Resolution of All the Paradoxes of Special Relativity

• Length contraction and time dilation are ways of regarding things and do not correspond to physical reality. (Max Born)
• There is not a single concept of which I am convinced that it will stand firm, and I feel uncertain whether I am in general on the right track. (Einstein)

The special theory of relativity SR presented by Einstein in one of his five articles from the "miraculous year" 1905, is viewed as a pillar of modern physics introducing new mysterious qualities of "strangeness" in the form of "space contraction" and "time dilation" asking for a fundamental revision of classical Newtonian concepts of space and time, by Einstein excused with: "Newton, forgive me!"

SR emerged from a realisation that there is no "unique aether" as an "absolute space" for the propagation of light. Instead, it appeared that there were "many aethers" as different space coordinate system for the expression of Maxwell's equations describing propagation of electromagnetic waves including light, which could be fixed to different observers/receivers of light moving with respect to each other.      

But SR is filled with paradoxes or contradictions: Ladder Paradox, Bug-Rivet Paradox, Cooling Paradox, Twin Paradox, Trouton-Noble Paradox,  Ehrenfest Paradox, Detonator Paradox,  Falling Rod Paradox,  Bar/Ring Paradox,...and many more. The most famous paradox is the Twin Paradox about two twins both ageing slower than the other according to SR, which has been subject to heated debate over more than 100 years without any real resolution, with only a proclaimed "consensus" that the paradox is just an "apparent paradox" resolved since long, since in fact only one of the twins is ageing slower than the other, then in contradiction to SR as an expression of an inevitable most real "strangeness" of modern physics.

In a series of posts and the book Many-Minds Relativity MMR I have analysed SR to find the true root of the paradoxes, because it is not easy to find in the literature.

Let me now present the result of the analysis in easily understandable concise form, which you can go through in a couple of minutes. That is, if you want to understand that the "strangeness" of modern physics based on SR is the result of a logical mistake by Einstein, which when removed allows rational progress without "strangeness".  If you are happy with the consensus of "strangeness" as an ultimate expression of the victory of modern physics, you can stop reading.

The basic question to be answered is thus: From where do all the paradoxes of SR come?

SR concerns observations of different observers $X$ using different space-time coordinate systems $(x,t)$ with space axes moving with constant velocity with respect to each other, with $x$ a Euclidean space coordinate, here for simplicity 1d, and $t$ a time coordinate, which are referred to as inertial systems. 

Einstein assumes all observers in SR to use identical  standard "rigid rods" to measure distance in space and identical standard clocks to measure progress of time. An observer $X$ equipped with an $(x,t)$-system thus scales the $x$-axis using a standard rigid rod and a standard clock to scale the $t$-axis. So far no paradox.

Einstein formulated the following basic Postulate of SR:

• All observers $X$ observe light signals to travel (in vacuum) with speed 1 (normalised) along the observer's $x$-axis independent of the movement of the source of light on the $x$-axis.

Or in the exact wording of the 1905 article:

• Light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body.

We see that the Postulate of SR allows a light source to be moving. But it says nothing about an observers motion, which can only be interpreted as implicitly requiring the observer to be at rest in the observer's system when receiving/observing the arrival of a light signal sent from some possibly moving source.

More precisely, allowing the observer to move in the observer's system would contradict the basic set-up of equipping each observer $X$ with an individual $(x,t)$-system. If observation by moving observers was included in the Postulate of SR, then it would be enough with just one inertial system, which could then act as an "absolute space" in contradiction to the basic motivation to develop SR as physics beyond Newtonian physics supposed to require some "absolute space" (cf. PS4 below).

An observer $X$ at rest in the observer's $(x,t)$-system can thus observe the propagation of light by receiving/observing light signals at different space locations emitted by possibly moving light sources at other locations on the $x$-axis.

The Postulate of SR takes the more precise form:

• All observers $X$ at rest on the observer's $x$-axis observe light signals to travel with speed 1 along the $x$-axis independent of the movement of the source of light on the $x$-axis. 

In particular, an observer $X$ will in the observer's $(x,t)$ system observe a light signal emitted at $x=0$ at time $t=0$ to follow the trajectory $x=t$. Another observer $X^\prime$ will in the observer's $(x^\prime ,t^\prime )$ system observe a light signal emitted at $x^\prime =0$ at time $t^\prime =0$ to follow the trajectory $x^\prime =t^\prime $. In the Postulate of SR there is no connection between signals in different systems. Even though the coordinates of emission $(x,t)=(0,0)$ and $(x^\prime ,t^\prime )=(0,0)$ are the same, the light signals will be different, one traveling in the $(x,t)$-system and another in the $(x^\prime ,t^\prime )$-system.

Note that the Postulate of SR is compatible with the null result of the Michelson-Morley experiment by allowing light signals to travel the same way independent of the motion/orientation of the arms.

Analogy can be made with elastic waves traveling in two identical parallel elastic beams moving with respect to each other, where it is evident that the waves are not the same even if launched the same way when the beams overlap. Each beam then represents a space axis, with the beams/axes moving with respect to each other. So far no paradox.

In Einstein's illustration with a stationary observer $X$ on a train embankment and another observer $X^\prime$ traveling inside a moving train with windows covered, the Postulate of SR does not admit $X$ on the embankment to observe light signals inside the train, nor $X^\prime$ inside the train to observe light signals in the system of the embankment outside the train. This is a key point!

Connecting to Maxwell's equations the Postulate of SR can be expressed in the form:

• All observers $X$ assume light propagates according to Maxwell's equations with speed of light normalised to 1 in a $(x,t)$-system in which $X$ is at rest. If the light source is moving on the $x$-axis with velocity $-1\lt v\lt 1$, then the frequency of received light signals is subject to a Doppler shift of $\frac{1}{1+v}$. 

This is also the basic Postulate of MMR. The "rigid rod" as measure of distance can then be replaced with the 1983 SI standard of defining meter as a certain fraction of light-second as the distance of light traveled in 1 second. So far no paradox.

Now we come the critical point concerning the origin of all the paradoxes of SR. This is the very  starting point for Einstein in his pursuit of SR considering two observers $X$ and $X^\prime$ using $(x,t)$ and $(x^\prime ,t^\prime )$-systems with space axes moving with respect to each other, in the form of the following question:

• What is the observation by $X^\prime$ of a light signal in the $(x,t)$ system? 

Einstein was quick in his 1905 article to point out that there was something contradictory in his preparation of SR, but then reassured himself that it was only an apparent contradiction ("another postulate, which is only apparently irreconcilable with the former").

In any case the question is really contradictory, because observations are assumed to be made by observers at rest on their respective space-axis, and so $X^\prime$ cannot observe light signals on the $x$-axis because $X^\prime$ is at rest on the $x^\prime$-axis, but not on the $x$-axis.

Einstein thus poses a question which is contradictory and thus can only have a contradictory answer, but Einstein nevertheless seeks to give an answer, in a heroic attempt to reveal new strange physics of space and time.

In Einstein's train example the question reflects breaking the Postulate of SR by asking $X$ on the embankment to observe light signals in the inside system of the train and $X^\prime$ inside the train to observe light signals in the embankment system outside the train, both involving moving receivers/observers. This is the origin of all the paradoxes.

To get around the contradiction of the Postulate of SR asking observation of real light signals by moving observers,  Einstein engaged in his famous "thought experiments" to replace real experiments, thus allowing $X$ on the embankment to peek into the train and there observe not real but imagined light signals in the system inside the train, or $X^\prime$ inside the train to peek through closed windows to observe light signals in the system of the embankment. Of course a train driver is able to observe a light signal emitted from a source on the embankment, but then at rest on a space axis on which the the light source is moving.

Einstein's answer to the contradictory question is the Lorentz transformation supposedly opening $X^\prime$ to observations in the $(x,t)$-system mediated by the coordinate transformation:

• $x^\prime =\gamma (x-vt)$, $t^\prime = \gamma (t-vx)$, $\gamma =\frac{1}{\sqrt{1-v^2}}$,

supposing that the $x^\prime$-axis of $X^\prime$ moves with velocity $0\lt v\lt 1$ with respect to the $x$-axis, so that the origin $x^\prime =0$ follows the trajectory $x=vt$ in the $(x,t)$-system.  

$X^\prime$ is thus invited "to observe" the light signal in the $(x,t)$-system in terms of $(x^\prime ,t^\prime )$-coordinates connected to the $(x,t)$-coordinates by the Lorentz transformation. This is in contradiction to the assumption that $X^\prime$ can observe only light signals in the $(x^\prime ,t^\prime )$-system. Nevertheless, the light signal in the $(x,t)$ system when transformed into the $(x^\prime ,t^\prime )$-coordinates according to the Lorentz transformation by Einstein is required to be a light signal in the $(x^\prime ,t^\prime )$-system represented by a trajectory $x^\prime =t^\prime$. 

Einstein thus derives the Lorentz transformation by requiring a light signal in a $(x,t)$-system to be observed by an observer $X^\prime$ using different $(x^\prime ,t^\prime )$-coordinates as a light signal in the $(x^\prime ,t^\prime )$-system. This forces $X^\prime$ to use different measures of space and time when "viewing" the light signal in the $(x,t)$-system,  than the rigid rods and clock belonging to $X^\prime$.

SR as based on the Lorentz transformation according to Einstein thus boils down to twisting the brains of an observer $X^\prime$ with an $(x^\prime ,t^\prime )$-system "to view" a light signal in another $(x,t)$-system with the $x^\prime$-axis moving with respect to the $x$-axis, as a light signal in the system of $(x^\prime ,t^\prime )$. The twisting involves forcing $X^\prime$ to use different measures of space and time when "viewing" a light signal belonging to a different $(x,t)$-system.

The necessity of twisting the brains of $X^\prime$ when "viewing"  light signals in the $(x,t)$ system, is seen in the basic example of a light signal in the $(x,t)$ system emitted at $x=0$ at time $t=0$, which at time $t=1$ can be observed by $X$ at $x=1$ as having traveled the distance 1 with respect to the $x$-axis but only the distance $1-v$ with respect to the $x^\prime$-axis moving with velocity $v$ with respect to the $x$-axis, which requires  $X^\prime$ to reset time to "observe" a speed of light of 1.

The difference between Einstein and Lorentz who wrote down his transformation before Einstein picked it up, is that Lorentz does not consider the transformed coordinates $(x^\prime ,t^\prime )$ to represent physical coordinates, only some form of "local (unphysical) coordinates". Lorentz thus does not consider the light signal in the $(x,t)$-system as "viewed" by $X^\prime$ to be a physical light signal in the $(x^\prime ,t^\prime )$-system. It was Einstein who took this step in contradiction to the Postulate of SR and then leading into the bushes of non-physics.

Without twisting of brains the truth buried in the Postulate of SR is that all observers use the same measures of space and time, which means that the coordinates between moving systems are connected by the Galilean transformation:

• $x^\prime =x -vt$,  $t^\prime =t$.

With a Galilean transformation the Postulate of SR carries no contradictions or paradoxes, and the consequences are explored in MMR. In a Galilean transformation there is no "space contraction" or "time dilation" and no "strangeness". 

Summary: All paradoxes of SR result from Einstein's question of asking for the view of an observer of the propagation of light in a system in which the observer is not at rest, which is contradictory because the Postulate of SR requires observations to be made at rest in the system for observation.

Einstein's question expresses a form of logical contradiction, in the sense of asking for something which is forbidden in the Postulate of SR. From one logical contradiction infinitely many contradictions can be derived, which is seen in the many paradoxes of SR.

The Postulate of SR is physical and without contradiction, but the contradiction of the Postulate as the Lorentz transformation is unphysical. Einstein's SR is thus unphysical.

The "consensus" of modern physicists seems to be that despite its logical contradiction from start and the resulting infinitely many paradoxes, Einstein's SR represents an advancement of physics of a magnitude never seen before with Newton's mechanics only a bleak incorrect first step.

The crisis of modern physics witnessed by so many in recent times can be seen as a consequence of the contradictions of SR.  To get out of this trauma blocking progress, a revision of SR is needed. Today professional physicists pay lip service to SR, while viewing SR as a dead subject of interest only to cranks presenting ever new resolutions of the paradoxes in fruitless attempts to come to grips with the "strangeness" of SR.

PS1 More aspects are found under the tag special theory of relativity.

PS2 The Postulate of SR in particular expresses that emission and reception/observation of a light signal is coupled process where the observer is connected to the emitter through a space coordinate system in which the observer is at rest.  The connection can be seen as being established by a standing wave satisfying Maxwell's equations with the light signal as a superimposed traveling wave.

PS3 The Postulate of SR is not compatible with light as a stream of "particles" or "photons", which is anyway unphysical.

PS4 The Postulate of SR is in some texts stated as allowing not only emitter but also receiver to be moving in the space system of observation, but Einstein did not include receiver and doing so does not make any sense as argued above.

• But one thing is the thought, another thing is the deed, and another thing is the idea of the deed. The wheel of causality doth not roll between them. Friedrich Nietzsche, Thus Spoke Zarathustra
• What I wanted to say was just this: In the present circumstances, the only profession I would choose would be one where earning a living had nothing to do with the search for knowledge. (Einstein’s last letter to Born)
• The question whether the Lorentz contraction does or does not exist is confusing. It does not really exist in so far as it does not exist for an observer who moves (with the rod); it really exists, however, in the sense that it can as a matter of principle be demonstrated by a resting observer. (Einstein 1911)
• But no Anglo-Saxon can understand relativity. Said at a dinner in 1910, teasing Ernest Rutherford, who replied: No, they have too much sense.