It depends if both the axis you choose are already parallel to each other.

Then rods will be parallel to both x and x' (but moving rod will appear shrinked).

Open this link to know more in detail: http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/tdil.html

http://hyperphysics.phy-astr.gsu.edu/hbase/relativ/tdil.html

Bhushan,

Thank you for your response, however I my question was about the rod moving perpendicularly to x while being at all times parallel to it. At some point it should cross x axis completely enclosing it. The link does not seem to cover this scenario.

Bhushan,

"shrunk" :-)

or even shrunken, depending on exact meaning

The issue whether rod shrinks or is larger is another matter worthy a separate discussion - may be a bit later. Now I want to focus on parallel lines first.

All I can say at this point that Johan's comment is consistent with unqualified statements in relativity that moving objects shrink. It seems logical to think that stationary objects should become relatively larger. Let's leave this for later.

Robert's comment is perfectly right in the formal sense of having transformed two simultaneous events of the rod passing the X axis into two "successive" (i.e. one clock state is greater than another) as per relative simultaneity principles. But the events are only successive if one thinks that two different clock times mean successive.

This is not true for example in different time zones on Earth.

With Lorentz transformation this is not that obvious.

My objection against the standard interpretation of transformed moving rod which in the stationary frame becomes co-linear with X axis and by definition, X is co-linear with X' axis, is:

(after a correction)

 The rod being wholly contained in X and X' cannot ever be non-parallel to it.

The effect of transformation gives two different times for each rod's end, by which we judge that two ends passed X successively. But we can have the same in the stationary frame if two identical accurate clocks at two ends are arbitrary shifted by an offset.

More formal reasoning I have put together in the attached pages. I hope it clarifies better my point of view:

(after a correction)

Robert,

Sorry, I have used common language here.

• From the philosophical point of view it is "contained" or aligned for one instance of its existence whatever that means
• In the presented mathematical model, "contained" when both ends become a part of X
• "contained" means the set of all points of the rod becomes a subset of X axis which is co-linear with X'axis.

Additionally I had to correct the attached pdf file in my previous post.

Robert,

I do not question what you are saying but I want to understand it better:

Lorentz Boost seems to be defined as  "rotation-free Lorentz transformation" and that is as simple as a single Lorentz transformation (the original Lorentz transformation) with constant velocity v. (I do not know why someone introduced the "boost" term as if LT was something else.)

My problem is in a single "boost", and non-accelerated rectalinear motion so I am not sure how the commutator of two Lorentz boosts is relevant and why do I need to introduce complication of curve-linear motions when I try to understand the simple(?) thing first.

The simplest way to think about this is to imagine the spacetime diagram. The blue strip in the figure represents the trajectory of a rod AB that is parallel to the x axis and moves perpendicularly to the x axis (ie, along the y axis) It is clear from the figure that there is no line in the blue strip parallel to the x′ axis.

Thank you Robert for clarifications.Thank you Eric for your reply.

I appreciate the time invested by both of you to my seemingly pseudo problem. I hope you do not mind that at this stage of my understanding I would still not fully agree.

The example, is a good illustration of the problem and it does indeed reflect what you can derive from Lorentz transformation. It describes physical reality in some way.

But is it a physical reality?

Is it an arbitrary conventional view that maps reality into a map which is a distorted one in some way?

We map the globe into flat maps and there is nothing wrong with it but we cannot argue the Earth is flat because our map demonstrates this.

You show clearly that X’ and X axes are not parallel and everything can be explained that way, but I have at least two reservation to this interpretation.

Consider Einstein in “On the Relativity Principle and the Conclusions Drawn from It,

Jahrbuch der Radioaktivitat und Electronik 4 (1907)”:

We chose the position of the x’-axis in such a way that it has, with reference to S, the same direction as the translational motion of S’ has with reference to S, then it follows for the reason of symmetry that the S-referred coordinate planes of S’ must be mutually perpendicular. We can and will chose the positions of the two coordinate systems in such a way that the x-axis of S and the x’ais of S’ coincident at all times., and that the S-referred y’ axis of S’ be parallel to the y-axis of S. Further we have chosen the instant at which the coordinate origins coincide as the starting time in both systems.

He still (1907) treats the world as 3 dimensional and the 4’th dimension seems to be an artifact of mathematical representation.The fact that axis X is not parallel to X’ while coincident with it and in the same direction as the velocity vector, does not seem to cross his mind.

I do not believe the 4-th spatial dimension deserves the ontological status of existence because it is simply not there.  Therefore we have a pseudo geometry (in terms of physical reality not pure mathematics) - still useful for many things same way as the map is useful in navigation.

The other reservation comes from a remotely possible physical scenario depictet on the attached drawing.

The scenario is that the X axis is the centre of a metal tube that has a piston moving through it.

In the centre of the piston there is an axially mounted wire extending along the tube axis.

I am asking under what speed the wire touches the wall of the tube since it will become not parallel to X and the tube walls?

The tube may have a window allowing the rod enter the tube and align with X axis. Why would not it also align with X’ axis then? 

Space or abstract geometry may be 4 dimensional but solid bodies seem to have only 3 dimensions that they stubbornly keep.

Andrew ~

Notice that in my diagram I omitted the t′ axis, for simplicity. Hence the diagram and the conclusion drawn from it are just as valid for the situation in Galilean relativity! The Galilean transformation is x′ = x − vt, meaning that a point with coordinate x in a reference system S has the coordinate x′ in a reference system S′ moving relative to S along the x-axis with velocity v. The trajectory of the origin of S′ (x′ = 0) is the line x = vt while the trajectory of the origin of S is the t-axis.

Einstein’s statement “the x-axis of S and the x′- axis of S′ are coincident at all times” is puzzling (he doesn’t always express himself clearly). However, if you draw in the figure (in my previous post) a line parallel to the t-axis, the two points x and x′ where it cuts the x-axis and the x′-axis represent the same spatial point (unmoving with respect to S) at different times. The x-axis and the x′-axis are in the same spatial direction (though not in the same temporal direction).

As you said, Einstein was still thinking of space and time separately at this stage. It was Minkowski who saw that Einstein’s theory indicated a 4-dimensional geometry that unified space and time. I cannot agree with you when you say “I do not believe the 4-th spatial dimension deserves the ontological status of existence because it is simply not there.” Of course time is "there" - just as "real" as the three spatial dimensions!

I’m attaching another diagram. In this case the blue strip represents the trajectory of a tube along the x-direction (stationary in the system S) and the line x′ is the trajectory of  a point on the wire that is travelling along it with velocity v. Every point on the wire has a trajectory parallel to this one. All these lines lie within the blue strip. If we move the line x along the t axis keeping it parallel itself we see the evolution of the system as the wire moves through the pipe.

Of course, if the tube is not stationary in S but is moving in the y-direction, the velocities of the points on the wire have an x component and a y component. Their trajectories would be represented by a family of parallel lines drawn on the blue strip of my first diagram.

Johann ~

x, y, z and t are coordinates - nothing more. So are t and t'. Coordinate systems are not physics. They are useful mathematical concepts that enable us calculate and make predictions about the behaviour of physical systems, without which we wouldn't be able to formulate physical theories.

Lengths and times are physical quantities. x, y, z, x', y' and z' are related to lengths, but conceptually are not lengths. t and t' are related to time, but are conceptually distinct from time (which is what is measured by a clock).

Hence I agree with you up to a point. But I disagree when you say that physics that makes use of coordinates in 4D Minkowski spacetime "must be scrapped".  That physics is Einstein's relativity, which is in excellent agreement with a century of experimental observations. Why scrap something that works so well in practice?

I see from your own publication "On the derivation of the Lorentz-transformation" that you have no quarrel with Lorentz transformations. Are you not contradicting yourself by accepting that transformation law while rejecting the physical theories that it implies?

Allow me to try your patience once more by coming right down for a moment to the level of school mathematics, in order to try to convince you of the logical consistency of Minkowskian 4-space as a useful means of representation. Please bear with me.

The motion of a point in one dimension is given by a function x(t). We can set up an x-axis and a t-axis and draw the graph of that function – a curve in “the x, t plane”. The speed of the point at any time, dx/dt, is just the slope of the graph. The trajectory of a light ray along the x-axis is represented by a straight line satifying dx/dt = c. This means of representation can be extended to the motion of points moving in a plane, described by two functions x(t), y(t); we can set up three axes x, y, and t and represent the motion as a curve in this mathematical 3-dimension space; light rays are represented by straight line with slope c, that is, (dx/dt)2 + (dy/dt)2 = c2. Nothing objectionable about the logic – its simply a useful means of representation.

By analogy, the motion of a point in three dimensions is given by functions x(t), y(t), z(t). We can mathematically (though not visually) set up four orthogonal axes x, y, z and t. The trajectory of the point is now represented by a curve in this 4-dimensional space, in which the trajectory of a light ray is a straight line satisfying (dx/dt)2 + (dy/dt)2 + (dz/dt)2 = c2. The transformations acting on this 4-space that map light rays to light rays (ie, straight lines of slope c to straight lines of slope c) are Lorentz transformations. The 4-dimensional space is Minkowski spacetime.

Dear Johann ~

I agree that there are logical pitfalls for the unwary when drawing physical conclusions from the mathematics underlying relativity. Even Einstein himself was not always immune. The relationship between the mathematics and its interpretation is subtle. My feeling is that the flaws you refer to are mistakes in interpretation, not in the theory itself.

However, I shall back out of the discussion now because it’s become clear that you and I are never going to reach agreement. Let’s just agree to differ – no hard feelings (-;

Best wishes

~ Eric

Eric,

The Galilean transformation is x′ = x − vt, meaning that a point with coordinate x in a reference system S has the coordinate x′ in a reference system S′ moving relative to S along the x-axis with velocity v.

It is correct and Lorentz transformation should be seen the same way

The trajectory of the origin of S′ (x′ = 0) is the line x = vt while the trajectory of the origin of S is the t-axis.

Functions like x= vt are not trajectories. The trajectory of the origin in 3 d is a segment of the straight line given by the intersection of the two planes: {y=0,z=0} and x E

Einstein’s statement “the x-axis of S and the x′- axis of S′ are coincident at all times” is puzzling (he doesn’t always express himself clearly).

A bit unfair to him in this particular case. I quoted Einstein only because it sometimes attracts attention when you show the highest authority making inconvenient statements. But practically I do not care what authorities say unless I agree with them. In this case I agree because of my second argument illustrated by the physical example with the tube. 

No matter what we draw and think, the moving wire will remain on the axis of the tube and the rod moving perpendicularly will hit it with all its surface from end to end. If the transformation equations show different then they show a distorted map of reality.

And there is a simple explanation for it, which I will give to save time :).

Forgive me some idealisation. After all they are thought experiments

As for your diagrams, the way I see them there is some problem. If you chose to show X’axis tilted in time then there should be many X’axes as the X’Y’Z’ real origin travels along X while the fictitious one the XYZt stays where it was at synchronisation. And X travels along t too. One can see this by putting the finger on the confluence point created by XYZ or XÝ'Z' intersecting axes and watching the state of the clocks ticking there.

(corrected diagram)

Andrew ~

“It [x′ = x − vt] is correct and Lorentz transformation should be seen the same way”

Yes, that’s what I’m saying. The gamma-factor in the Lorentz case is not relevant to the question you asked.

“Functions like x = vt are not trajectories”

Yes, I know. y = 0 and z = 0 were to be understood, of course. I use the term “trajectory”  to mean a curve in spacetime given by functions x(t), y(t), z(t).

“But practically I do not care what authorities say unless I agree with them. In this case I agree [with Einstein's statement] because of my second argument illustrated by the physical example with the tube”

I was not disagreeing with Einstein. It wasn’t immediately obvious to me what he meant, that’s all.

“No matter what we draw and think, the moving wire will remain on the axis”

My spacetime diagrams do show that the moving wire remains on the axis. That was their purpose. The point is that the tube remains parallel to the x-axis as it constantly moves in the t-direction "as time goes by". (An animated diagram would have made this clearer – I had to leave that to imagination!)

The rest of your post, and your diagram, are in total agreement with what I was trying to convey (and with what Einstein meant in the statement you quoted).

Thank you Eric.

I really appreciate your comments.

The first quote “It [x′ = x − vt]...

It was more a remark to myself then the answer to you. Should not be there in fact.

Trajectory: This is only because I call trajectory the path in space in 3,2 or 1 dimensions covered in time.

I don't use it for 4D  because it is out my direct experience. For distance vs time functions because it would be ambiguous with 3 d path, but these are minor terminology bias no - real issue.

Einstein: I was under impression you find the quoted statement incorrect. Thanks for clarification.

I may have misinterpreted your graph that triggered my modification. All I should see is that the tube wall upper and lower edge also are at the same angle as X' lines and lie in the blue strip.

You say you agree with all the rest of my post but from your first post it appears you would not agree with:

7. connecting different rod position at different moments in history creates mathematical illusion of the rod being tilted while in reality it is not.

In my opinion if you tak 4D the rod is no parallel andIagree. Like in 3D a line segment may not be parallell to X being parallell to xy plane because is tilted in z.   But it's projection on x naturally is.

Alternatively if a line segment could only live in xy in a flat universe plane it could still have z coordinate in the extra dimension for whatever reason, and the segment in 3D would be a segment non parallel to X but it would be some kind of an image not the object as it exist by definition in xy plane only.

Johan,

In this particular example:

The event coordinates are: 

| c t1’ |

| 0     |

| 0     |

| 0     |

In standard framework the primed system observes the unprime moving in negative X direction

After Lorentz transformation with –v instead of v

              | c t1’ |          | c gamma t1’ |

              | 0     |          | v*gamma t1’ |

 L(-v)      | 0     |  =       |         0         | 

              | 0     |          |         0         |

where L(-v) is inerse  Lorentz transformation matrix

Now if you assume that was O’ the system which has accelerated, then its clock calculated a lower t1’ number of ticks than that in O, at the corresponding position.

So in this case t1= gamma t1’ and t1 is gamma times greater as expected of older clock. 

The position then is vt1 as expected.

You however need to believe that the accelerated system was O’ and accelerated clocks run slower. With this assumption all is logical.

Dear Johan ~

My “feelings” in this matter are not vague hunches. They come from a lifetime of study of the facts relating to the mathematics and the physics. My experience has been that whenever it has seemed to me that there was some inconsistency in the theory of Relativity, arising either from the underlying mathematical structure or its interpretation, application of logical thinking (sometimes involving quite hard work) has always cleared the matter up for me.

You may think I’m a “mainstream physicist” unwilling to think beyond what I’ve been taught and blindly accepted. That is not so. Had it been so I wouldn't have entered into a dialogue with you.  I'm open minded and always willing to give serious consideration to ideas that seem strange to me [one of my favourite quotes, which I think you will like, is “What gets us into trouble is not what we don't know. It's what we know for sure that just aint so.” - Mark Twain]. But that doesn't mean that I can be persuaded that a flaw or an inconsistency exists when I can plainly see that it does not (a case in point being the idea that Minkowski spacetime, as a mathematical construct, is nonesense – such an idea is, quite frankly, ludicrous).

I’ve enjoyed discussing with you, Johann, but it is time-consuming (I have a life outside ResearchGate) and it’s been leading nowhere, for either of us. My backing out has nothing to do with “ducking and diving the issues”.

Time to quit – I thought you’d agree.

‘Bye and good luck ~ Eric

Andrew ~

I’m afraid I have to admit that I was mistaken earlier when I said “Notice that in my diagram I omitted the t′ axis, for simplicity. Hence the diagram and the conclusion drawn from it are just as valid for the situation in Galilean relativity!” That is wrong! Because, by definition, t = 0 along the x-axis and x = 0 along the t-axis (and similarly for x′ and t′ ), the Galilean transformation x′ = x − vt, t′ = t implies that, in the Galilean case the x-axis is unchanged and it’s the t-axis gets tilted, not the other way round! I confused myself and hope I didn’t confuse you.

According to the Galilean relativity principle t = t′ = 0 is the plane xy for all observers. All “simultaneous” events lie in a plane parallel to that one. The orientation of a moving rod can be determined by making simultaneous observations of its end points. It follows that the Galilean relativity principle implies that all observers will agree about the orientation of the rod.

According to the Einsteinian relativity principle, t = 0 is the plane xy for the observer whose reference system is S, and all simultaneous events for that observer lie in a plane parallel to that one. But from the Lorentz transformation t′ = 0 when x = c2t/v. The line given by that equation along with y = 0 is the x′ -axis. Simultaneous events for the observer whose reference system is S′ lie in a plane parallel to the plane t′ = 0, ie, the x′ y plane. It follows that the Einsteinian relativity principle implies that the two observers will not agree about the orientation of the rod. (In terms of my first diagram, an instantanous observation of the rod by the first observer corresponds to the intersection of the blue strip by a plane parallel to the the plane t = 0 (the xy plane). For the second observer, an instantanous observation of the rod corresponds to the intersection of the blue strip by a plane parallel to the plane t′ = 0 (the x′ y plane). The conclusion is that for the first observer the rod is oriented parallel to the x-axis, but for the second observer, it is not - it has a y-component.

I hope that answers your remark “…it appears you would not agree with: 7. connecting different rod position at different moments in history creates mathematical illusion of the rod being tilted while in reality it is not.”

The implication of my first post was that the two observers would not agree about the orientation of the rod. The observations differ because the meaning of "simultaneous" is observer dependent. Of course, there is a sense in which the "actual" orientation is the one measured by a third observer, who travels with the rod, in which case the rod is "at rest". That observer would conclude that it's parallel to the x-axis.

It seems here is again some discussion that goes on without concrete results. And again that is a consequence of the problem, which one can meet in yet many cases here: to discuss the physical situations where the notions of the space and of the time are essential is necessary to understand – what is the space and what is the time. If the understanding absents, the opponents never come to concrete - and correct - conclusions.

When the space/time problem is rather simple and it was cleared in other threads a number of times; for this case is sufficient to understand that the space and the time are absolute, are components of the absolute 4D “Cartesian” spacetime; that neither space nor time depend on any material object and on any “observer”. And, besides, that the Pythagoras-Lorentz transformations relate to the showings of concrete clocks and rules, which constitute rigid system that is called “reference frame” and only rather implicitly the transformations relate to some spacetime points.

So in the thread’s problem the rod, which is oriented in a frame K along x-axis and moves, say, along y-axis, falls on the x-axis simultaneously flatwise. As well as – if a frame K’ moves so that its x’-, y’-axes are collinear with the x-,y-axes, in the reality the rod falls on the x’ axis simultaneously flatwise also. That is evident.

Though in the K’ an observer can conclude from its clocks data that the rod’s ends crossed the x’ axis in different times and so the rod crossed the x’ axis at some angle to the axis. But that is if the clocks were synchronized in accordance with the SR, let – by “Einstein synchronization”. In this case clocks in the frame are placed on known distances and register time moments when the light of a light pulse, say in middle of the clocks’ line. After that the clocks’ showings, Ti, are established under condition that Ti =Li/c, where Li is the distance between i-clock and pulse point and under condition that at the pulse the “Pulse ”clock showing was equal to the zero.

It is evident, that in this case for the clocks, which are in the x’ axis [relating to the pulse point] along K’ motion, the time moments will in reality be larger relating to the clocks that are in opposite [to the K’ speed] direction relating to the pulse point. So the “Einstein synchronization” is, in certain sense, some artificial. But in many case, for example if one measures the speed of a body in the K’ frame, he obtains quite adequate result that allows, say, adequately estimate a mechanical interaction of a couple of bodies.

However one can synchronize other set of the K’ clocks by using the low transport. If at that these clocks don’t constitute with the K’ x- axis a rigid system, then their showings will differ relating to the clocks of the first set. The set-2 clocks data, if are used to measure, say, the speed of a body that moves in the K’ frame, lead to incorrect result. But in this [thread problem] case the set-2 clocks show correct result – the rod falls flatwise, as that the K frame clocks show.

Though the K’ observer can obtain the correct result by using an other method – he can put on the x’-axis a basin with plastilin, and see that the rod has fallen flatwise, when his clocks show that the rod must thrust in in the basin by one end…

Cheers

Sergey ~

“…the space and the time are absolute, are components of the absolute 4D “Cartesian” spacetime that neither space nor time depend on any material object and on any observer”

The "spacetime" – Minkowskian 4D space − is absolute and independent of any material object or any observer. But “distances” and “times” are measurements made by observers. They are dependent on the relative motion of the observers that measure them. That is the very essence of the theory of relativity.

“…the rod, which is oriented in a frame K along x-axis and moves, say, along y-axis, falls on the x-axis simultaneously flatwise. As well as – if a frame K’ moves so that its x’-, y’-axes are collinear with the x-,y-axes, in the reality the rod falls on the x’ axis simultaneously flatwise also. That is evident.”

No, it is not evident. It is wrong. A flaw in that argument is the use of the phrase "in reality" when what we are discussing are observations. Another flaw is the use of the word “simultaneously”. For the observer whose reference system is K "simultanously" means t = 0; for the observer whose reference system is K′, it means t′ = 0. When t′ = 0 the two ends of the rod have different y values (simply because the rod is moving in the y direction). That’s all there is to it. “Simultaneity” is not absolute in relativity, it is observer-dependent. That is the point.

Of course, the observer with the reference system K needs a method of determining the set of events occuring at t = 0 (and similarly for the observer in K′ and the events occuring at t′ = 0). Which is why you bring “clocks” into the discussion. But, with all due respect, that is not relevant. If the two observers ascertain the positions of the end points of the rod “simultaneously” − whatever their chosen method for doing that − the observed orientation of the rod is different for the two observers.

Dear Eric,

Your “The "spacetime" – Minkowskian 4D space − is absolute and independent of any material object or any observer. But “distances” and “times” are measurements made by observers. They are dependent on the relative motion of the observers that measure them. That is the very essence of the theory of relativity.”

- is evidently incorrect.

The Minkowskian 4D space depends on an observer, or – on the observer’s reference frame that is a set of material rules and material synchronized clocks.

So, for example if there are two observers that are in a relative motion, let with the speed V, then there are two Minkowskian 4D spaces, where, for example, the t-axes (and some others axes) are differently directed; in different frames distances between the same points can be spacelike and timelike – i.e. can be real or imaginary – simultaneously, etc. So this space isn’t absolute by any means.

Besides the Minkowskian 4D space cannot be real, it is a mathematical model, and so, as any other model, it can be only be adequate in some limits to the spacetime that exists in the reality. In the reality the spacetime is 4D Euclidian (“Cartesian”) manifold, where, e.g., any distances between points are real. So the Minkowskian 4D space also is adequate to the reality in some very little extent.

So the using of Minkowskian formalism without limitations – what the SR (and you) do, leads in some cases to incorrect implications. An example – your “…A flaw in that argument is the use of the phrase "in reality" when what we are discussing are observations.”

Any observation in any science has a sense only if allows to obtain correct information about material process that, in turn, allows to an observer to make such interpretation of the experimental outcomes that is adequate to the reality.

In other cases – i.e. if observations show some illusion that isn’t adequate to the reality – such “observations” have no sense, and, if an experimenter thinks that they are true, he can measure many rather magic physical results – as in this experiment, for example as (if the observers are true believer of the SR) “…the observed orientation of the rod is different for the two observers.”

And such situation isn’t something exclusive; an other well known example follows from the SR postulate about total equivalence of all reference frames. From this directly follows that ,say, the two observers above  simultaneously obtain theoretically evidently absurd result that their vis-à-vis’s clocks tick slower; and – if they make experiments to measure the vis-à-vis’s clocks tick rates by using their frames’ clocks and rules that are arranged in accordance with the SR, - they both experimentally obtain the same absurd results.

Something like as in this case, where in the reality the rod falls on both, x and x’ axes, flatwise – in contrast to the implication of a true SR believer.

And if the observer in the K’ frame is indeed the experimenter and so attempts to measure something always by using a number of independent methods, then, using the experimental technique that consists of a basin and plastilin, (s)he will obtain correct outcome…

Cheers

Sergei ~

"The Minkowskian 4D space depends on an observer"

NO! Minkowskian 4D space is a mathematical concept. It is not physics. It is not about "observers".

Einstein's theory of Special Relativity is about observations. That theory can be elegantly expressed in terms of the mathematical language of  Minkowskian 4D space.

"So the using of Minkowskian formalism without limitations – what the SR (and you) do, leads in some cases to incorrect implications"

NO! Special Relativity, properly understood, does not lead to incorrect predictions about observations.

You imply that some observations are illusions that don't correspond to "reality". Special Relativity correctly predicts what those observations are. They are true observations. "Reality" is not the point. The various observations do not in fact contradict each other; they are interrelated through Lorentz transformations. 

"two observers simultaneously obtain theoretically evidently absurd result that their vis-à-vis’s clocks tick slower; and – if they make experiments to measure the vis-à-vis’s clocks tick rates by using their frames’ clocks and rules that are arranged in accordance with the SR, - they both experimentally obtain the same absurd results."

Yes, that is what they will observe. But there's nothing "absurd" here. What you have stated here is "Dingle's paradox", which arises from a naive misunderstanding of the theory.  If you and I stand at a distance from each other, you will look smaller to me and I will look smaller to you. Both observations are correct. You will agree that there's nothing absurd about that. Observations of moving clocks are analogous. The observations of moving clocks, correctly predicted by SR, are, likewise, not "absurd". Similarly, the fact that the observed orientation of the rod is different for the two observers is not absurd. The two observations are related through Lorentz transformations and are ipso facto mutually consistent.

You speak of "SR believers".

It's not a matter of "believing". It's a matter of asking, not whether SR is "true", but what SR predicts for the outcome of particular observations by particular observers..

If you are claiming that SR as a theory of physics is absurd, then we part company - there is in that case nothing to be gained by further discussion.

Dear Eric,

“...If you are claiming that SR as a theory of physics is absurd,” sorry, but that isn’t so. I don’t “claim” anything, including – because I cannot allow to myself such a luxury (not, of course literally; that is an idiom in Russian).

So I can write and so I write only rigorously grounded posts; moreover – only evidently grounded. Including that relates to the SR – this theory indeed is outside rational thinking since it claims such strange things as, for example, that real spacetime in Universe is practically the Minkowskian 4D space – or more “correctly” pseudo-Riemannian space, which also has partly imaginary metric, etc.  – 99% mainstream cosmology is/ are with accordance with this postulate. (So you aren’t true when write “...NO! Minkowskian 4D space is a mathematical concept. It is not physics.”)

And – next time - there are a number of analogous “strange” relativistic assertions, as, e.g. “time dilation”, “space contractions”, etc. that have no relations to the reality

– there are in the reality only different, depending on their speed in the absolute Euclidian 4D spacetime and on relative speeds concrete tick rates of concrete clocks and concrete projections of concrete lengths of concrete material bodies in the plain (ct,x) – lengths, not “distances”; etc., etc., etc.

And – again – the claims as “...They are true observations. "Reality" is not the point…” are, regrettably, a next fallacy; that – as I see - is rather common for the believers in the SR.

When that is evidently non-scientific; the aim of any science is the study just of Reality, not of illusions.

You touched the Dingle problem and repeated a popular – among true believers of the SR “explanation” of the problem:

“..."Dingle's paradox", which arises from a naive misunderstanding of the theory. If you and I stand at a distance from each other, you will look smaller to me and I will look smaller to you. Both observations are correct. You will agree that there's nothing absurd about that. Observations of moving clocks are analogous.”

Such “explanation” is in other (about this problem) thread in the RG and I commented it. But since that thread is rather spammed, I [practically] repeat the comment here

– “...you will look smaller to me and I will look smaller to you.” – that is a trivial illusion and so hasn’t any relation to the Dingle problem. It doesn’t differ in this case, for example, from the case when a big number of observers observe how a man saws trough in twain poor girl in a circus scene. And these both (as well many other) illusions can be always explained without any logical or physical problems.

But the observers’ illusions in the Dingle problem principally differ from the illusions above; they aren’t avoidable in the SR principally, because of they are postulated in the theory.

Though the Dingle problem – as well as all the rest “relativistic paradoxes” quite evidently disappear if a physicist isn’t a true believer in the SR and understands, that in depth the SR is simply inconsistent, including for this case; that in reality all reference frames are non-equivalent; that there aren’t “relativistic effects” above and a number of others; and that in Reality there exists [indeed] absolute 4D spacetime, etc.

More - see at least http://viXra.org/abs/1311.0190

Cheers

Dear Sergey ~

“…I write only rigorously grounded posts.”

So do I − at least I try to!.

To a mathematician Minkowskian 4D geometry is is an interesting abstract object; it belongs to a family of various kinds of non-Euclidean geometries. It says nothing about “reality”. However, it turns out that it can be a useful tool in physics – just as many other aspects of “pure” mathematics are useful (and necessary) in theoretical physics.

I agree that some of the terminology of Relativity theory is unfortunate. There is a lot of confusing muddle in the literature of Relativity. I agree with you entirely when you say

“…there are a number of analogous “strange” relativistic assertions, as, e.g. “time dilation”, “space contractions”, etc. that have no relations to the reality”

What is “reality”? All we can know about “reality” is what we observe. It can be argued that physics is not about reality, it’s about observations. (In quantum mechanics the relation between reality and observation is even more puzzling and bizarre than it is in relativity…). You see what you think are absurdities in the observations that Relativity theory predicts and dismiss those predicted observations as “illusions”. In a sense you are right. But an illusory observation (or perception) is “real” in the sense that it is a real illusion that can actually be observed.

If you like you can resolve Dingle’s “paradox” by saying that when the two observers are in motion relative to each other, they are experiencing “illusions” when they observe each other's "clocks". I wouldn’t object to that interpretation. Referring to so-called “time dilation”, you say “But the observers’ illusions in the Dingle problem principally differ from the illusions above; they aren’t avoidable in the SR principally, because of they are postulated in the theory.”

They are not “postulated”. They are a consequence of the Lorentz transformations, which are a consequence of the postulate of the constancy of light speed, which in turn is a consequence of Maxwell’s EM theory.

Let the “clocks” of the two observers in Dingle’s argument be beams of monochromatic light which they use to signal to each other. If they are receding from each other each will see the signal from the other “red shifted”. Where’s the “paradox”?? “Time dilation” is nothing more than a misleading way of referring to a Doppler effect.

Thank you Sergey, Eric and Johan for continuing this discussion from different perspectives. It goes beyond my "simple" question but it is relevant and interesting.

Firstly, for the first time I see someone who supports my point of view. That is Sergey's:

"Though in the K’ an observer can conclude from its clocks data that the rod’s ends crossed the x’ axis in different times and so the rod crossed the x’ axis at some angle to the axis. But that is if the clocks were synchronized in accordance with the SR", and everything that follows. 

That is exactly the point. With the absence of infinite fast signal, Einstein's synchronisation makes simultaneous events non-simultaneous by the clock indication.

We can do the same without relativity within an inertial system by deliberately shifting two clock times by an arbitrary offset, then assuming linear gradient of time in our own frame, and voila.... a rod crossing the line as parallel, but according to equations of motion of its ends in this time convention, it seems to be tilted, and it is longer longer than expected. A mathematical illusion in this case.

But I would not bet on the absence of instantaneous synchronisation. It is a very popular discussion these days about faster than light laser spot  sweep. It is true that it cannot carry information from point to point but instead it can provide two synchronisation signals at much shorter time than by any light signal. Given this is a thought experiment, one can arbitrarily increase the distance from the emitter and the sweep rate until two clocks in two points are synchronised almost instantaneously within accepted experimental error. But such synchronisation would not conform to Lorentz transformation and the same speed of light in two opposite direction in general cannot be valid.

One interesting thing is that whether you you use Lorentz or Galilean transformation, the speed of light measured in the moving system by timing a round trip of a light pulse, is the same for the same units of time and length.

Johan ~

"Time-dilation" is NOT a consequence of the Lorentz-equations"

Depends what you mean by "time dilation". I think that that term "time dilation" is confusing and misleading anyway. I understand it to refer to any of the physical predictions that are deducible from the Lorentz equation t′= γ(t − vx/c2) as conventionally interpreted.

"They [t and t′] are NOT different times that occur simultaneously on both clocks"

Yes, I know. It's a charactistic feature of Einstein's theory that "simultaneous" doesn't have the same meaning for different observers. Nevertheless, the observed rate of the passage of time given a clock at rest (relative to the observer) and by a clock that is moving (relative to the observer) are not the same. That is simply a consequence of the fact that information can be transported only at finite speeds.

Dear Eric,

It seems that you again repeat assertions that were discussed here already; that seems as not too productive, at least to me,

Again - “…Minkowskian 4D geometry is an interesting abstract object; it belongs to a family of various kinds of non-Euclidean geometries. It says nothing about “reality”…”

– again, in the SR and further in the GR Minkowskian /pseudo Riemannian spacetimes are considered as the real spacetimes in our Universe – that is well known in physics, including the cosmology.

Again – “What is “reality”? All we can know about “reality” is what we observe. It can be argued that physics is not about reality, it’s about observations”

– again, the reality is something that exists, interacts and evolves objectively, without any relations to – there are some observers or not; at that the physics, if it is a science, studies just the objective reality, not some illusions of some observers. That is evident to anybody who isn’t a true believer in some exotic idealistic philosophy. If you are, say, a subjective idealist, then, sorry, I cannot explain you anything – any mainstream philosophy outside the informational conception is, in fact, only some religion and it is useless to discuss anything with fanatics.

“…Dingle’s “paradox…you say “But the observers’ illusions in the Dingle …are postulated in the theory.”

They are not “postulated”. They are a consequence of the Lorentz transformations, which are a consequence of the postulate of the constancy of light speed, which in turn is a consequence of Maxwell’s EM theory”.

Again, the observers illusions in the Dingle problem, when one of the observers, if (s)he is true believer in the SR and acts in accordance the SR, necessarily makes false implication that his vis-a-vis’s “time is dilated”, are postulated, since that is direct consequence of the SR postulate about the total equivalence of all [inertial] reference frames and, further, so reckless application of the Pythagoras-Lorentz transformations at the theoretical estimation and clocks and rules of his reference frame and at experimental measurement of the vis-a-viz’s clock rate.

What, again, principally differs from other illusions, errors, etc. – see examples above.

Including “Let the “clocks” of the two observers in Dingle’s argument be beams of monochromatic light which they use to signal to each other. If they are receding from each other each will see the signal from the other “red shifted”. Where’s the “paradox”?? “Time dilation” is nothing more than a misleading way of referring to a Doppler effect.”

- sorry, but that has no relation to the Dingle problem, that is only confirms that the relativity principle is valid at mechanical and EM interactions (though – the Doppler effect doesn’t relate to the “time dilation”, the “time dilation” is – in reality – a change of the rate of internal processes in the T-particles/bodies, i.e. those that are created at temporally directed impact and so always having non-zero speed along the t-axis).

The problem of relativistic theories isn’t in that they use the relativity principle, the problem is in that in these theories the validity of this principle is absolutesed up to the absurd. That was excusable in early 1900, when physicists didn’t know practically interactions besides mechanics, EM and gravity, but now seems as something non-understandable. Though even in this case the declaration that the Lorentz transformation describe just the time and the space – when authors didn’t understand what is the time and what is the space (what father resulted in appearance of those strange “relativistic effects”) – isn’t not too understandable also.

And - an example relating in some sense to the problem and your example with Doppler effect.

Let a spaceship moves somewhere. In the ship’s ends there are two observers (back – oA, front - oB) and grow two cactuses – cA and cB. The cactuses can live in the green light only, in the middle of the ship a green lamp radiates the light.

It is evident, that - since the momentums of radiating lamp’s electrons sum up with the photons’ momentums, in the reality photons, that move ahead, will be blue; moving back - red. But since when photons hit in a cactus the corresponding momentums of cactus’s electrons and photons again sum up, the cactuses “see” the green light and are satisfied. Analogously the observers see the green light also.

But a human should differ from a cactus -?

Cheers

Sergey ~

I must object to the idea that the Doppler effect and the “time dilation” effect in SR are unrelated. They are frequency shifts due to motion. They are different aspect of essentially the same phenomenon.

“ …“time dilation” is – in reality – a change of the rate of internal processes in the T-particles/bodies”

You surely can’t intend to mean what this seems to mean – that the internal processes of a particle or a body depend on its velocity? That’s a violation of any kind of relativity principle.

The meaning of the final paragraph (about cactuses and green light in a ship) eludes me. It appears to be a description of a totally static situation. If a lamp and an observer are moving relative to each other, of course there is an observed frequency shift. Whether the observer is a human or a cactus, and whether you prefer to call the shift a “Doppler effect” ot a “time dilation effect”, makes no difference. The frequency shift is real.

Dear Johan ~

I think I’m understanding what you are saying. An observer cannot in fact observe events “simultaneously” because information does not travel instantaneously from an event to an observer. If (as usually presupposed in such discussions) information is gathered from light signals, then an event taking place at a distance D at a time t from an observer will not be observed until a further time D/c has elapsed.

Hence, when deducing how the rod in the original question is oriented, each observer will, when interpreting the results of his observations of the two ends of the rod, have to take the finite speed of light into account and calculate where the ends were at some particular time. That is implicit in the problem. On that understanding, it turns out that the observed orientation of the rod is different for the two observers.

Johan,

Event takes 0 time in any frame and therefore does not move as far as I am aware.

...the set of Lorentz-transformations cannot be treated as if they form a group... ... the mathematically inverse equations ... are NOT modelling actual physics...

With all due respect, Johan, this is nonsense. Of course, the Lorentz transformations form a group and, of course, each Lorentz transformation has an inverse. This is not 'physics-nonsense' but a simple mathematical fact. Do you really want to hide your lack of understanding SR by overriding mathematical logic?

...mathematics does NOT determine what is physically possible.

Sorry, Johan, that is not the point. My objection is that you use pseudo-physical arguments to override mathematics. 'Length contraction' is derived by purely mathematical means from the constancy of c. So you might put into question the interpretation of the Michelson experiment (and, at the same time, of numerous other experiments), but you cannot question the mathematics used in SR - unless you explicitly prove by purely mathematical means that five generations of physicists and mathematicians have been using wrong mathematics.

@Johan: Just a difficulty of language: an event is a point in space *at a given instant in time*. True, realistically we should perhaps relax these requirements, and say: an event is a region of space and time, the spatial and temporal extent of which can, in the given context, be disregarded. Using this definition it is not clear how one can define the stationarity of an event. It certainly is not a concept ordinarily used in relativity. It seems to me that the very essence of the proinciple of relativity consists in saying that, given any event (say a spark being emitted at a given point x,y,z at a given time t, or the collision of two billard balls), it is not possible meaningfully to assigna reference frame to it in which it would be stationary. Rather, all reference frames that move in uniform rectilinear motion one with respect to some inertial frame are equivalent.

Johan,

I do not understand your position regarding inverse transformations. Linear transformations tend to have inverse ones.

So for the train example you gave, a bit of calculation in the attached document shows that they work both ways as expected. Once we know what was in the train we obtain what is wrt the platform, and what we have calculated for the platform, we transform back to the train only to obtain what we have started with.

@Johan: Clearly you are right about saying that no transformation is needed for the observer on the train to measure the coordinates of an event in his reference frame (I refuse to make the distinction you make about whether the event happens on the train or not: whether the ball hangs from the roof of the train compartment or from that of the waiting room in the station is irrelevant. It matters for the further evolution of the ball, but not for the description of the *instantaneous event* of the thread's being cut). In both cases the train observer, call her Alice, can measure the event's coordinates, say x, y, z, t. So can the station observer, call him Bob. He will measure coordinates x', y', z', t'. 

If Alice actually measured x, y, z, t, well and good. Suppose she did not, on the other hand, and she only knows Bob's results x', y', z', t'. Then she needs to do a relativistic rtansformation to obtain the results she would have mesured. Similarly, if Bob only know Alice's measurements, he also needs to perform a transformation. The two transformations are inverses of each other, and both have their usefulness.

Dear Johan and F Leyvraz, 

With your different approach to SR related matters, do you believe that the rod in my main question is not parallell to the moving observer X'axis? 

Lorentz transformations do of course indicate otherwise. As Einstein said in the 1907 paper you quoted, the whole solution of the conundrums posed by the electrodynamics of moving bodies lay in identifying a ``local time'' (``Ortszeit'') earlier introduced by Lorentz, to be the actual time. Essentially, we are of course free to imagine the possibiliity of some true time, which would in some way resolve our issues. But we cannot measure it, for if such a true time, different from the time calculated by the Lorentz transformation, were observable, then so would the discrepancy between true time and Lorentz time. We could then define a preferred reference frame in which true time and Lorentz time would coincide, thus leading to a concept of absolute rest, the very antithesis of what the relativity principle is about.

In fact, as was pointed out by Mermin, the Lorentz transformation can be derived without any reference to light, solely from the assumption that uniform rectilinear motion is unobservable. Of course, it cannot then be shown that the limiting speed is the speed of light (EM waves). But the existence of a limiting speed and the form of the Lorentz transformation can all be derived from the unobservability of recilinear uniform motion, together, of course, with the experimentally well established fact that the actual transformation is not Galilean.

Thus any result contradicting the formal apparatus of Lorentz transformations brings you to a difficult choice: you must either assume the world to be entirely Galilean, in stark contrast to observations, or else you must allow that rectilinear uniform motion can be detected. This is why I believe that your original problem has the solution given by the formal application of the Lorentz transform: the rod is indeed not parallel in the moving frame.

Let me finally add that the paradox is essentially two-dimensional. The different one-dimensional versions you have made are quite different. Clearly, for example, if there is no transverse motion (no motion in y direction), the rod remains always parallel to the x, and of course also to the x' axis.

Finally, yet another way to understand what is happening. Your rod is, in the x system, moving strictly in the y direction. In the x' system, it is thus moving obliquely in some direction. Determining this direction would be non trivial (vector relativistic addition of velocities) but it is irrelevant: the direction surely exists and is oblique. Draw two new axes, xi and eta. xi is the direction in which the rod moves, and eta the perpendicular direction. Of course, now the rod is slanted. But now, from standard Lorentz transform, we know the rod's contraction will be all in the xi direction, none in the eta direction. Since the rod is slanted wrt the xi-eta system, this invlves a change of orientation of the rod wrt the x axis, hence non-parallelism.

I hope this may help.

The citation: Mermin, N. David. "Relativity without light." American Journal of Physics 52.2 (1984): 119-124.

Rather surprisingly the discussion about falling flatwise rod in two [stationary and moving observers; the second moves parallely to the rod’s axis] cases continues; when the answer is evident and seems well explained (see the SS posts above) – the rod falls equally flatwise for both observers.

And, again, the non-understanding by somebody of this fact follows from the same delusion, which follows, in turn, from the somebodies’ rather non-understandable belief in total validity of the SR.

Though the SR is evidently self-inconsistent – that directly follows from the Dingle problem, including – in this [Dingle’s] case one has an example of the senseless experimental outcome when both SR observers experimentally obtain the same senseless result.

This example with Wutke’s rod in this thread is simply a next example, when the SR theoretical and experimental results are non-adequate to the reality; that’s all.

And again – the “relativistic delusion” that is, as it seems, widely spread among the SR’s true believers, follows, first of all, from the other delusion – that some people, who don’t understand what is the space and what is the time can (and have a right) to make some implications about the space and the time.

When such a situation is evidently nonsensical; as the consequence – all declarations in the SR (as well as in the GR, though) about space, time, and spacetime have rather “indirect” relations to these fundamental notions.

To understand – what these notions are – see http://viXra.org/abs/1402.0173 , http://viXra.org/abs/1311.0190

Besides there were detailed discussions in the RG, for example https://www.researchgate.net/post/Did_the_Big_Bang_create_a_parallel_universe_where_time_goes_backwards#view=548f193bcf57d7d4178b4658

The examples here:

“As Einstein said in the 1907 paper you quoted, the whole solution of the conundrums posed by the electrodynamics of moving bodies lay in identifying a ``local time'' (``Ortszeit'') earlier introduced by Lorentz, to be the actual time.”

- Yes, Einstein identified Lorentz’s “local time” with the “actual time”, but without any reasonable explanation – why these “times” are the same? Moreover – Lorentz’s “local time” is much more adequate to the reality comparing the SR’s “actual time”, which in the SR exists in whole Universe and, at that, totally depends on concrete observer and his “reference frame”; from what immediately follow evidently nonsensical implications.

“Essentially, we are of course free to imagine the possibiliity of some true time, which would in some way resolve our issues. But we cannot measure it…[if we can] We could then define a preferred reference frame in which true time and Lorentz time would coincide, thus leading to a concept of absolute rest, the very antithesis of what the relativity principle is about.”

- That isn’t so. The measurement of “some true time” – more correctly – the measurement of the absolute spatial speed of, say, Earth, in the absolute 4D Euclidian spacetime (what is the real spacetime in our Universe; the Minkowskian spacetime is nothing more then some mathematical instrument, which isn’t even a model of the real spacetime) – is possible; that is rather simple experiment, again see http://viXra.org/abs/1311.0190

“Thus any result contradicting the formal apparatus of Lorentz transformations brings you to a difficult choice: you must either assume the world to be entirely Galilean, in stark contrast to observations, or else you must allow that rectilinear uniform motion can be detected.”

- That isn’t so. The world is neither Galilean nor “Lorentzian” (and nor “GRian”) – the world’s spacetime is absolute, it doesn’t depend on any material object inside and “governs” every material object only implicitly, it is more the set of possibilities then the set of rules. And rectilinear uniform motion can be rather easily detected – see the link above

“...original problem has the solution given by the formal application of the Lorentz transform: the rod is indeed not parallel in the moving frame.”

- That isn’t so – again, the rod is indeed parallel in the moving frame – see above. And into the basin with the palstilin…

Cheers

Andrew ~

The original question is a very simple and straightforward question that deserves a simple and unambiguous answer. It can be analysed into separate constituents:

(1) Is the theory of Special Relativity mathematically self-consistent?

(2) What conclusions do calculations based on the mathematics of the theory imply will be the simultaneous positions of the end points of the rod, for the two inertial frames?

(3) Do the predictions of SR correspond to physical reality?

There is no ambiguity about the answers:

(1) Yes, the mathematical apparatus of SR is self-consistent.

(2) The instantaneous orientation of the rod is different in the two frames.

Note that the answer (2) has nothing to do with whether we accept SR as “true” or whether we “believe” in the theory. It is simply a mathematical theorem that follows from given axioms and propositions. It is a theorem in 4D Minkowskian space, just as the theorems we learnt at school are theorems in 2D or 3D Euclidean space (to which we give assent to the logical consistency of their proofs by writing "QED"…).

That has nothing to do with physics, or with “reality”, and has certainly nothing to do with “belief”.

Possible physical relevance is brought in by question (3). The answer is:

(3) Empirical evidence from a century of experiment and observation suggests overwhelmingly that, yes, SR is a valid physical theory.

Contributions to the discussion from those who don’t “believe” the theory, because it seems to them to be “absurd” or “senseless”, arise from misunderstanding of the fundamentals of the theory. Those contributions are irrelevant and have created unnecessary obfuscation and confusion.

In full agreement with Eric Lord above. I would just like to add a short remark on the role of mathematics in the answer to question (3): we have principle, well supported by a considerable amount of empirical evidence, namely that rectilinear uniform motion is unobservable. As examples, there are a large number of  optical experiments made in the XIX century to detect the motion of the Earth (aberration in water-filled telescopes, Michelson, and several more). Further we have a sizable body of experimental evidence that shows that the rate at which clocks go---which is the only way we have of knowing how time flows---is different in different frames when these move at (large) speeds with respect to each other. From these two data, it is then a simple matter of logic (or perhaps I should say algebra) to deduce the actual form of the Lorentz transformation. I do not state that physics is algebra. But once you have, on physical grounds, accepted some postulates of a mathematical nature, you must then surely accept the consequences which arise from them through elementary algebraic manipulations. One might quote Galileo: ``Philosophy is written in that great book which ever lies before our eyes — I mean the universe — but we cannot understand it if we do not first learn the language and grasp the symbols, in which it is written. This book is written in the mathematical language, and the symbols are triangles, circles and other geometrical figures, without whose help it is impossible to comprehend a single word of it; without which one wanders in vain through a dark labyrinth.'' (The Assayer).

Thank you everybody for participating in this discussion, and I would happy for it to continue until some resolution is acceptable to everybody. But it may not be possible because the root of the controversy is a belief in what time is and whether it exists at all. Beliefs enter into the equation when there is no absolute certainty.

As Einstein said in his “Geometry and Experience”:

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.

This thread had many twists and turns and there are two opposing answers to my original question. I want to briefly re-state the problem to eliminate the need of going to multiple posts.

In the standard Special Relativity framework, in the stationary frame there exists a rigid rod of length L such that:

a) It is parallel to X axis at some distance from it, at time t=0 in the negative XY quadrant

b) It moves at low speed vy towards the X axis in the direction of the Y axis

c) It continues in parallel motion until both ends coincide with the X axis at time t=tau, and it is then co-linear with X.

The problem:

In the stationary system both ends of the rod hit the X axis when time t=tau everywhere. Then, all points of the rod coincide with the X axis. The question arises whether the rod is co-linear with the X’ axis of the moving system.

In general:

Is the rod parallel to X’ axis in the moving system?

The improved picture illustrates this scenario.

The two answers:

1) Lorentz transformation of the two ends of the rod from the stationary to moving frame indicates that two ends of the rod coincide at two different times t’ in the moving system, therefore one can say the rod is not parallel and does not align with X’ axis

2) Since the rod by definition is co-linear at some stage with X axis and X axis is co-linear with X’ axis, then the rod is co-linear with X’ axis.

There is no doubt the answer 1 is consistent with the result in the O’X’Y’Z’ frame after Lorentz transformation, and I do not dispute this.

But I support answer 2. By doing so I do not a’priori question the validity of Lorentz transformation. I do question however, the interpretation of time after the transformation. When two ends of the rod show different times while the rod is co-linear with the X axis identical with X’ (their tick marks do not align), then it means the clocks are phase shifted. I acknowledge their relative rates may be different than those in the OXYZ, but all the same within their own frame.

The answer 2 is also supported by Sergey based on his theory of time and space. I am still hoping to find more support.

For all supporting the answer 1, I appreciate your explanations and references, so I see all possible contra-arguments against my view.

I do not have my theory of time and space, but I think the existing ones are sufficient to present a clear and detailed reasoning, being powered by all the above discussions. I will attempt that sometime in the future in this thread.

It seems that dears Eric and Leyvraz didn’t read my last post (a few hours ago).

Besides – some evident common remarks.

(1) - any empirical fact doesn’t prove any physical theory, any experiment with an outcome “in accordance with the theory” is only a fact, that the theory in this concrete case is true, and nothing too much else. But obtaining at some experiment of one outcome that contradicts with the theory is sufficient to conclude that the theory is wrong or, at least, has some limitations.

Though in the case of the SR that is redundant, it is simply self-inconsistent logically. On the other hand the gedanken experiment with the Wutke’s rod in this thread is rather convincible realization of this inconsistence.

(2) Again, the mathematics is, at a description of processes in Nature, very effective instrument, but no more; it only guarantees that mathematical results rigorously follow from given postulates. If a system of postulates is true, the results are true, if a system of postulates is false, the results are false. The SR formalism isn’t by any means a exclusion.

Cheers

Eric,

It is a good to do the overview you have proposed above

(1) Is the theory of Special Relativity mathematically self-consistent?

It depends how you define self-consistent. According to Goedel even the arithmetic of integer numbers is not self-consistent.

Any physical theory has two components:

a) mathematical representation and

b) physical semantics.

Once you allocate meanings to the symbols correctly at the foundation of the theory, you can forget physics for a while and stick with mathematics until you transform to some result and de-construct the reality from the result.

If deconstructed reality corresponds to empirical facts and the mathematics is correct, then the physical theory is self-consistent in my opinion.

SR (as in 1905) has 2 postulates. One about the laws of physics being the same in all inertial frames, the other is a stipulation [Einstein 1916] that velocity of light is the same in all directions independent of motion of the frame. For the reality it assumes rigid 3D coordinate frames moving wrt each other at a constant velocity.

For time can only be measured by clocks, it proposes a particular synchronisation method of distant clocks in order to be able to define velocities and measure them experimentally. Lorentz transformations are derived from that.

As long as you stick with the clock being the only way to physically indicate time and its synchronisation method, the theory appears mathematically self-consistent.

The most unexpected outcome of the assumptions was that clocks must run at different rates. But you need reality checks to be sure,

(2) What conclusions do calculations based on the mathematics of the theory imply will be the simultaneous positions of the end points of the rod, for the two inertial frames?

The conclusion is that the at time t both ends and therefore the whole rod become co-located with the X and therefore with the X’ (which is the same line differently and dynamically scaled). Clocks at both ends of inertial frame indicate different times in the moving system while they indicate the same time in the stationary one.

This is the result of assumption regarding clock synchronisation method which is the only method for which SR equations are valid.

Orientation of the rod is a matter of 3D geometry to which time does not apply. You can see that every two time independent parallel lines in the stationary system transform to two parallel lines in the moving system. It is quite easy to prove.

(3) Do the predictions of SR correspond to physical reality?

Yes if the symbols are correctly interpreted. If you stick to original Einstein definition of time:

Time is then defined as the ensemble of the indications of similar clocks, at rest relatively to K, which register the same simultaneously.[Einstein 1923]

Keeping that in mind you can easily absorb the concept that clocks can have two different values at two ends yet remain temporally coincident. This is because two real clocks must co-exist together at every instant of their existence no matter what they indicate. So there is no multiple time realities-just clocks shifted by an offset.

I admit this is not the main stream interpretation, but if you think about it you will find using your skills that such approach explains more without destroying your innate temporal logic.

I think it is better to invest time in exploring this possibility than defending old theory which creates controversies for more than 100 years due to incorrect interpretation of time. And that is what I try to do within my abilities. Once you get time concept right, all oddities disappear. I have a draft publication for this purpose, but is probably too long and not so mature to be read by other people. This discussion helps me to refine my understanding.

It is a theorem in 4D Minkowskian space, just as the theorems we learnt at school are theorems in 2D or 3D Euclidean space (to which we give assent to the logical consistency of their proofs by writing "QED"…)

I prefer 1905 Einstein’s work which has some roots in reality, to 4D Minkowskian.

(3) Empirical evidence from a century of experiment and observation suggests overwhelmingly that, yes, SR is a valid physical theory.

It is a useful theory but its limits are not clearly highlighted.

Thank you for you participation in this thread. It is good to exchange views from different positions. Without them the discussion would be pointless.

Johan

A Single event requires the laws of physics to occur: The laws of physics are the same within all inertial reference frames. So the event can occur within one of them NOT within ALL simultaneously.

I have hard time to agree with such generalisation.

Not a single atom is in the same frame as another, as they move in their own right,

Yes, the events are caused by material bodies and all inertial observers are generally not there, and coordinate systems don't have to exist.

Yet the events they cause are simultaneously present in any inertial frame and can be located, observed/used and interfered within any reference coordinate system they establish. Physical space as a container is shared by all, only measurements differ for moving observers.

Saying that I see also a good value in your position. When an observer conducts experiments within his inertial frame - which is where they initiate. The others can only observe this. The causality chain is started by the experimenter, not to by external observers. That is meaningful

The originator however may attempt to calculate what others may see and the others may derive from what they see what the originator was doing. This is they have correct theory and measurements.

@Andrew:

"It depends how you define self-consistent. According to Goedel even the arithmetic of integer numbers is not self-consistent" Not really. Gödel showed that you cannot *prove* self-consistemcy. He certainy did not show that there were contradictions, nor did anyone else. Furthermore, the quite elementary linear algebra involved in deductions within SR is quite certainly selfconsistent.

"Once you allocate meanings to the symbols correctly at the foundation of the theory, you can forget physics for a while and stick with mathematics until you transform to some result and de-construct the reality from the result." Agreed. If the final results do not coincide with reality, however, it means that your original hyptheses did not either, or that you did not do your math right.

"The most unexpected outcome of the assumptions was that clocks must run at different rates. But you need reality checks to be sure," They exist. See Wikipedia on Time Dilation. The two most striking to my mind are the experiments involving cosmic muons, produced at relativistic speeds in the upper atmosphere and arriving to the lower atmosphere, even though their decay time would not allow them to go further than a few hundred meters even ath the speed of light. The other is the Hafele-Keating experiment, where Cesium clocks were carried on commercial jets around the globe. Mathematically, this is a bit messy to understand, but it is delightful, since it really is a direct check withe something we can call clocks. There is a lot more, such as the Ives-Stillwell experiment, as well as modern experiments showing time dilation with velocities of the order 10 m/s. The reality check is there.

Another kind of reality check, not directly involving time dilation, is kinematics. Momentum and energy conservation take a different form in relativity, and the analysis of collisions at relativistic speeds requires these concepts. In accelerators, such analyses are routinely made and agree with SR to high accuracy. In a way, this measn tha milloions and millons of high accuracy checks of SR have been and are performed every day.

Of course Johan is quite right in saying that ``But obtaining at some experiment of an outcome that contradicts with the theory is sufficient to conclude that the theory is wrong''. All those experiments can be viewed as contradicting Galilean relativity, so that should, by Johan's own words, be discredited. Again, he is right in saying that experiments do not prove the theory. But they do display with clarity the most salient and, as you correctly say, the most controversial feature of the theory.

"Keeping that in mind you can easily absorb the concept that clocks can have two different values at two ends yet remain temporally coincident." There I must finally admit that my knowledge of the English language is quite insufficient to give this, admittedly pleasing, sentence any kind of non-contradictory meaning. Try "Keeping that in mind you can easily absorb the concept that lengths can have two different values at two times yet remain spatially coincident"...

"you will find using your skills that such approach explains more without destroying your innate temporal logic. " About innate temporal logic, and innate ideas in general: surely we are not born, for example, with the concept of rotationally invariant space. If we were, and if we were to take it seriously, it would be essentially impossible to keep one's balance in the non-rotationally symmetric word that is relevant to our everyday experience. Nevertheless, after some degree of abstraction, we may imagine space divorced from the up/down direction (t took a while: Aristoteles never did) and reach the concept of homogeneous rotationally invariant space. My point about innate temporal logic is much the same: we never have any real need to think carefully about simultaneity etc..., because the effects do not appear in everyday life. Language is forged by everyday life, and what we consider possible in the framework of a certiaain language is strongly influenced by everyday life experience. Saying that time dilation contradicts innate temporal logic is entirely true. It is also similarly true to say that the existence of bacteria contradicts my innate visual logic. Both statements are correct, but neither is a valid objection either against bacteria or against time dilation. Microscopes reveal the former, and the above mentioned experiments the latter.

"I think it is better to invest time in exploring this possibility than defending old theory which creates controversies for more than 100 years due to incorrect interpretation of time." Possibly. But I would suggest you also consider the possibility that the usual view is correct. The controversy, as raised amongst others by Dingle, never succeeded in formulating anything like a contradiction, or in proposing an experiment in which difficulties in the intepretation of SR would be raised. The controversy to some extent consists of people who keep repeating that SR is illogical, without adducing a shred of evidence. The crucial element is always a belief in something evident beyond SR, as well a disinclination to look at the logical arguments which show which parts of SR need to be proved. Essentially, as I have said before, Mermin showed that all you need for Lorentz transformations is the unobservability of uniform rectilinear motion together with the fact that Galilean transformations are incorrect.

Thank you F. Leyvraz 

Your response requires me to spend some time to fully respond. But now some immediate clarification, because I do not want to confuse anyone. My non-native English may not always be the strongest element in discussions to say the least. I do not want to look as a nut case here.

"Keeping that in mind you can easily absorb the concept that clocks can have two different values at two ends yet remain temporally coincident." There I must finally admit that my knowledge of the English language is quite insufficient to give this, admittedly pleasing, sentence any kind of non-contradictory meaning.

Simultaneity and clocks are deeply engraved in minds of contemporary scientists. But before accurate clocks were in existence people were already discussing simultaneity...

I distinguish temporal coincidence from simultaneity because the later is these days associated with having the same time (as per clock indication). 

There is an infinite number of ways clocks can be synchronised for many different reasons. Equal clock times at different locations does not necessary mean events there-then are temporally coincident.  All depends how the clocks are synchronised.

Now, since simultaneity is said to be relative, I argue that temporal coincidence is not. This is to avoid the conflict of terminology.

With the moving rod case, and without any relativistic observer, within our inertial frame:

if we assume both ends align with X axis simultaneously to the best experimental ability using GMT, we acknowledge parallel orientation on X axis at some time the same for both ends.

We can also have two sundials at the opposite ends of a very long rod in East-West direction and conclude the ends are passing the X axis at different times according to two independent observers. Taking one crossing point as the origin we can derive the equation of motion of the rod by which it is not parallel to X even though it is.

This is only a simplistic case, and the relativistic case has a lot of subtle issues which I do not want to touch right now. I only want just to clarify the issue: clock-simultaneous events may not be temporally coincident depending on synchronisation method.

"Try "Keeping that in mind you can easily absorb the concept that lengths can have two different values at two times yet remain spatially coincident"...

This intended sarcasm which I do not hold against you is not as silly as you think.

My tape measure is scaled in centimeters and inches on the other side. Each end of it may be in two different time zones :), yet it is spatially and temporally coincident with the rod it rests on.

No sarcasm was intended: I just really did not unerstand your meaning and translated the sentence from time to the more familiar space to make it clear why I found it obscure. That issues of units are important is undeniable, but they are surely not quite as fundamental as you say: 12 in and the corresponding number of centimers are not, in any real sense, different values. Similarly, time zones are a matter of convention, whereas the flow of time at a given point is not.

Are there any consistent ways of synchronizing watches other than Einstein's, or methods equivalent to Einstein's? I am not aware of any. Requiring

a) synchronization shoudl be doable without using resources such as instantaneous signals

b) synchronization should be an equivalence relation (if A is synchronous to B, then B is synchronous to A; if A is synch to B and B to C then A to C)

are fairly stringent conditions. If you have a suggestion for an alternative, this could be of interest.

Andrew ~

There are many kinds of “geometry” studied by mathematicians (Lobatchevski’s “non-Euclidean” geometry, projective geometry, finite geometries, etc…) Minkwskian 4D geometry is just one of them. Each of these geometries is studied for its intrinsic interest – mathematicians are concerned only with logical consistency, not with whether the concepts correspond to the real world of physics.

Minkowskian geometry is self-consistent in that sense. Minkowski’s important contribution was to recognise that this 4D geometry describes the kinematics of Einstein’s Special Relativity if we identify the time elapsed on a “clock” whose trajectory connects two points with the geometrical “length” of the curve that represents the trajectory of the clock. Even if it had turned out, from emprical tests, that SR doesn’t correspond to the real world, it would still be mathematically true that Minkowskian geometry and the kinematics of SR are equivalent. Only that the physics of SR would in that case be the physics of an imaginary universe, not this one.

You say “I prefer 1905 Einstein’s work which has some roots in reality, to 4D Minkowskian.”

I accept that. It's a subjective preference – we each have our own way of thinking. Mathematically it makes no difference. In my first answer to your question (on page 2 of this now very lengthy discussion) I drew a spacetime diagram. That diagram is nothing more than a convenient shortcut to the answer to the rod question. Anyone who is not comfortable with this way of thinking about SR is welcome to do the calculations without referring to Minkowski spacetime. The result will be the same.

To my mind, the original question about the rod is simply this:

The orientation of the rod is determined by asking what are the positions of its end points at a given instant. That is, the determinations of positions of the end points have to be simultaneous. As we all know, the concept of “simultaneity” in SR does not have the absolute meaning that it had in Galilean relativity – it is observer dependent. In the reference frame of the first observer all simultaneous events are events with the same value of t. For the second observer, they are all events with the same value of t′ . Because the rod is moving in the y direction, the orientation of the rod aquires a y-component for the second observer. To anyone who can’t believe that I say do the calculation. Mathematics isn’t about “believing”!

You said “Orientation of the rod is a matter of 3D geometry to which time does not apply. You can see that every two time independent parallel lines in the stationary system transform to two parallel lines in the moving system.”

Yes, in the three-dimensional geometry the X and X′ axes, and the rod, are parallel. But time cannot be ignored when we are talking about a moving rod and a moving observer!

Of course, for any observer, determining the position of two simultaneous events is not a direct observation because information can be transferred only with finite speed. We need to presuppose that the inertial observers take that into account when using, for example, light signals to interpret their direct observations. (That, I assume, is essentially what you mean by “clock synchronization”).

My view of the original question is that it is asking what can be deduced from the postulates of SR. It is a purely mathematical question, to be answered by mathematical calculation, not by hand-waving discourse about “interpretations” of SR and “beliefs” about the physical validity of the theory.

Johan ~

"I thought all along that we are doing physics! Not pie-in-the-sky mathematics."

I don't know what "pie in the sky mathematics" is! Mathematics is mathematics.  I answered Andre's question by stating the answer according to the mathematics of SR. I did not concern myself with the question of whether the physics of SR is right or wrong.

Johan ~

Andrew asked a question concerning the kinematic implications of Einstein's Special Relativity. My answer is readily established by a straightforward application of the Lorentz transformations.

Calculations based on Minkowski spacetime and calculations based on the Lorentz transformations are, in fact, equivalent. I'm well aware that you regard Minkowskian spacetime as mathematical nonesense. But we've been through all that before - let's not get into it again!  (-:

Dear Johan ~

Yes, of course I agree, if the coordinates are real numbers. But the Minkowski expression is x^2 + y^2 + z^2 − w^2 = 0. The minus sign is crucial. But, as I said, "let's not get into that again".

At this point I understand Johan's concerns. As much as complex numbers are useful and play definite role in physics, they can be deceiving in physical interpretation.

If a parabola equation x=at2/2 a>0 describes a motion whereby at negative times a material point was decelerating towards x=0 touching x=0 at t=0 only to accelerate back towards positive x, we can say the parabola (in t,x plot) must then have had common point with x axis (x=0).

if the equation of motion is x=at2/2+3, we say the parabola has no common points with x axis because the equation at2/2+3=0 has no solutions in R.

But someone could argue that x has been intersected at t= sqrt(6/a)i. Most of us would agree the intersection does not occur in the real world even though imaginary solution exists.

I want once again to thank all the participants for their contribution to resolving my dilemma, no matter what orientation you represent. It is all very useful and stimulating. Please continue this discussion.

I will take a break for some time because thanks to this exchange of ideas - I think I have found the way to prove that the rod remains physically parallel at x,x' axes for all observers.

This could be summarised in two sentences, but I will rigorously check this concept first before writing here. I do not want to introduce untested ideas to revoke them later.

For those seeing this as a mathematical impossibility I say:

There is no need to stretch the math or the SR, or to introduce extraordinary elements. All within the classic (Einstein 1905) SR, and acceptable physical semantics used/introduced there.

The semantic from which some thought experiments are proposed is the key to it.

If I do not succeed I will concede a defeat.

Johan ~

I, personally, never use i = √(−1) when dealing with the Minkowski formulation. That's a very old-fashioned notation that, so far as I know, has been long abandoned. The fourth coordinate in modern notation is ct, not ict. The path of a light ray emanating from the origin is x2 + y2 + z2 − (ct)2 = 0. As is easily checked, that expression is invariant under Lorentz transformations.

So why bring in the number i ... ?

Johan, remember who was the person who brought in this number some 16 hours ago by writing w=ict? In the Minkowski expression x^2 + y^2 + z^2 − w^2 = 0 (invariant under the action of Lorentz transformations) the relation is w=ct. There is no i, which means w and t are real numbers as they should be. Writing w=ict means in fact

... fudging mathematics to fit unfounded pre-conceived ideas!

It seems as rather strange that the discussion here continues so long, when the answer is evident – the rod falls on both, X and X’ axes, flatwise; the outcome of “clocks observation” (if the K’ reference frame is arranged according to the SR and clocks in the frame were synchronized, say, by using “Einstein synchronization”) is incorrect. Again – that isn’t an unique mismatch between the SR “experimental observation” and the reality, more evident case is called “ the Dingle problem”.

As to using in the SR imaginary values – the first application was by Poincare, who introduced the famous invariant interval and “ict”; further Minkowski used this suggestion when developed “pseudoeuclidian” formalism. The main difference between the authors above was – if Poincare suggested some convention, Minkowski declared that real Universe 4D spacetime is really “pseudoeuclidian”, having imaginary either time or space.

This seemed rather strange and “covariant/ contravariant” formalism was suggested as some “development” of the theory, where all spacetime dimensions became “real”; but even in this case imaginary units weren’t (that is impossible in this case, though) removed from the theory – for example distances in the [same] Minkowskian space in some frame can be “spacelike”, i.e., imaginary, at that in other – real; what again seems as rather strange for a physical theory.

As to this discussion, one can add that there is useful to remember also that the Lorentz-Pythagoras transformations were, nonetheless, obtained by (Voigt-Larmor-FitzGerald….-) Lorentz - Poincare – Einstein just for Euclidian spacetime and for the case , when the X and X’ axes are collinear.

Moreover, in most of contemporary textbooks the same scheme is used – though the SR posits that the Euclidian spacetime is unreal, the real spacetime is Minkowskian. But in the Minkowskian spacetime the Lorentz  transformations realize themselves as rotations principally, two frames at a relative motion cannot have collinear X-axes. Such a situation seems as rather strange again, but that is only a next strangeness of this theory…

And so – in the Wutke’s rod case here – the incorrect “ the falling rod cannot be collinear X and X’” – some members attempt to “prove” rigorously following the Minkowskian- (and “lateEinstainian”) mainstream now version of the SR.

Cheers

Sergej:

... is really “pseudoeuclidian”, having imaginary either time or space.

Pseudo-Euclidean: YES. Imaginary time or space: NO.

Coordinates in Minkowski space are real. You can convert an Euclidean form x2 + y2 + z2 + c2t2 into a Minkowskian form by replacing t by itM with a real(!) tM. This gives you the correct pseudo-Euclidean metric x2 + y2 + z2 - c2tM2, with a real time coordinate. This is a mathematical trick, which is not really needed to define the Minkowski space.

... can be “spacelike”, i.e., imaginary ...

'Spacelike' does not mean 'imaginary'. Physically, 'spacelike' means: two points in 4D with a spacelike distance cannot be connected by a signal, even if the signal travels with the speed of light. There is nothing strange about this. Mathematically, Minkowski space is not a metric space. Therefore the familiar representation of distance by the norm of a vector does not apply.

Johan:

The Minkowski expression x^2 + y^2 + z^2 − w^2 = 0 is by definition a relation in 4D, connecting space and time coordinates. So it cannot be the Pythagoras in 3D - if you remember the contents of the Pythagoras. In fact, the expression defines a light cone in 4D. Of course, you could not know that, because there are no light cones in Galilean 3D geometry. Maybe some refreshment in 4D geometry will help.

Johan, whenever somebody (mathematician or physicist) tells you that this is a well-defined relation in 4D, you argue that he has no brains (Minkowski, Einstein, ...). Therefore, I have some problems in finding a person that fits your description of a 'mathematician with brains'. So why don't you try by yourself to convince me that this relation in 4D is impossible? I mean, such an exercise would give you the opportunity to demonstrate your superior mathematical capabilities to your audience.

Dear Johan,

Thank you for the extensive explanations. I am beginning to understand what your problems with SR are.

... the Lorentz transformation cannot be a "rotation" in a 4D manifold ...

The group of Lorentz transformations include 'rotations' in a 3-dimensional Euclidean subspace of the 4D manifold and 'pseudo-rotations', which 'rotate' a space coordinate into the time coordinate. A 'pseudo-rotation' has some mathematical similarities with a rotation in a Euclidean space (therefore 'pseudo-rotation'), but it is definitely not the same. I cannot believe that any theoretical physicist would have refused to acknowledge this fact. 

... because a coordinate rotation requires a SINGLE origin around which the rotation of coordinates occurs.

This is not quite correct: you can apply a 'coordinate rotation' to two coordinate systems with different origins O and O' at the same time. For example, you may apply a Lorentz transformation that equals a rotation around O in a 3-dimensional Euclidean subspace of the 4D manifold. This transformation keeps O fixed, but moves O'.

A coordinate rotation from (x',y',z',w') into (x.y.z.w) demands that when x'^2+y'^2+z'^2+w'^2=0, these coordinates MUST be the coordinates of the origin around which the rotation occurs, and which are given by x'=y'=z'=w'=0.

If you write x'^2+y'^2+z'^2+w'^2, then I understand that you define x,y,z,w as coordinates with respect to an origin O' of a coordinate system in an Euclidean 4-dimensional space, which means that x'=y'=z'=0 denotes O'. However, there is no 'MUST' concerning rotations. You can choose any other point to perform a rotation around that point. 

Now if for r' not zero r'^2=x'^2+y'^2+z'^2-w'^2 (NOTE NOW THE MINUS SIGN) then it is OBVIOUS that one has that r'^2+w'^2=x'^2+y'^2+z'^2 and that therefore s'^2=r"^2+w'^2 is the distance from the origin in a 3D manifold: Since when s'=0 one has that x'=y'=z'=0, which gives the coordinates of an origin within a 3D manifold.

Let me treat only the case r'^2=0. Then your equations read w'^2=x'^2+y'^2+z'^2 and s'^2=w'^2. In the second equation you have simply renamed the variable w' into s'. It does not make s' a distance. To use your own words: calling a cat a dog does not make a cat a dog. In Minkowski space the interpretation of the first equation is: the (absolute) value of the time coordinate equals the value of the distance (not: 'is the distance') in 3D from the origin. This describes the kinematics of a massless particle passing through the origin.

Furthermore, the Lorentz transformation involves TWO 3D origins O' and O moving with a speed v relative to one another. So how is it possible that with the help of time they can form a SINGLE 4D origin? 

Nobody claims that this is possible (cf. above.) If you apply a Lorentz transformation to a given coordinate system, then O and O' (trivially) coincide at the instant when you apply the transformation. But this does not mean that you cannot apply the same Lorentz transformation to more than one coordinate system.

In order to believe this one must believe that a single instantantaneous-event in space can occur simultaneously at two or more different instants in time. This is obviously absurd since "simultaneous" has ONLY ONE meaning and that is "the same instant in time which means " the same time."

As you certainly know, the notion of 'simultaneity' is central to SR. For good reasons SR uses this word with great care. Your understanding of 'simultaneous' simply ignores the problems in the concept of simultaneity. You will never understand SR, as long as you do not understand what simultaneity means in relativistic physics. There are excellent text books on this subject, and last not least Einstein's original paper. Additional remark: SR is not about some diffuse notion of 'time'. It is about what clocks measure. In other words: time in SR is defined by clocks.

Now is it possible that O' can be a distance D'=vt' away from O while O is simultaneously another distance D=vt from O'. It is obviously absurd. One must have that simultaneously D'=D and this means that t'=t; and that the clocks MUST keep time at the same rate!!! Where is your "time-dilation" now?

Here again you are using the word 'simultaneously' without being aware of what it means. If you do not specify how you understand 'simultaneously', then the equation D'=D does not make any sense. 

And since the event at O' is moving away from to O, the time rate is recorded to be slower owing to the Doppler shift in phase-time.

This is an unsubstantiated claim. How do you want to prove it? The Doppler Effect does not enter into the derivation of the Lorentz transformation, which allows uniquely calculating the time rate in a system moving relative to an observer.

Minkowski-space is based on the impossible absurd assumption that a SINGLE instantaneous event can occur at different simultaneous times LOL!

Minkowski would never have told you such nonsense, if you would have asked. The 'LOL!' at the end is your real problem. You have a preconceived opinion and you defend this opinion by all means. You obviously refuse to learn anything about SR and laugh at those who could have helped you to understand SR.

Johan:

I think I have already fully answered your last post with these two sentences of my previous post:

You have a preconceived opinion and you defend this opinion by all means. You obviously refuse to learn anything about SR and laugh at those who could have helped you to understand SR.

Yes, I have heard about Dingle. A very sad story.

And that is all, what you have heard about Dingle?

Cheers

Sergey, if you want to know more about Dingle, you may have a look into http://en.wikipedia.org/wiki/Herbert_Dingle and the references therein.

No, Johan, not all over again! You have had your chance to explain in what kind of logic you believe.