Einstein's Unphysical? Derivation of the Lorentz Transformation

Einstein's special theory of relativity SR boils down to the Lorentz transformation connecting the space-time coordinates $(x,t)$ and $(x^\prime ,t^\prime )$ in two (1d space) inertial systems with parallel space-axes moving with constant velocity $v$ with respect to each other:

• $x^\prime =\gamma (x - vt)$, $t^\prime =\gamma (t - vx)$,
• $x =\gamma (x^\prime + vt^\prime )$, $t =\gamma (t^\prime + vx^\prime )$,

where $\gamma = \frac{1}{\sqrt{1-v^2}}$ assuming the speed of light is 1 and $\vert v\vert \lt 1$. See that the origin $x^\prime =0$ of the $x^\prime$-axis follows the trajectory $x=vt$ in the $(x,t)$-system, and vice versa.

Let us now analyse Einstein's derivation of the Lorentz transformation. Recall that Lorentz, who presented his transformation before Einstein picked it up, did not view the two coordinate systems, the unprimed and the primed systems, to have the same physical significance. If the unprimed coordinate system had a physical meaning, the primed did not according to Lorentz concern real physics of space and time, only some form of fictitious or apparent ”local” space and time. It was Einstein who took the brave step to give both systems the same physical significance as an expression of full symmetry between the systems or complete relativity with no system more physical than another,  and so Einstein gave birth to SR as a fundamental pillar of modern physics.

Einstein's special ability was to derive far-reaching conclusions about concrete physics from some general assumption concerning mathematical form. Einstein thus formulated the following Postulates of SR:

• Laws of physics take the same form in all inertial systems. (Relativity)
• The speed of light (normalised to 1) is the same in all inertial systems. (Constant Light Speed)

We see that the Postulate of Relativity is an assumption about mathematical form and not about any concrete physics. Despite the very limited physics input in the form of Constant Speed of Light, Einstein was able to uncover deep truths about the very nature of space and time, including space contraction and time dilation,  as most surprising consequences of the simplest possible form of motion as translation with constant velocity. Amazing, but was it too good to be true? Let's see.

Note the important difference between the view of Lorentz (see PS below), as the inventor of the Lorentz transformation, with the primed coordinates expressing apparent physics with space contraction and time dilation as fiction, and the view of Einstein with the primed coordinates as real physics with space contraction and time dilation as real physics.

The difference is expressed  in the twin paradox with different rates of ageing only apparent according to Lorentz but most real according to Einstein. The question is then: Lorentz or Einstein?  

To seek an answer let us recall Einstein's derivation of the Lorentz transformation:

1. Consider two (1d) inertial systems $(x,t)$ and $(x^\prime ,t^\prime )$ with the $x^\prime$-axis sliding on top of the $x$-axis with velocity $v$ with its origin $x^\prime =0$ at position $x=vt$ at time $t$ thus establishing a connection between the systems.

2. Consider a light signal $L$ emitted from a stationary source at $x=0$ on the $x$-axis at $t=0$ viewing the emission to be an event with coordinates $(0,0)$ in the $(x,t)$-system. Conclude that $L$ follows the trajectory $x=t$ in the $(x,t)$-system, because the speed of light is assumed to be equal to 1 in the $(x,t)$ system.

3. Consider similarly another light signal $L^\prime$ emitted from a stationary source at $x^\prime =0$ on the $x^\prime$-axis at $t^\prime =0$ viewing the emission to be an event with coordinates $(0,0)$ in the $(x^\prime ,t^\prime )$-system. Conclude that $L^\prime$ follows the trajectory $x^\prime=t^\prime$ in the $(x^\prime ,t^\prime )$-system, because the speed of light is assumed to be equal to 1 in the $(x^\prime ,t^\prime )$ system. So far we have two light signals $L$ and $L^\prime$ in two systems moving with respect to each other.

4. Now take the step to identify $L$ with $L^\prime$ on the ground that their emission event coordinates $(0,0)$ agree, and conclude that $(x,t)$ and $(x^\prime ,t^\prime )$ describe the trajectories of one and the same light signal, so that $x=t$ if and only if $x^\prime =t^\prime$.

5. Make the Ansatz $x^\prime =\gamma (x-vt)$ with $\gamma$ a positive constant recalling that $x^\prime =0$ follows the trajectory $x=vt$. Conclude by identifying $x=t$ with $x^\prime = t^\prime$ according to 4. that $t^\prime =\gamma (t-vx)$ and $\gamma = \frac{1}{\sqrt{1-v^2}}$, which gives the Lorentz transformation. Recall that new strange effects in the form of space contraction and time dilation are consequences of the Lorentz transformation.

Let us now take a closer look at Einstein's argument.

The crucial step is the identification in step 4. Before identification there are two independent light signals, one emitted in the unprimed system and another emitted in the primed system from stationary sources at the origins, two light signals without any connection. But after the identification the primed signal can be viewed according to 1. to have  presence in the unprimed system as a signal emitted by a source at $(x,t)=(0,0)$ moving with velocity $v$. There are thus in total four light signals, two in each system with one emitted from a stationary source and the other from a moving source, all identified to be one and the same light signal.  The key question is now if from physical point of view it is correct to identify:

• (i) two signals in the two systems on the ground that their emission events have the same coordinates? 
• (ii) two signals in the unprimed system, one from a stationary source and the other from a moving source, and vice versa for the primed system?  

Concerning (i), one may ask from where the need arises to introduce two systems and then seek a connection between the two? For the description of propagation of light signals, it is enough to consider just one spatial coordinate system with the source fixed at the origin. But this was not enough for Einstein, who wanted to uncover deep secrets of space and time by comparing descriptions in different systems of what Einstein considered to be one and the same light signal.

Concerning (ii) we ask if a stationary source emits the same light signal as a moving source? The answer is no, since the frequency changes according to the Doppler effect, so the signals cannot be the same. Moreover, since a light signal is extended in space, the initialisation of the primed signal as viewed in the unprimed system is different from that of the unprimed signal, because the Lorentz transformation mixes space into time so that initialisation at $t=0$ and $t^\prime =0$ do not have the same spatial form.

We conclude that the two signals in the unprimed/primed system cannot be identified if frequency and initial wave form counts, and without this identification the two signals in the two systems cannot be made either and so the connection by the Lorentz transformation cannot be made.

Summary: Einstein's derivation of the Lorentz transformation between the coordinates in two inertial systems is based on an identification of two different signals, which can be questioned on physical grounds. Einstein's conclusion that the Lorentz transformation connects coordinates in two systems with the same real physical significance, thus can be questioned. For Lorentz the question does not have the same weight, since if only appearance is sought, the argument can be weaker. 

So what do you think: Is the identification anyway correct in some sense, if not concerning frequency and initial wave form, so that anyway the Lorentz transformation gives correct information about deep stunning secrets of real space and time, and not only appearances?

Or do you say that there is no reason to spend effort on a question like this, since all professional physicists know that the Lorentz transformation correctly describes real physics and thus does not require any justification a la Einstein or anyone?

If the view of Lorentz is the one that is correct, then much of modern physics qualifies as fiction.
In order for Einstein to be correct, a physically correct derivation of the Lorentz transformation is needed. Is there any?

PS1 Lorentz expressed his view on his transformation:

• ...a transformation of the time was necessary, so I introduced the conception of local time which is different for different frames of reference which are in motion relative to each other. But I never thought that this had anything to do with real time. This real time for me was still represented by the older classical notion of an absolute time, which is independent of any reference to special frames of coordinates. There existed for me only one true time. I considered my time transformation only as a heuristic working hypothesis, so the theory of relativity is really solely Einstein's work.

We read that Lorentz takes his hands off from Einstein's relativity theory based on misunderstanding Lorentz local time to be real time. Lorentz or Einstein?

PS2 From physical point of view with the $x^\prime$-axis gliding on top of the $x$-axis with constant velocity $v$, the only possible connection between the coordinates is that of a Galilean transformation:

• $x^\prime = x- vt$. 

It is unthinkable that there mere translation with constant velocity of  the $x^\prime$-axis on top of an $x$-axis, can change the scale of the $x^\prime$-axis vs that of the $x$-axis in any real sense. Unthinkable.

PS3 Recall that the Postulates of SR can as well serve as the postulates of Many-Minds Relativity MMR in which there is no need of the Lorentz transformation, and all the mysteries of SR resulting from Einstein's misinterpretation of the Lorentz transformation simply evaporate.

PS4 Recall that Einstein in his 1905 article introducing his special theory of relativity started by admitting that his Postulates appear to be contradictory:

• ...the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (the purport of which will hereafter be called the “Principle of Relativity”) to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body.

It was not a good start..

PS5 The Constant Light Speed Postulate was in Einstein's 1905 article formulated as follows:

• The speed of light in empty space is always 1 independent of the state of motion of the emitting body.

Einstein viewed this postulate to be in apparent contradiction to the Postulate of Relativity.  Why? Because the  notion of "empty space" and the qualification "state of motion of the emitting body" appear to give "empty space" the role of a "stationary aether" to which a "state of motion" can be related, which would be in conflict with the Postulate of Relativity interpreted as stating non-existence of any preferred (inertial) system.

On the contrary, the Constant Light Speed Postulate can be viewed to be a consequence of the Relativity Postulate, and as such carrying no contradiction.

PS6 Einstein (but not Lorentz) got hooked up by the idea of a necessity to describe propagation of a light signal in different inertial systems moving with respect to each other. But why not be content with one description, a description where the source or the receiver/observer is stationary? This is the set-up in MMR, where each observer is stationary in his/her chosen inertial system, while the source may be moving. If there are more than one observer, then the question is to what extent different observers will agree, which is analysed in MMR. There is then no need to ask about a description from the point of view of a moving observer required to be in full harmony with that of a stationary observer, as the (unreachable) objective of SR. There is then no need to go into SR with all its contradictions, and modern physics can be liberated to focus on realities instead of shadows.

PS7 SR can be described as follows: 

• All inertial systems are equally valid and there is a common description (of light signals) mediated by the Lorentz transformation.

MMR can be described as follows:

• All inertial systems are equally valid and there is no common description. 

MMR naturally connects to an idea that "all is relative", but not so for the idea in SR of a common description (of a light signal). MMR is like a many-gods religion (polytheism) without common god in harmony with full relativity, while SR is like a common-god religion (monotheism) in contradiction to full relativity.