “*s*”. You will multiply “*c*” into the final equation at the end. - Find the actual half–difference and replace the half–difference in the subtraction mean equation, resulting in:
*t – vx’ / (c^2 – v^2)*. - Replace the “
*t*” in the subtraction mean equation with*x’ / (*c – v*)*and simplify. - Multiply the result by “
*c*” and confirm that you have arrived at the same average equation:*ξ = c^2 * x’ / (c^2 – v^2)*.

While finding the average using the subtraction mean equation involves more steps, you will arrive at the same answer that is found using the addition mean equation. Now for a critical question: If you use the subtraction mean equation to find an average, but fail to recognize that you’ve done so, do you get to redefine the equations and expressions any way that you choose? You should agree that the answer is no.

Please keep the subtraction mean equation’s final equation and intermediate expressions in mind as you (re)read section 3 of Einstein’s 1905 paper: *On the Electrodynamics of Moving Bodies*. Hint: Einstein mathematically performs every step I asked you to perform (above) to find the average, but fails to recognize his use of the subtraction mean equation.

If during your review of Einstein’s paper you said “Hmmm?”, then you will find the remaining insights in the series interesting.

––––

**Steven B. Bryant** is a researcher and author who investigates the innovative application and strategic implications of science and technology on society and business. He holds a Master of Science in Computer Science from the Georgia Institute of Technology where he specialized in machine learning and interactive intelligence. He also holds an MBA from the University of San Diego. 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.

Images courtesy of Pixabay.

]]>