Essay:E=MC² and the Frogton Universal Force Law

 Preamble 

This article (with slight alterations) was deleted from Conservapedia after a few days by the Scafly, after one of Conservapedia's editors launched this tirade of religious hatred:

“I have put a delete notice on your article. I find it impossible to believe that this isn't a joke and that you aren't a jokester. I, and other people at Conservapedia, take the topic of relativity very seriously.”

It is essentially impossible to view the article as anything other than a joke. This "Universal Force Law" is named after an effectively unknown person (no Google hits, for example) who seems to exist only as a new Conservapedia author. And he takes a concept that is actually quite well understood by scientifically literate people, ind goes into paragraph after paragraph of nonsense trying to make it sound confusing.

Relativity denial is conducted at a rather high level here at Conservapedia. People wishing to jump on the bandwagon would be well advised to do their homework. For example, by looking at the many articles on the subject listed on my user page at. I particularly recommend the article.

SamHB(talk) 22:10, August 16, 2021 (EDT) I have removed the "delete" template. It appears that this person actually purported to "publish" a "book", visible on Amazon. He (Isaac Frogton) therefore needs to be taken a little bit more seriously than I was taking him yesterday.

I have been taking on relativity denial here at Conservapedia for at least 10 years now, and am quite good at it, as is my colleague AugustO. If Mr. Frogton wants to raise his objections to E=MC², or any other aspects of relativity, the Counterexamples to Relativity page is the place for it. He will be number 53. While the equation E=MC² is just difficult enough to be found mystifying by many people, it is actually well understood by scientifically literate people. There is much discussion here at Conservapedia and elsewhere, explaining the equation. Just throwing around nonsense of one's own doesn't make one's point that the equation is nonsense.

An example (of many) of the confused nature of Mr. Frogton's writing may be found in in the book excerpt on Amazon:

This book sets the record straight, ... explaining things way beyond the comprehension of physicists, such as the the fact that E=(3/4)MC² couldn't possibly be correct, as it would mean no force could ever accelerate an object beyond √(3/4) of the speed of light, which rather makes a mockery of the concept of photons having momentum.

No one proposes such an outlandish "straw man" notion, or tries to explain how such a notion could limit speeds of objects, or could affect the concept of photons having momentum.”

I wonder if Rationalwiki editors will find it equally unacceptable?

E=MC² and the Frogton Universal Force Law 

Abstract

This paper shows that E=MC² is not a law of nature, as it is dependant on the historic definition of kinetic energy, and debunks other myths surrounding the equation and Einstein's 1905 derivation. Finally it explains why the Frogton Universal Force Law is the proper description of nature.

Introduction

E=MC² is one of the greatest stories in physics; not because it is a great revelation and a testament to the genius of Einstein, but rather because it is the most meaningless of equations, and the least of Einstein's achievements.

Meaning

The perennial answer to the question “what does E=MC² mean?”, is that it means energy equals mass times the speed of light squared. A more useless and meaningless answer is hard to imagine.

What E=MC² means, was set out by Einstein at the end of his famous 1905 paper 'Does the inertia of a body depend upon its energy-content?', where he stated “If a body emits energy L, then its mass will decrease by L/C²”, he clarified this by adding (in modern units) that if its energy changes by 1 joule or kg m²/s², then its mass will change by 1/[9×1016] kilograms.

Although it seems certain from his 1905 derivation, that Einstein did derive the collection of symbols M=L/C², in his mind at least, it doesn't actually appear in his paper. Instead he merely describes the result of his investigation by saying in effect L joules = L/C² kg. Thus 1 joule = 1/C² kg, 1 kg = C² joules, M kg = MC² joules. That is how E=MC² is most commonly illustrated today, with people saying that in the bomb that destroyed Hiroshima, 0.0007 kg of mass were converted to 0.0007 C² joules of energy.

Whilst it is normal to express the relationship between units in the same way as 1 kg = C² joules, for example 1 kg=2.2 pounds; the very nature of units means that the same relationship can instead be described by saying “to convert kg to pounds, you multiply the number of kg by 2.2 to obtain the number of pounds, P=2.2K”, and E=MC² adopts this approach. So E=MC² means “to convert kg to joules, you multiply the number of kg by C² to obtain the number of joules”.

Although E=MC² is not the most intuitive way of describing the relationship between kg and joules, it is consistent with the fact that the symbol M in physics denotes a number times a unit of mass, E denotes a number times a unit of energy, and C denotes a number times a unit of speed. In general terms, E=MC² means that if you multiply a number of units of mass by the speed of light squared, you get the number of units of energy, where the unit of energy is such that the units on each side of the equation match.

This interchangeability between these two units of mass/energy, does mean that energy has mass, but the conversion is only a mental one in terms of units; the popular claim that mass can be turned into energy, or energy into mass, can only be based on a perverse definition of the words 'mass' and 'energy'.

Imagine that you were to compress a long spring by exerting a force of 1 newton over a distance of 1 metre, and then clip it shut. By definition you would have transferred 1 kg m²/s² of energy from your body into the spring. By M=E/C², if you were to put the spring on an incredibly accurate weighing machine, it would weigh 1/C² kg more, and your body 1/C² kg less; so you could just as easily argue that you had transferred 1/C² kg of mass from your body into the spring. But to argue that you had somehow turned your energy into mass in the spring, or that you had turned mass from your body into energy in the spring, it is necessary to define the words 'mass' and 'energy' in a way that absurdly complicates the universe.

Energy and mass

When we are not dealing with E=MC² or M=E/C², then the words 'mass' and 'energy' do tend to have different connotations. The concept of mass is quite intuitive; we can hold a 1kg weight in our hand and feel the force of gravity pushing it towards the centre of the earth, or we could feel its inertia resisting our attempts to accelerate it. On the other hand the concept of energy requires a certain amount of education; students are taught that if you are at the top of a hill on your bike, you have potential energy, then as you roll down the hill the potential energy is turned into kinetic energy, and if you crash your bike at the bottom, your kinetic energy is turned into the energy of heat, sound, and even a few sparks of light.

Had physicists chosen to define the unit of energy solely in terms of heat, light, or sound, then the unit of energy would not be the kg m²/s², and therefore E=MC² would not exist. For example, the calorie was defined as the energy required to raise 1 kg of water by 1 degree, and has no association with E=MC². This is your first clue that E=MC² is not an equation that of itself describes the universe, since it is dependant on the historical definition of the unit of energy.

Historic development

Perhaps the first scientist to really get to grips with the concept of energy, was Leibniz. He must have considered Galileo's law of falling bodies, which states that near the surface of the earth, where the force of gravity is nearly constant, the speed of a body is proportional to the square root of the distance it has fallen; for instance if a body falls 9 times as far, it will hit the ground at 3 times the speed. Since raising a body 9 times as far logically implies using 9 times as much energy, Leibniz reached the conclusion that the energy of a moving body is proportional to the square of its speed; this is a law of nature, though only accurate if the body is not moving near the speed of light. Leibniz expressed this in terms of the equation E=MV²; but E=MV² is not itself a law of nature, but rather a valid definition of a unit of kinetic energy in terms of kg m²/s².

As it happened, physicists did not adopt Leibniz's formula, instead they decided to define the unit of energy foremost in terms of force × distance, E=FD, which integrates to give the now standard kinetic energy formula E=½MV². Had physicists stuck with Leibniz's E=MV², then they would have had to adjust E=FD to E=2FD, but there is no experiment that could ever be done to prove that either combination is better than the other, it is purely a matter of definition. However had physicists sided with Leibniz, then Einstein would have had to have come up with E=2MC²; which again demonstrates that E=MC² is not an equation that of itself describes nature, as it depends on the historic definition of the unit of energy.

Derivation

In physics an equation can only be derived by making certain assumptions. The 2 assumptions necessary to derive E=MC², are firstly that the unit of energy is defined by the Kinetic Energy Equation E=½MV², and secondly that the mass dilation equation M(total)=M(rest)/√[1- (V/C)²] is the correct description of nature.

In the 1600's, Galileo came up with his principle of relativity, which stated that somebody below deck on a ship, would be unable to perform any experiment to determine whether the ship was stationary in port, or was moving across a tranquil ocean at a constant speed. In the 1800's, it was shown that this principle of relativity must apply to all experiments, because experiments involving light gave exactly the same results throughout the year, even as the speed of the earth varied relative to the fixed stars, as the earth orbited the sun.

The only way to explain this was with the 3 Lorentz transformations, length contraction, time dilation, and mass dilation. At low-speed the mass dilation equation M(tot)=M(rest)/√[1- (V/C)²] can be mathematically approximated by M(tot)=M(rest)+½M(rest)(V/C)², thus the kinetic energy of a moving body, measured in the units of mass, is ½M(rest)(V/C)² kg, which can be equated to the kinetic energy measured in joules, ½MV². ½MV² joules=½M(V/C)² kg, simplifies to 1 joule = 1/C² kg, so 1kg=C² joules, which is what E=MC² means.

Alternatively we can say the mass of the kinetic energy, M(e)=½M(rest)(V/C)², thus M(e)C²=½M(rest)V², and since ½M(rest)V² is by definition E, we get M(e)C²=E, that is E=MC².

That is the only proper way to derive E=MC², because that is all it means. As such E=MC² is largely meaningless, because it is not an equation that describes nature, it merely describes the relationship between the historic definition of the unit of energy and the equation that does describe nature, M(tot)=M(rest)/√[1- (V/C)²].

M(tot)=M(rest)/√[1- (V/C)²] is full of meaning; because the fact that kinetic energy has mass, means that all energy must have mass, by the principle of conservation of mass. If an object is brought to a halt by a spring, the mass of the kinetic energy must be transferred to the spring. If a moving object heats up some water, the mass of the kinetic energy must be transferred to the water. If a moving object crashes and emits sparks, some of the mass of the kinetic energy must have been transferred to the light.

General misunderstanding of E=MC²

There is a common misconception amongst the general public that nuclear reactions can only be explained by saying mass is converted into energy, and that ordinary chemical reactions do not involve this.

However in his 1905 paper Einstein derived E=MC² by considering a body emitting energy in the form of light, without in any way specifying the source of the energy. At the end of the paper, Einstein added; “It is not impossible that with bodies whose energy-content is variable to a high degree (e.g. with radium salts) the theory may be successfully put to the test”, and this may have been the origin of the misconception that mass being converted to energy, is peculiar to nuclear reactions.

The misconception was surely fuelled by the front cover of Time Magazine in 1946, where it depicted a pensive Einstein alongside a mushroom cloud from a nuclear explosion with E=MC² written across it. And it was reinforced by the 2005 PBS Nova documentary 'Einstein's Big Idea', which dramatised Lisa Meitner and Otto Frisch attributing the energy released in the nuclear fission of uranium to mass being converted into energy, via E=MC², despite mentioning that the source of the energy is the electrical repulsion between the protons in the uranium nucleus.

Nowadays competent physicists are well aware that all energy has mass. In 2005 PBS Nova asked 10 famous physicists to describe Einstein's equation to curious non-physicists, and Sheldon Glashow pointed out “When an object emits light, say, a flashlight, it gets lighter”, which clearly illustrates that a torch is as much a weapon of mass destruction as an atom bomb, because if you claim that an atom bomb converts mass into energy, you must accept that a torch does the same thing. This wisdom seems to have filtered down to the lower echelons of the physics community, with the 2009 book 'Why does E=MC², and why we should care' pointing out that when you burn coal in oxygen, the mass of the carbon dioxide produced is less than the original combined mass of oxygen and carbon, due to the heat and light energy emitted. However it may not have reached the bottom-feeders, as Neil deGrasse Tyson told the same 2005 PBS Nova survey “It's something that doesn't happen in your kitchen or in everyday life”.

What physicists don't appear to have acknowledged, is that E=MC² is peculiar to kinetic energy, because the units of E are the units of kinetic energy.

Physicists' misunderstanding of the meaning of E=MC²

There is evidence that some physicists have, at some times, acknowledged that E=MC² is merely an equation for converting one unit of mass/energy into another unit of mass/energy. However the prevailing sentiment seems to be that E=MC² was a great revelation, not only in popular science, but also amongst the scientific community.

The overriding evidence that physicists failed to grasp the true meaning of E=MC², is that, even in the 21st century, they were carrying out experiments to test the accuracy of the equation. This is utterly absurd, because, since E=MC² is based on a definition, it is either true by definition, or false by definition; or, once we define energy as force × distance, exact by definition or inexact by definition.

The Lorentz Mass Dilation Equation M(tot)=M(rest)/√[1- (V/C)²], can of course be tested experimentally; but it has to be considered above suspicion, because if it was not exactly correct, the universe would not abide by the principle of relativity. So the exactness of E=MC² depends on the definition of the joule.

If we define the joule with KE=½MV², then 1 kg will always be slightly less than C² joules, because the kinetic energy measured in kg is given by M[{1/√[1- (V/C)²]}−1], which by the Taylor expansion is M times (1/2)(V/C)²+(3/8)(V/C)4+(5/16)(V/C)6..... Indeed as V gets ever closer to C, the joule actually becomes a larger unit of mass/energy than the kg.

To give the joule a precise definition using KE=½MV², we need to specify the mass of an object, and hence its speed. If we were to define the joule as the kinetic energy of a mass of 1 kg moving at √2 m/s, then the kg would be equal to C² joules to within about 1 part in 1017. To make 1 kg exactly equal to C² joules, we would need to define the joule as the kinetic energy of a mass of M going at √[2/M] m/s, in the limit as M tends to infinity.

However physicists have taken a different approach. Instead they have taken the kinetic energy measured in kg, as given by the Lorentz Mass Dilation Equation: M[{1/√[1- (V/C)²]}−1], and just tagged C² on the end to give KE=M[{1/√[1- (V/C)²]}−1]C², and called that the correct version of the Kinetic Energy Equation. Having redefined the joule with the sole intent of ensuring that 1 kg always equals C² joules by definition, to then carry out experiments to find out whether 1 kg does equal C² joules, is utterly daft.

Einstein's 1905 derivation

Einstein's derivation is blighted by verbosity, mathosity, misdirection, and obfuscation, but this is the basis of his derivation expressed in the simplest possible way:

You have a stationary apparatus that emits identical photons of total energy L in opposite directions, thus the apparatus remains stationary. You now propel the apparatus through the lab at speed V; by the principle of relativity it must continue at the same constant speed V even as it emits its photons; but this time, according to Einstein's relativistic Doppler effect, derived in a previous paper, the total energy of the 2 photons increases to L/√[1- (V/C)²], which is mathematically approximately equal to L+½LV²/C², provided V²/C² is a small fraction. Thus the extra photonic energy emitted due to the apparatus moving is ½LV²/C².

So where did this extra photonic energy come from? It must have come from some of the kinetic energy you gave the apparatus when you accelerated it to speed V. Specifically it must have come from the difference between the kinetic energy of the apparatus before emitting the photons, and the kinetic energy of the apparatus after emitting the photons. Thus, after much beating about the bush, Einstein ends up with the equation K0−K1=½LC²/V², where K0 and K1 are those kinetic energies. At this point, Einstein abandons maths, and goes straight to his conclusion; however it is clear that, in his head at least, he must have rewritten K0−K1 as ½M(difference)V², to give the equation ½MV²=½LV²/C². In order for that equation to be valid, the units on each side of the equation must match, so L must be in the same units as kinetic energy, that is joules or kg m²/s². Dividing both sides by ½V² leaves M=L/C², which is equivalent to M=E/C², which rearranges to E=MC².

A more explicit way to do the derivation is to imagine the apparatus has a device like a spring, which pings apart to produce photons of total mass X kg, when the apparatus is stationary; thus you must add X kg when you compress it. Once the apparatus is accelerated to speed V, the X kg of energy embodied in the spring, must have kinetic energy of ½XV² joules, as given by the Kinetic Energy Equation. This kinetic energy can then be equated with the extra ½LV²/C² kg of photonic energy dictated by Einstein's relativistic Doppler effect; thus we have ½XV² joules = ½XV²/C² kg, which cancels down to give 1 joule = 1/C² kg; which corresponds to Einstein's conclusion “If a body gives off the energy L in the form of radiation, its mass diminishes by L/C²”.

Max Planck apparently criticised Einstein for approximating of the Lorentz factor 1/√[1- (V/C)²] to 1+½V²/C²; but that just illustrated Planck's ignorance of the true nature of E=MC² as being merely a formula for converting units, where the joule itself was defined by an approximate formula.

Herbert Ives famously criticised Einstein's derivation in a 1952 edition of the Journal of the Optical Society of America, on the grounds that Einstein merely assumed what he set out to prove, petitio principii, or 'begging the question'. That is also an unfair criticism, because Einstein derived M=L/C² by comparing the Lorentz factor in his relativistic Doppler effect, to the Kinetic Energy Equation, which is the only way of deriving E=MC² without petitio principii.

Certainly Einstein deserves criticism for making his derivation ridiculously complicated; but the real problem relates to the illogicality of trying to derive E=MC², an equation that is all about energy having mass, from KE=½MV², an equation which is based on the assumption that energy does not have mass.

E²=M²C4+P²C²

A popular claim amongst modern physicists, is that E=MC² is just part of the full equation E²=M²C4+P²C².

As if E=MC² was not bad enough, by changing the simple kg into the more complex kg m²/s², E²=M²C4+P²C² goes a stage further, with the units kg²M4/S4.

Physicists derive E=MC² from E²=M²C4+P²C², by pointing out that if the object is stationary, then the momentum, P, is zero, so you are left with E²=M²C4, and by taking the square-root, you prove that the rest-energy of an object is given by the equation E=MC².

A more blatant example of assuming what one sets out to prove, petitio principii, or 'begging the question', is hard to imagine, because E²=M²C4+P²C² can only be derived by first assuming that E=MC².

To derive E²=M²C4+P²C², we need to start with the Lorentz Mass Dilation Equation M(tot)=M(rest)/√[1−(V/C)²]. First you square it: M²(tot)=M²(rest)/[1- (V/C)²] Then multiply by [1- (V/C)²]:  M²(tot)[1- (V/C)²]=M²(rest) Multiply out the bracket:  M²(tot)−M²(tot)V²/C²=M²(rest) Add M²(tot)V²/C²:  M²(tot)=M²(rest)+M²(tot)V²/C² Multiply by C4:  M²(tot)C4=M²(rest)C4+M²(tot)V²C² Rename M²(tot)C4 as E², rename M²(rest)C4 as M²C4, rename M²(tot)V² as P² to get E²=M²C4+P²C².

Contrary to what physicists claim, E²=M²C4+P²C² cannot tell us any more about how nature works than M(tot)=M(rest)/√[1−(V/C)²]: if we rearrange it to M(tot)√[1−(V/C)²]=M(rest), we immediately see that for an object travelling at C, the rest-mass must be 0.

The idea that the momentum of a photon is given the equation P=E/C, can only be derived by assuming E=MC² to be valid, renaming MC as P to give E=PC, and then dividing by C to give P=E/C.

The Frogton Universal Force Law

It is fairly intuitive to view E=MC² as the relationship between the Kinetic Energy Equation and the low-speed approximation of the Lorentz Mass Dilation Equation; ½MV² joules = ½MV²/C² kg, thus 1 joule = 1/C² kg, 1 kg = C² joules. However KE=½MV² is itself derived from E=FD, so we can explore the C² relationship at a more fundamental level.

The idea that energy is proportional to force × distance, is the almost inescapable result of logic. Raising a 1 kg mass 1 m against a constant force of gravity on two occasions; really must be equivalent to raising 2 kg mass 1 m, or raising a 1 kg mass 2 m; thus we can write the equation E=KFD where K is a constant. Historically physicists just took K to equal 1; which they could get away with, because if you define a joule to be the energy of a force of 1 m/s/s acting over a distance of 1 m, then E=FD is true by definition, and there is no experiment anybody could ever do to contradict it. .

However once we acknowledge that energy has mass, the constant can be determined experimentally. Clearly energy = FD does not work if we measure energy in kg, because if we compress a spring by exerting a force of 1 kg m/s² over a distance of 1 m, then we will not find it has increased in mass by 1 kg.

From E=MC² we know that if we exert a force of 1 kg m/s² over a distance of 1 m, the spring will increase its mass by 1/C² kg; hence we get the equation: energy measured in kg = 1/C² FD. What this actually means, is that if a constant force imparts a momentum of 1 kg m/s each second, to each of the two objects it acts between, then it will also impart a a total of 1/C² kg to the two objects when it acts over a distance of 1 m; or put another way, if a force imparts 1 kg of mass/energy when it acts over 1 m, then it will impart C² kg m/s of momentum when it acts for 1 second; and that is the Frogton Universal Force Law.

Clearly the Frogton Universal Force Law is linked to E=MC², because they both contain the same C² relationship. However the Frogton Universal Force Law explains nature of itself and is exact in all regards; whilst E=MC² is meaningless unless you already understand the kinetic energy equation, and is based on the low-speed approximation.

To understand why the Frogton Universal Force Law accurately describes nature, we need to show that it integrates to give the Lorentz Mass Dilation Equation. To do this it is advantageous to express the Frogton Universal Force Law by saying that 'if a force produces a total of 1 kilogram of energy when it acts over a distance of 1 light-second, it must also impart a momentum of 1 kilogram C to each of the bodies it acts between in 1 second', that way force has the same numerical value in both its forms. It is also easier to define V as the speed measured as a fraction of the speed of light, rather than writing V/C each time.

If we apply a constant force between a stationary very heavy object and a stationary very light object, then the mass of the light object after it has gone a distance D will be M(rest)+FD by definition, as given by the Frogton Universal Force Law. After time T the momentum of the light object, M(total)V will be FT, again by definition from the Frogton Universal Force Law. Thus we get the equation: FT=[M(rest)+FD]V, which calling M(rest) simply M, becomes FT=[M+FD]V.

Distance is defined as speed × time, or in the case of variable speed by ʃVdT between T=0 and T=T; so we end up with the equation FT=(M+FʃVdT)V. The solution to that integral equation is V=FT/√[M²+(FT)²]. With a bit of algebra, this rearranges to give T=MV/F√(1−V²). We can then put that value for T into our original equation F=M(tot)V/T, and with a bit more algebra we get M(tot)=M/√(1−V²), which is of course the Lorentz Mass Dilation Formula, where V is the speed of the object expressed in light-seconds per second, or, in other words, as a fraction of the speed of light.

The fact that the universe abides by the principle of relativity, is partly the result of the Lorentz Mass Dilation Equation, the Lorentz Mass Dilation Equation is the result of the Frogton Universal Force Law, and the Frogton Universal Force Law is most probably the result of the fact that forces propagate at the speed of light, but that remains to be demonstrated.