Special Relativity: Is There a Problem1?

Anton Vrba
Ryde. UK

ABSTRACT: This paper reconstructs the argumentation of Einstein that resulted in the iconic equation E=mc² and then continues the thought process to show that there is no unique solution to the mechanics of photons reflecting off moving mirrors that satisfy observations from any moving reference system. This not only contradicts the principle of relativity and consequently the special theory of relativity, it also challenges the explanation given for the Doppler effect for electro-magnetic radiation.

Introduction

The pivotal 1905 paper “On the Electrodynamics of Moving Bodies” [cite source=’doi’]10.1002/andp.19053221004[/cite] defines the theory of special relativity. For the purpose of this paper we recall two results:

The first result is from § 8; it concerns the energy of light ¹ $L$ , as measured in a stationary system, and its transformation to $L'$ when measured in a moving system

(1) $\begin{equation*} \frac{{L'}}{L} = \frac{{1 - \cos \phi \cdot v/c}}{{\sqrt {1 - {v^2}/{c^2}} }}, \end{equation*}$

Einstein remarked “It is remarkable that the energy and the frequency of a light complex vary with the state of motion of the observer in accordance with the same law.” as in the previous section, § 7, he derived the same ratio for the frequency of light observed in the relative moving reference systems. Indeed, from a contemporary point of view, Einstein demonstrated that the Planck energy-frequency equivalence $E = hf$ transforms in accord with the special theory of relativity.

The second result, from § 10, is that the kinetic energy of an electron, or any ponderable mass is expressed as

(2) $\begin{equation*} W = m{c^2}\left( {\frac{1}{{\sqrt {1 - {v^2}/{c^2}} }} - 1} \right). \end{equation*}$

The iconic equation $E = m{c^2}$ , defining the mass-energy equivalence, is derived in the fourth annus mirabilis paper of 1905 “Does the inertia of a body depend upon its energy content?” [cite source=’doi’]10.1002/andp.19053231314[/cite]

This paper begins with the premise that a body in a stationary system has energy ${E_0}$ and in a relative moving system has an energy ${H_0}$ . This body emits simultaneously two opposing light waves, each having energy $\tfrac12 L$ as measured in the stationary system. After the emission of light the body has energy ${E_1}$ and ${H_1}$ in the respective reference systems. According to Einstein the following holds true:

(3) $\begin{equation*} {E_0} = {E_1} + \frac{1}{2}L + \frac{1}{2}L, \end{equation*}$

(4) $\begin{align*} {H_0} &= {H_1} + \frac{1}{2}L\frac{{1 - \frac{v}{c}cos\phi }}{{\sqrt {1 - {v^2}/{c^2}} }} + \frac{1}{2}{\rm{L}}\frac{{1 + \frac{v}{c}cos\phi }}{{\sqrt {1 - {v^2}/{c^2}} }}\\ \nonumber \\ &= {H_1} + \frac{L}{{\sqrt {1 - {v^2}/{c^2}} }}. \end{align*}$

Einstein asserts that the difference in ${E_0}-H_0$ and ${E_1-H_1}$ reduces to the difference in kinetic energy ${K_0}$ and ${K_1}$ of the body as measured in the two reference systems. Therefore

(5) $\begin{equation*} {K_0} - {K_1} = L \left\{ \frac{1}{\sqrt {1 - {v^2}/{c^2}} } - 1 \right \} \end{equation*}$

from which, by comparing (2) and (5), it follows trivially that $E = m{c^2}$ The paper [cite source=’doi’]10.1002/andp.19053231314[/cite] concludes “If the theory corresponds to the facts, radiation conveys inertia between the emitting and absorbing bodies.”

At this point we recap and establish that Einstein states that an observed photon in the stationary reference system which has an energy $L$ , and when the same photon is observed from a moving reference system its energy increase or decrease depending on the relative direction motion of the photon as expressed in (1) and used in (4).

We also conclude that the total energy of an observation, that is the sum of the mass energy, kinetic energy and photon energy, transforms according to (2) when observed from a different moving reference system.

A further conclusion is that the change in frequency of a photon in different reference systems, is the principle behind the relativistic Doppler shifts and when applied twice as in a reflection, i.e the operating principle of a police radar, reduces to the classical Doppler shift

(6) $\begin{equation*} \frac{{{f_r}}}{{{f_t}}} = \frac{{c - v}}{{c + v}} \end{equation*}$

for a receding target.

Thought experiment: Reflections from moving mirrors

Using all of above and applying the results obtained, we conduct following thought experiment:

Steve, in the stationary reference frame, observes the two-photon decay of a particle of mass $m$ . The Planck relation $E = hf$ determines the frequencies of the photons that he observes, i.e.

${f_{S1}} = {f_{S2}} = {f_0} = \frac{{\delta m{c^2}}}{{2h}},$

where $h$ is the Planck constant, $\delta m$ is the particle’s loss of mass that is converted to photonic energy, and $c$ is the speed of light. Let the direction of the photons oppose each other and be parallel to the $x$ -axis of Steve’s reference frame.

The total energy before and after the two-photon decay remains constant at

(7) $\begin{equation*} E = m{c^2}. \end{equation*}$

Monica is in a moving reference frame with constant velocity $v$ parallel to Steve’s $x$ -axis. According to § 7 of [cite source=’doi’]10.1002/andp.19053221004[/cite] first equation page 912, she observes the frequencies of the two photons as

${f_{M1}} = {f_0}\sqrt {\frac{{c - v}}{{c + v}}} {\qquad\rm{and}\qquad}{f_{M2}} = {f_0}\sqrt {\frac{{c + v}}{{c - v}}} .$

The sum of the energy of the two photons is

$({f_{M1}} + {f_{M2}})h = \frac{{\delta m{c^2}}}{{\sqrt {1 - {v^2}/{c^2}} }}.$

Thus in the moving reference system, as asserted in [cite source=’doi’]10.1002/andp.19053231314[/cite] the total energy before and after the two-photon decay also remains constant at

(8) $\begin{equation*} H = \frac{{m{c^2}}}{{\sqrt {1 - {v^2}/{c^2}} }}. %(2.2) \end{equation*}$

At this point, we note that we have not deviated from the methods pioneered by Einstein in [cite source=’doi’]10.1002/andp.19053231314[/cite] that resulted in $E = m{c^2}$ . We now continue the thought process using the same methods, and introduce a reflection: Monica now uses two ideal mirrors ² to reflect the photons towards their source. She does not observe a change in energy in the photons and the mirrors remain stationary for her. In the moving system, the photon frequencies before and after the reflection are unchanged. The energy remains

(9) $\begin{equation*} H' = \frac{{m{c^2}}}{{\sqrt {1 - {v^2}/{c^2}} }}. \end{equation*}$

On the other hand, again by use of § 7 of [cite source=’doi’]10.1002/andp.19053221004[/cite], Steve observes the frequencies of the returned photons as

${f'_{S1}} = {f_0}{\mkern 1mu} \frac{{(c - v)}}{{(c + v)}}\qquad{\rm{ and}}\qquad {f'_{S2}} = {f_0}{\mkern 1mu} \frac{{(c + v)}}{{(c - v)}}.$

Thus, in the stationary system, the energy after reflection calculates to

(10) $\begin{align*} E' &= (m - \delta m)c^2 +h f_0 \nonumber \left(\frac{c - v}{c+v}+\frac{c+v}{c - v}\right)\\ &= mc^2 + \frac{{2\delta m{v^2}}}{{(1 - {v^2}/{c^2})}} \end{align*}$

In Equations (9) and (10) only the energy was addressed, a similar calculation for the momentum shows that following change in momentum still needs to be accounted for in the moving system

(11) $\begin{equation*} \Delta \varrho' = 2\frac{{{f_0}h}}{c}\left( {\sqrt {\frac{{c + v}}{{c - v}}} - \sqrt {\frac{{c + v}}{{c - v}}} } \right) = \frac{{2\delta mv}}{{\sqrt {1 - {v^2}/{c^2}} }}, \end{equation*}$

and in the stationary system the momentum of the photons changed by

(12) $\begin{equation*} \Delta \rho' = 2\frac{{{f_0}h}}{c}\left( {\frac{{c + v}}{{c - v}} - \frac{{c - v}}{{c + v}}} \right) = \frac{{4\delta mv}}{{1 - {v^2}/{c^2}}} \end{equation*}$

Conclusion

In deriving (7) and (8), we have followed the same method that Einstein pioneered in [cite source=’doi’]10.1002/andp.19053231314[/cite] and we continued to use his method to derive (9) and (10).

The principle of relativity, as asserted in axiom 1, § 2 of [cite source=’doi’]10.1002/andp.19053221004[/cite] (page 895), quote “The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of co-ordinates in uniform translatory motion.” implies that the system state, before and after an interaction, changes the same for both Steve and Monica.

However, this affirmation is contradicted by the result
(8) $=$ (9) and (7) $\neq$ (10).

The principle of relativity requires that:
IF (8) $=$ (9) THEN (7) $=$ (10),
which clearly is not the case.

Also, the discrepancies in momentum calculated in the stationary system (12) has no logical meaning when compared to the moving system (11).

The mathematicians call this reductio ad absurdum (Latin: “reduction to absurdity”), or argumentum ad absurdum (Latin: argument to absurdity), which is a common form of argument which seeks to demonstrate that a statement is false by showing that a false, untenable, or absurd result follows from its acceptance. (Or, to demonstrate that a statement is true by showing that a false, untenable, or absurd result follows from its denial.)

Experience has corroborated the energy mass equivalence, inertial mass dilation, clock rates dependent on velocity, etc. Has Einstein obtained the right answers for the wrong reasons? For physics to progress explanations are needed that withstand mathematical rigour and corroboration by experience—that is all there is to it.

Additionally, results (7) and (10) violate the law of energy conservation, thus challenging the explanation given for the Doppler effect for electro-magnetic radiation.

Discussion

Nobody, so far, has ventured an answer to my argumentation above. Most academics ignore my approaches, some have agreed with the mathematics and reasoning presented, but kept it open by stating that “there are many subtleties in relativity” and one has to think about it. Nobody has said “you are correct” or “you are wrong” to me. Frankly, what should they answer? Either way is a problem; it is hard to admit that the special theory of relativity may be on unsound footings, nor can they admit that observers from two different reference systems have different outcomes for an experiment that contradicts the most fundamental law of energy preservation.

I will keep this in the category “Demonstrably Nonsense” until someone can make an argument demonstrating that I err in my logic.

Note that Einstein used the symbol E in [1]. This paper uses the symbol L representing the energy of the light or photon, for reason to have a compatible syntax throughout as Einstein switched to the symbol L in the second referred paper [2].
An ideal mirror is characterized that there is no loss in reflecting a photon, its mass approaches infinity thus its kinetic energy is unchanged before and after a reflection, thus the act of reflection does not change the energy of the system.

Betelgeusean Physics

Introduction

Thought experiment: Reflections from moving mirrors

Conclusion

Discussion

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