Essay:Rebuttal to Logical Flaws in E=mc²

This is a rebuttal to the points in the Logical Flaws in E=mc² article.

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1. The energy of a mass would change rapidly as the speed of light changes..

   The speed of light cannot change, by definition. The speed of light is a defined quantity. It is possible for units of length and time to change (e.g. the metre and the second), it is not possible for the speed of light to change, by definition.

2. How is "m" defined?

   Many formulas are in terms of symbols and concepts that the reader is expected to be familiar with. This includes things like

   

   or

   

   It is simply not practical to put in an explanation for all the symbols. This is especially true for equations that are "eye-catching", which this one certainly is.

   Since this formula says something that is not straightforward in layman's terms, leading to silly things like saying that the formula says that "mass and energy are equivalent", it is worth being careful here.

   "m" is mass, obviously. Really. That's all. It's what you measure if you put something on a scale. There has historically been some extra complexity and confusion involving the terms "rest mass" and "relativistic mass", but those terms are obsolete. The old term "rest mass" is what we use in the present day, calling it just "mass". If you put a charged flashlight battery on a scale and weigh it, that's it's mass. If you put a spent battery on a scale and weight it, that's it's mass. They will differ by about a picogram, by the way, due to E=mc².

   "c" is the space-time calibration factor of the Lorentz transform and the Minkowski space. This calibration factor has a very deep meaning in relativity. It's everywhere. You need to understand it as the calibration factor. It is a speed, 3x10⁸ meters per second. It happens that light travels at that speed. The work of James Clerk Maxwell explains why. Because of this, it is commonly called the "speed of light", but it doesn't actually involve light. "Speed of light" is just easier to say, and has a meaning that is more accessible to the layman, than "space-time calibration factor". But E=mc² is not about light. It's true even in the dark.

   "E" is energy. It's not easy to give a good description of this from first principles. But it's really what you learned about in junior-high and high-school science classes. It's that thing that is conserved. It can be "potential" or "kinetic". It's the ability do work, for example, to make something move. A battery has energy (potential) because, if connected to a motor, it can make things move. A coiled spring does the same. A speeding bullet has kinetic energy, and moves whatever it hits.

   The fact that an equation has been presented without an explanation of the terms appearing in it does not constitute a "logical flaw" in the equation. It constitutes at best a "pedagogical flaw" in the book in which it was presented.

3. The formula falsely implies that it is impossible for matter to increase its energy without increasing its mass in direct proportion.

   It does not imply this. An object's mass is the energy the object has when it is standing still (to be more precise, it is the energy a system has in a reference frame were its energy is at a minimum). If you increase an objects energy, e.g. by speeding it up, it does not mean it increases in mass.

4. Why hasn't the formula led to anything of value?

   This has led to many things of value, for one example of a huge number PET scanners.

   Mere equations should not be evaluated by their direct usefulness in making products. In particular, failure of an equation to provide actual things of value does not constitute a logical flaw in the equation. The logic of something transcends its immediate usefulness. An equation can be criticized for not leading to useful products, but that isn't a "logical flaw".

5. The formula implies existence of a unified theory of mass and light, when no such theory is possible.

   First, saying that "no such theory is possible", on any subject, is a rather slippery slope.

   Second, relativistic quantum electrodynamics is very much a "theory of mass and light". Physicists deal with this subject matter all the time. Perhaps the simplest example of the interaction of matter and light at the quantum-mechanical scale is Compton scattering.

   Finally, as noted above, the "c" in the equation is commonly called the "speed of light", but it is really just the calibration factor. The equation is not about light. Just mass and energy. The equation is true in the dark.