This is a re-post of a old one of mine, from this thread originated by Sylvia Else, two years ago:

Why E=mc² is wrong
https://groups.google.com/g/sci.physics.relativity/c/bBhcSfznBB0/m/CsOncDEKAgAJ

In this post I tried to show that the 3 pages paper, which uses elementary math, has false conclusions, only
is valid for low values of v/c (an approximation of γ on its first two terms of its series expansion), is full of fallacies
and circular reasoning, and Einstein invested his next 40 years (six attempts) to make it right without success.

Yet, the formula E=mc² has been adopted by consensus on its application for energy at rest, in the same way that
c = 299792458 m/s has conveniently FIXED as a standard value (in vacuum). Eo=mo.c² brought a convenient conversion
between mass and energy (at rest), making the life of scientists happier (physicists, chemists, bio and molecular sciences, etc.)

Now the post where I tried to dissect the three pages, analyzing the problems with math, relativity and fallacies. I edited it
a bit to facilitate comprehension, but I didn't altered any fundamental content:

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I'm not discussing the validity of Einstein's paper on SR.

I'm specifically discussing that his derivation for E=mc² is FALSE and based on fallacies, as his next FIVE PUBLIC ATTEMPTS to correct the paper in the next 40 years were a failure.

There is NO THEORETICAL PROOF that support the validity of E=mc² as DEVELOPED by Einstein, and no other attempt other than his failed ones to prove it.

E=mc² was just ACCEPTED by the physics community, because it's the base of further developments at physics (de Broglie, Schrodinger, Dirac, Feynnman, etc.). It didn't play any rol in the development of nuclear physics, but it proved to be very convenient for the branch of particle's physics.

Here follows MY PROVE that 1905 Einstein's derivation is FALSE. It's very detailed and require to have a copy of the english translation at hand, to perform fact-cheks.

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I addressed the topic around E=mc² in several threads, particularly on this one:

“So, we are clear now? Einstein never proved E=mc² is due to loss of mass.”

https://groups.google.com/forum/#!topic/sci.physics.relativity/eoxLarQJFyg%5B76-100%5D

In that thread, months ago, I analyzed step by step how the Sept. 1905 paper fail to prove the link between the ALLEGED loss of mass in the object located at the origin with the transformation of such a small loss (L/c²) into electromagnetic energy L (visible light in this case).

The paper “DOES THE INERTIA OF A BODY DEPEND UPON ITS ENERGY-CONTENT?” (By A. EINSTEIN. September 27, 1905) is very simple and also very short (2 ½ pages), with initial references to his prior paper (June 30, 1905) : “ON THE ELECTRODYNAMICS OF MOVING BODIES”.

I’ll prove here, AGAIN, that the approximation developed by Einstein to obtain (L/c²) is FALLACIOUS and mathematically INVALID, and that the extension of this approximation to a wide range of velocities “v”, lower than “c” makes the extrapolation of such result to the entire mass of the body located at the origin is FALSE, and so is any further attempt to use this failed attempt as a general expression E=mc², with disregard of the range of the velocity “v”.

The paper was almost immediately criticized by Planck (1906) as a wrong approximation, based on a circular reference introduced almost at the beginning (petition principii fallacy), because it introduced early on what was being to be proved. Even this interpretation by Planck is incorrect, because even when the main fallacy is there, hidden at plain sight, there IS NOT any physical connection between the energy L of the pair of light’s beams and ANY PROVEN DECREASE in the mass Mo of the object, as narrated by Einstein.

Einstein tried, publicly, six more times in the next 40 years, failing each time, until he dropped the subject by 1945. By then, the equation E=mc² was being pushed very far into the body of knowledge of relativistic physics, and ACCEPTED without discussions even for velocities “v” almost close to “c”.

It was late, by then, to reverse the tide of indoctrination that permeated physics at every statement. That such aberration could be published at Annalen der Physik required complicity, particularly from Wien (Chief Editor) and Planck (Chief Consultant on Theoretical Physics, representing the German Association of Physicists).

--- ANALYSIS OF THE 1905 PAPER (it requires having a copy at hand) -----

FIRST, I QUOTE EINSTEIN’S PARAGRAPHS:

“Let there be a stationary body in the system (x, y, z), and let its energy—referred to the system (x, y, z) be E0. Let the energy of the body relative to the system (x’,y’,z’) moving as above with the velocity v, be H0.

Let this body send out, in a direction making an angle φwith the axis of x, plane waves of light, of energy ½ L measured relatively to (x, y, z), and simultaneously an equal quantity of light in the opposite direction. Meanwhile the body remains at rest with respect to the system (x, y, z)..

The principle of energy must apply to this process, and in fact (by the principle of relativity) with respect to both systems of co-ordinates. If we call the energy of the body after the emission of light E1 or H1 respectively, measured relatively to the system (x, y, z) or (x’,y’,z’) respectively, then by employing the relation given above we obtain…………..”

I STOP HERE THE LITERAL COPY&PASTE (except for the use of (x’,y’,z’)).

I introduce some simplifications (not violating any premises from the quoted paragraphs:

1) As it’s valid for any angle φ, I select φ=0, which makes cos φ=1. Now the beams of light are parallel to the horizontal axis. I LET IT GO that Einstein wrote, before the quoted text:

“Let a system of plane waves of light, referred to the system of co-ordinates (x, y, z), possess the energy l”.

This is utterly incorrect, as it would require that the semi-spherical beams of light be measured very far away from the source. But this is a minor license that can be tolerated, given the gross developments that follow.

2) I introduce the HIDDEN MASS Mo of the stationary object that resides at the system (x, y, z), which has been mentioned only once at the beginning of the paper, but without giving it an expression Mo.

3) Einstein introduces the terms E and H to express the ENERGY of the body (Mo, even when he purposely omit it), being that E is the energy of Mo observed from the RESTING frame (x, y, z), and H as the energy of Mo observed from the MOVING frame (x’,y’,z’), in such a way that:

E0:Energy of the body with mass Mo BEFORE the emission of light, from (x,y,z).
E1: Energy of the body Mo AFTER the emission of light, from (x,y,z).
H0: Energy of the body Mo BEFORE the emission of light, from (x',y’,z’).
H1: Energy of the body Mo AFTER the emission of light, measured from (x’,y’,z’).

4) I’ll be using the letter γ to represent the Gamma Factor.

Now, I write what Einstein expressed after the quote ending in “we obtain..” as:

Eo = E1 + ½ L + ½ L
Ho = H1 + ½ L.Y + ½ L. γ
= H1 + γ . L

By subtraction we obtain from these equations

Ho – Eo - (H1 - E1) = L ( γ – 1)

The equations from above are LITERAL equations written by Einstein (using φ = 0 and γ as Gamma).

The FALLACY (circular reference criticized by Planck) is in the very first equation. Einstein INTRODUCES his “petitio principii” fallacy by STATING that the body Mo already has the energy embedded into its mass, as it results if I read the equation as E1 = Eo – L.

So, in this in dishonest, deceptive way, he introduces what he wants to prove: that L was WITHIN Mo before the light was turned on.

Einstein continues with his manipulation of the difference of energy between TWO DIFFERENT SYSTEMS, as if they formed a CLOSED system, which is not because light beams travel toward infinity. It’s a very different approach than that of 1904 Hasenhorl, where he used a CLOSED SYSTEM (a cavity with planckian caps).

Then, he subtract the energy from two different systems, one in motion, and express:

Ho – Eo = Ko + C
H1 – E1 = K1 + C

and assert: “since C does not change during the emission of light. So we have”

Ko – K1 = L ( γ – 1)

Now, decomposing Y into a series and discarding high power terms, by asserting that v << c, he gets:

Ko – K1 = ½ (L/c²) . v²

Quoting:

“From this equation it directly follows that:—
If a body gives off the energy L in the form of radiation, its MASS DIMINISHES by L/c². The fact that the energy withdrawn from the body becomes energy of radiation evidently makes no difference, so that we are led to the more general conclusion that

The mass of a body IS A MEASURE OF ITS ENERGY-CONTENT; if the energy changes by L, the MASS CHANGES in the same sense by L/9×10^20, the energy being measured in ergs, and the MASS in grammes.”

And this is the conclusion of his paper, which is FALSE by several reasons:

1) FALSE: He introduces a value Eo for energy at (x,y,z) before the light is turned ON. This is FALSE, as the body with mass Mo is at REST on his reference frame, so Eo = 0.

2) FALSE: He introduces (his main fallacy) what he wants to prove: That Mo = Mo’+ dM, where dM will be transformed into energy L when the light is turned ON. He, in a slicky way, never mention Mo, and writes E1 = Eo + L, with an innocent attitude at the beginning. The correct expression is E1 = L, as Eo = 0, which is FALSE, as he doesn’t explain from where L is provided.

3) He, conveniently, suppressed any reference to Mo and the value of energy Ho. It happens that Ho = γ . (½ Mo. V²), which is the kinetic energy perceived from the moving frame, when observing the behavior of Mo in the resting frame. So,

Ho – Eo = Ko + C = γ . (½ Mo. V²) + C

4) FALLACY: Now he introduced the perceived energy H1 as having a decrease of “L” respect to Ho, so he can conveniently assert that

H1 – E1 = K1 + C = γ . (½ Mo. V² - L) - L + C

which results in his final value of difference of energies:
Ko – K1 = L. ( γ – 1), which is approximated to Ko-K1= ½ (L/c²).v²

And this is FALSE and mathematically wrong (criticized by Planck).

CONCLUSIONS:

1) Einstein NEVER proved that dM=L/c² is a decrease of mass in the value Mo from the resting object.

2) His extrapolation E=mc², even when based on very low velocities “v” is FALSE.

3) So, the general and widespread use of E=mc² is UNDUE as the equation is FALSE.

4) NEW: The adoption of 2 first values of the expansion of γ in series (alleging low v/c ratios):

γ = 1 +½ (v/c)^2 + 3/8 (v/c)^4 + 15/48 (v/c)^6 + ......

is increasingly wrong for values of (v/c) > 0.1 as the influence of powers higher than 2 can't be discarded.

Then, the extraction of the partial value 1 +½ (v/c)² to be used in Ko – K1 = L ( γ – 1) ≈ ½ (L/c²) . v²
and the unproven assertion that L/c² is the mass lost by the emission of light with energy L is a double fallacy,
as it's not proven mathematically, is artificially extracted from a higher level formula, and it's only a casual coincidence
that the result resembles the expression for kinetic energy.

It doesn't matter if further attempts to get a better value for any range of γ, as in E =mc² . (γ - 1), being m the mass
value at rest. The extension of dM=L/c² to define m at rest value as an INFINITE CHUNKS of dM decomposition is
UNTRUE, because if it represent a bit of matter at rest, a value of v = 0 should have be used.

Not a bit of matter "at rest" in the interval v ≤ 30,000 Km/sec AND (mandatory) v > 0 Km/sec.

E =mc² has been settled as an absolute and unquestionable truth (valid anywhere in the universe) in the same way
that c = c = 299792458 m/s has been settled at any Bureau of Standards.

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