Dear Christian, accurate as always! 

Indeed, one of the papers quoted by the reviewer states that the object is perceived not contracted, but rotated as you said.  However, every one the other quoted papers says something different...  

Concerning Einstein, to be honest, I prefer to trust him, than anyone else...   

In fact it's not rotated-a Lorentz boost isn't a rotation. Quotations don't matter, calculations do. And the statement about the sphere, or any object, is that , once more, it doesn't make sense. The only statements that make sense refer to Lorentz invariant quantities and these are local-they are computed in any given frame, but are independent of the frame. 

Since x^2 + y^2 + z^2 isn't a Lorentz invariant quantity, i.e. depends on the frame, any statement about it doesn't make sense, outside a given frame. So different observers will reach different conclusions, precisely for that reason. But the differences aren't meaningful, precisely, because they aren't related with Lorentz invariant quantities.

However, dear Stam, 

when I (hypothetically) watch a sphere moving at light speed, I do not care of Lorentz invariant quantities, I just watch an object with my eyes, by my standpoint.  Also Einstein makes statements about the object by an observer's standpoint (in the quotation he clearly states that: he says the object sqeezes along one of its dimensions).  How do I, simple observer unaware of Lorentz invariants, see the sphere? squeezed, rotated, unchanged, smaller, of what?  

The question is less stupid than it seems, because I, observer, may be totally unaware of my speed: if the object truly squeezes when I watch it, I could understand his relative speed (compared with mine).  Theoretically, if I watch a moving sphere that resembles a long cigar, I could understand that I''m going at speed light, while the sphere is not moving.  

Dear Dr. Arturo Tozzi,

No people can tell you a valid result for this.

Very unfortunately, not only Einstein’s theory of relativity is just a faked story or pseudoscience, but also the so-called Standard Model should result in pseudoscience if we cannot rightly treat it. May you see:

https://www.researchgate.net/publication/317014509_Anti-ethics_and_pseudoscience_On_Albert_Einstein%27s_theory_of_relativity

I am spreading this article. If anyone should help to this spreading, I should thank him/her very much.

Data Anti-ethics and pseudoscience: On Albert Einstein's theory o...

Please Yin, stop it.  

I just asked a serious question that troubles me, I do not want to listen pseudosceintific claims at all.  If you want to make promotion of your ideas, try elsewhere, but not here.   

Dear Dr. Arturo Tozzi,

You have to discuss and argue the theory of relativity with scientific standard.

As you asked a question about this theory, you have to know what it is first. If you think the theory of relativity cannot be argued with scientific standard, it just show that it is pseudoscience. 

Dear Christian, 

thanks a lot! the paper you suggested is amazing! Therefore, Einstein was right...

Concerning my example, take the Figure 14a (attached below): if you are an observer and you do not know your speed, you may take for real what you see in the landscape... 

Ray tracing optics in relativity is a fascinating subject. I remember this result from college.  The boosted description of the sphere is shortened but the ray tracing gives one an image of an unchanged sphere.  I didn't ever do the calculation myself but remember reading it.  

Dear Cliff, thanks a lot!  Look just  at the pictures from the intriguing paper suggested by Christian: it seems that the sphere's contour gets unchanged, while other geometrical figures are badly deformed.  

The statement about watching a sphere means that one does care about Lorentz invariant quantities-the statement ``watching a sphere'' doesn't make sense otherwise. It's not recognizing this relation that is the source of all such questions on this and other fora. 

In fact the only difference between the non-relativistic limit and the relativistic case is that, in the former, a rigid body is a well-defined object, in the latter it isn't. However this doesn't mean that there is any invariant quantity that could describe the deformation of the sphere-one can't obtain any such effect by a change of coordinates, that corresponds to a choice of an inertial frame; and the pseudoforces one would measure by using a non-inertial frame don't do any work, since they're perpendicular to the velocity, so can't induce a deformation, either-that's the short answer. 

Dear Stam, 

very clear.  Thanks a lot.  However, even if the deformation, of course, does not truly occur in the moving object, the observer perceives the sqeezing (it is a clear Einstein's statement).  The picture above (the apparent tunnel in the street) seems to me very useful, in order to understand the perception of an observer totally unaware of Lorentz transformations, invariant quantities and Einstein.     

No, the observer doesn't ``perceive'' anything of the kind,  precisely for the reasons given. Nor does it make sense to say that the observer is ``unaware'' of Lorentz transformations-the consistent description of nature doesn't require any awareness on the part of anyone. If an observer doesn't use the appropriate description, (s)he is going to get a  meaningless-result, that's all. And that the result is meaningless can be given meaning by checking whether it is or not invariant under Lorentz transformations.  In the case at hand the statement about the sphere being squeezed can be checked to be meaningless, precisely because it can't be associated to any Lorentz invariant quantity. 

The standard exercise, that's in all textbooks on electromagnetism, which shows how this question can be formulated properly, is the calculation of the electric and magnetic fields of a moving spherically symmetric charge distribution-e.g. 

https://www.u-cursos.cl/usuario/6c35e35ec55a71af969f59a168a300a7/mi_blog/r/Electricity_and_Magnetism_-_Purcell_01_-_100_-_ConiF.pdf chapter 5.

or http://www.feynmanlectures.caltech.edu/II_26.html chapter 26. 

The fields, however, are not Lorentz scalars-they're components of a Lorentz tensor. So the Lorentz invariant quantities must be deduced, once more. This isn't stressed as strongly as it should be, unfortunately.

Therefore, Eistein's phrase is wrong...

Dear Stam, page 807 of your (marvelous, thanks!) book: 

The famous Lorentz contraction. Loosely stated, lengths between

fixed points in F^1, if parallel to the relative velocity of the frames, are

judged by observers in F to be shorter by the factor 1/γ . This statement

remains true if F and F^1 are interchanged. Lengths perpendicular to the

relative velocity measure the same in the two frames.  See the Figure in attachment.  I'm sorry, but Eistein was right, also according to your book.  

We often forget that special relativity allows the same deformation of coordinates and GR.  Even more often we forget that classical mechanics allows the same sort of freedom.  One still has problems in CM with what you see rather than what you use as an "instantaneous" description.  If one thinks in terms of acoustic images instead of ones made by light the effects become more dramatic.  Of course there is no Lorentz contraction but your images are affected and distorted! 

One can consider coordinates that mix space and time even in classical mechanics.  Einstein was so concerned how to make sense of things in a global sense he gave very explicit instructions on how to build a set of synchronized rods and clocks.  Special invariant global frames are very handy and define our notion of what "is" vs what we see.  The reality is only defined by the equivalence class of all such constructions.  If you want to use "light front" coordinates you are closer to what will describe the images you will actually see. 

It's not sufficient to read something-understanding it is, also, necessary. There exist quantities that transform in a certain way under Lorentz transformations. And there exist quantities that remain invariant. The latter are constructed from the former. The latter only matter. And it doesn't matter what Einstein himself, or anyone else, claimed or is thought to have claimed-the mathematics is, now, trivial. 

Dear Stam, I'm not performing a delicate experiment of sophisticated astrophysics, where I need to take into account just the invariant quantities (for example, the rest mass in the case of special relativity), but I'm simply looking with my eyes at a moving object in the sky, and I need to assess also the not-invariant quantities.   

I do not understand, Stam...  EINSTEIN himself, and the same book YOU suggested me, state clearly that an observer sees a squeezed moving object, and you insist that it is not true.  Therefore, Einstein and your influential book say bollocks.  Difficult to digest...  

   Dear Arturo,

   The picture of Christian is very easy to understand if you consider that the space coordinates transform in an anisotropic form because the coordinate parallel to the velocity is the only one to contract while the others two are kept invariant. Let us to choose the x direction as the one of the inertial motion

                      X’=γ (x-vt)

                      Y’= y

                      Z’=z

   And obviously if you had initially the buildings orthogonal, you could know your relative velocity just knowing the angle of inclination.

   Another thing very different is that the Lorentz transformations are just rotations and pseudo-rotations belonging to the special orthogonal group SO(3,1).

Dear Baldomir, the above Figure (from the book Stam kindly sent me) states the same wise concepts you clearly explained.  Therefore, does the observer see the fast object (say a bar, not a sphere, that is more complicated) unvaried or squeezed along (just) one of its axes (the one parallel to the relative velocity) that contracts? 

Dear Dr. Tozzi,

The recent work of Rau, Weiskopf and Ruder presents results obtained using the elementary geometrical methods of relativistic polygon rendering

http://www.vis.uni-stuttgart.de/~weiskopf/publications/vismath97.pdf

The images of a moving sphere (with parameter \beta=0.95) are shown in figure 3. They look elongated. I hope that this result is useful in resolving the debate described in your question.

Kind Regards,

J. Pudykiewicz

Dear dr. Pudyiewicz, thanks a lot for the PRECIOUS material.  

Amazing!

The paper is exactly what I was looking for, together with the other material kindly suggested by the all of you.  I think my paper will be rejected the same, but somebody has to say to such alleged "experts" that they are wrong!  

    Dear Arturo,

    It is simple. The only coordinate that changes is the one parallel to the velocity and one obverver sees

        X’=γ (x-vt)

    and the another

       x=γ (X'+vt)

    for one observer there is contraction ,while for the another there is expansion of the coordinate. But if you calculate the length you can see very straightforwardly that it always contract for both of them. That is a simple exercice for beginners in relativity that you can find in any text.

Dear Daniel,

thanks a lot also to you!   It is uncanny and unclear why the reviewer made such an assumption.     

Because the only quantity that matters isn't any particular coordinate, that's why. It's completely pointless to discuss the meaning of quantities that are not invariant under Lorentz transformations. Under a Lorentz boost x isn't the only coordinate that changes, t, changes too. But these changes only serve to show that x^2-(ct)^2 = (x')^2-(ct')^2. And, similarly, the electric and magnetic fields of moving charges only serve in determining the Lorentz invariant combinations thereof. 

So it doesn't make sense to ask how two points on a spatial sphere are transformed under Lorentz transformations, in general,  because the spatial sphere isn't an invariant object. Something more is needed to define the problem.

Points that belong to a spatial sphere in one frame belong to something else in another frame. The shape is completely determined-but it's a fiction of the transformation, since it's not invariant under Lorentz transformations. So, instead of wasting time on issues that are meaningless, it would be better to learn the issues that are-namely what are the Lorentz invariant quantities for the specific problem that one is interested in. Asking about a space-like sphere isn't enough-that's what Lorentz invariance implies. 

Dear Arturo,

In the figure attached to your question, Einstein wrote: >.

Moreover, in the second paragraph of the introduction to document "J. Terrell. Invisibility of the Lorentz contraction", it is written that the difference between by a reference frame or a coordinate system and by a camera or a human eye must be distinguished.

I do not think we should attach too much importance to single words, but rather to a set of words and context. Words are not natural, they are invented to express ideas. 

If a reference frame is a three-dimensional Euclidean space (cartesian and rectangular coordinate systems can be defined here) to which we associate a temporal real variable qualified as natural, then "observing" an object means considering all the spatial positions it occupies at a given moment in the reference frame, while "seeing" an object is to consider its form as it appears on a photograph (the result depends necessarily on the position of the camera that will not see the whole object ). 

For example, if in the reference frame two material points move on the same half-line at the end of which there is a camera, then the camera will only see one material point while the reference frame observes two material points.

The transformation of Lorentz and the mathematical contractions concern what is observed in the reference frame. And after determining what is observed, it is necessary to make calculations to predict how this observation will be seen by a particular camera.

And to predict how a given camera will see an object that has already undergone a Lorentz contraction, it must be taken into account that the rays of the lights that form the photograph were not necessarily emitted at the same time by all the parts (visible by the camera) of this contracted object otherwise they would not arrive at the same moment on the photograph. And when the contracted object is moving, the fact that these visible parts are not illuminated at the same moment modifies its apparent form, which is the result of the non-simultaneous illumination of the different parts.

Dear Rommel, if, in theory of course, an observer sees in the sky an object moving at speed light, it is to much squeezed that the ligth rays are emitted almost from the same part of the object.  Furthermore, another issue: how could be sure that an object at speed light could have illuminated parts? It could be illuminated just by light rays which display its same.speed...

Dear Stam, 

my question is less stupid than you may imagine.  There is a thought beyond.  

Eistein's special relativity is based on invariant and non-invariant quantities (time and the axis parallel to the moving object), the latter being quantities that may change according to the subjective standopint of the observer. While physicists (like you) dispise the non-invariant quantities and focus their accurate calculations on the "hard" data coming from the invariant frame, we proposed that it could be the opposite for mental functions. Indeed, our (TESTABLE!) theory depicts the individual's brain as able to assess the non-invariant quantities of a relativistic-like inertial frame, giving rise to the astonishing differences in mental activity experienced by various individuals.

For further details about our model of special relativity applied to the brain:  

https://www.researchgate.net/publication/315113848_QUANTIFYING_MENTAL_IDEAS_THROUGH_SPECIAL_RELATIVITY_AND_BEKENSTEIN-HAWKING_FORMULAS

I know, it is speculative, but it is testable, and, I believe, is more stimulating to try to find novel (TESTABLE!) bridges, instead of simply apply special relativity more, and more, and more....

Preprint QUANTIFYING MENTAL IDEAS THROUGH SPECIAL RELATIVITY AND BEKE...

Once more:  invariant quantities are constructed from non-invariant quantities. And it doesn't make sense to discuss non-invariant quantities by themselves. The statement about mental functions is simply meaningless, since, precisely any testable prediction only can refer to invariant quantities. 

That people can construct models that are meaningless is true-and not, only, about special relativity. But that's (a) nothing new and (b) a waste of time, since it's understood what are the meaningful questions that can be addressed in special relativity-and that questions about space-like objects alone aren't among them. 

Therefore, dear Stam, summarizing your thoughts:

a) You are right, Einstein is wrong.

b) I'm meaningless.  

Every reader may easily draw his own own conclusions, at least concerning the point a).

'nuff said.   

Arturo,

since the topic of noninvariant quantities come up, it is important to emphasize that any give measurement device of collection of observers define a special system in the universe.  It is much like discussing light in a dielectric.  A solid body (to the extent it can exist in relativity) has a preferred frame.  Once can, of course, describe it in invariant terms but the dynamics will be most simple in the frame of the object.  

In rheology we often talk about "objectivity".  This is a classical hyrdo analog of relativity.  I suspect that you are looking at noninvariants for a reason.  Even when there is some microscopic invariance, the particular global situation you have in mind may make some of these quantities of special important interest.  Technically, the distance from Maine to Florida is ambiguous because it is not the same in all frames.  However, we all recognize that we are pretty much all living in the same one.  People talk about the "age" of the universe.  How can they do that if we can only give invariant distances between "events"?  It is because there is enough uniformity to the universe's behavior to give a number that works for almost everyone.

Dear Clifford, 

Your observation is perfect when you talk about physics, in particular about the rather linear cosmic evolution or planet orbits.  However, if you start to move towards biology, the things start to become more complicated.  The invariances start to be locally broken, and the non-invariant quantities become more relevant.  To make an example, a biological niche, or our brain, locally modify the rather stable cosmic conditions, leading to tiny zones of anisotropy and non-ergodicity.  Therefore, the schematizations of the rigid physicians does not allow to achieve in biology accurate calculations of the subtending invariant quantities. Two electrons behave in the same way, but Trump and Putin do not.  If you assess Trump and Putin just in terms of invariant quantities, you cannot understand how they move.

Dear Stam,

Perhaps determining what you name invariant quantities may sometimes require modeling based on basically relative measures. For example, the Doppler effect (which is measurable and physically meaningful) is predicted by suggesting that the source has had the possibility of moving in the three-dimensional and immutable Euclidean space of the receiver between two transmissions of a signal.

I would have said that the Doppler effect is a covariant quantity but I do not seek a war of words since it is necessary to adopt the vocabulary of the majority to communicate.

Dear Rommel, of course, I fully agree with you.  

...not to talk about the measurements in quantum mechanics, of course...

"Invariant quantity" is a very weird concept, here, because of the non-commutativity of the Heisenberg's terms...

On the other side, it is not true that invariance is necessarily required, in order to study some physical phenomena.  

"Physicists discuss the time-reversal invariance of local and/or macroscopic descriptions of physical systems, independent of the invariance of the underlying microscopic physical laws. For example, Maxwell's equations with material absorption or Newtonian mechanics with friction are not time-reversal invariant at the macroscopic level where they are normally applied, even if they are invariant at the microscopic level; when one includes the atomic motions, the "lost" energy is translated into heat."

Quotation from: https://www.revolvy.com/topic/T-symmetry&uid=1575

That Maxwell's equations in macroscopic media aren't invariant under time reversal means, precisely, that the system isn't closed-the medium reacts, generates electrc and magnetic fields, and this has consequences about what are the effects that can be described-no problem. If a transformation isn't a symmetry of the physical system it's not very useful to look for quantities that transform in that particular way, that's all. But if it's known that certain transformations are symmetries of the physical system, then it doesn't make sense to study quantities that aren't invariant under them, by definition. And in the case at hand there's no way to give meaning, in a Lorentz invariant way, to the statement that the sphere is squeezed under a Lorentz boost. A Lorentz boost, since it relates two inertial frames, doesn't describe any additional forces that can do the work that would lead to the term ``squeezing of a sphere'' acquiring any meaning at all. And if one does compute Lorentz invariant quantities, one will find that they take the same value in both frames. If one computes Lorentz non-invariant quantities, they will, generically, be different-but that doesn't mean anything. Providing initial data that aren't Lorentz invariant means that the problem isn't uniquely defined and it's that ambiguity that's the result of the confusion. 

Regarding the Doppler effect: it's just the result of a Lorentz transformation, in which one is focusing on a particular component-the time-like component-of the frequency, wavevector 4-vector. With waves other than electromagnetic waves what matters is that the elastic medium, in which they propagate, defines a privileged frame, in which the medium is at rest-the effective Lorentz invariance of the wave equation breaks down when the medium's constituents-that can be described by their equations of motion, when the waves can, too,  can be resolved. For electromagnetic waves in vacuum this isn't possible (more precisely, the constituents, the photons, whose superpositions describe electromagnetic waves, aren't described by classical equations of motion, when the waves are) and that's why calling the corresponding effect the Doppler effect is, in fact, misleading. It can become, of course, meaningful, in a medium, that, then,  defines a rest frame, which means Lorentz invariance is broken. This doesn't imply, however, that ALL Lorentz non-invariant quantities become meaningful, since HOW the symmetry is broken does matter-and this is known.

One should refrain from mixing up history and physics. 

Science isn't personal-so it's meaningless to quote verbatim.  That some person claims something doesn't imply that the claim is true or false, because of the person-but relies on the content of the claim. And can and should be checked impersonally.

  Let me just add some few references concentrated in the measurement of the time dilatation on muons. I could give you also measurements of kaons or meny other particles in the modern LHC. I think that this shows clearly how no invariants can be fundamental and important in Physics to be measured

1.Rossi, B.; Greisen, K.; Stearns, J. C.; Froman, D. K.; Koontz, P. G. (1942). "Further Measurements of the Mesotron Lifetime". Physical Review. 61 , 675–679, (1942)

2.Rossi, B.; Nereson, N. "Further Measurements on the Disintegration Curve of Mesotrons". Physical Review. 64, 199–201,(1943)

3.Frisch, D. H.; Smith, J. H. (1963). "Measurement of the Relativistic Time Dilation Using μ-Mesons". American Journal of Physics. 31, 342–355, (1963)

4. Coan, Thomas; Liu, Tiankuan; Ye, Jingbo. "A Compact Apparatus for Muon Lifetime Measurement and Time Dilation Demonstration in the Undergraduate Laboratory". American Journal of Physics. 74, 161–164, (2006).

5.Durbin, R. P.; Loar, H. H.; Havens, W. W. . "The Lifetimes of the π+ and π− Mesons". Physical Review. 88, 179–183,(1952).

6.Eckhause, M.; Harris, R. J., Jr.; Shuler, W. B.; Siegel, R. T.; Welsh, R. E. . "Remeasurement of the π+ lifetime". Physics Letters. 19, 348–350,(1967).

7.Greenberg, A. J.; Ayres, D. S.; Cormack, A. M.; Kenney, R. W.; Caldwell, D. O.; Elings, V. B.; Hesse, W. P.; Morrison, R. J. . "Charged-Pion Lifetime and a Limit on a Fundamental Length". Physical Review Letters. 23. 1267–1270.(1969).

8.Ayres, D. S.; Cormack, A. M.; Greenberg, A. J.; Kenney, R. W.; Caldwell, D. O.; Elings, V. B.; Hesse, W. P.; Morrison, R. J.. "Measurements of the Lifetimes of Positive and Negative Pions". Physical Review D. 3 (5): 1051–1063. (1971).

9.Burrowes, H. C.; Caldwell, D. O.; Frisch, D. H.; Hill, D. A.; Ritson, D. M.; Schluter, R. A. (1959). "K-Meson-Nucleon Total Cross Sections from 0.6 to 2.0 Bev". Physical Review Letters. 2 , 117–119. (1959).

Said differently: Under a Lorentz boost, for instance, in the x-direction, x^2 + y^2 + z^2-(ct)^2 becomes (x')^2 + y^2 +z^2 -(ct')^2 (the two expressions are equal in value).  The spatial sections of both are spheres-spherical symmetry isn't broken by a Lorentz boost-which is what ``squeezing of a sphere'' means. Observations and measurements are always local-within a given frame.

And the reason, in general,  is obvious: a Lorentz boost preserves the Minkowski metric (by definition), whose spatial part is spherically symmetric: 

x^2 + y^2 + z^2 - (ct)^2 = (x')^2 + (y')^2 + (z')^2 - (ct')^2

The spatial sections in both frames are spherically symmetric.

So a sphere isn't perceived as squeezed in special relativity. 

    Dear Stam,

    You are right that

        ds2=gμν dxμdxν

is invariant under the Lorentz transformations, but what is wrong is to assume that

       ds2=gμν dxμdxν=0 

is one sphere. This is not at all one sphere and the motion of one observer is on an hyperboloid where the transformation of the coordinates depend strongly on the velocity (not only on its magnitude but also of its direction).

   In other words, the previous distance is one Lorentz scalar but each coordinate is a tensor of rank 1 which transforms following the Lorentz transformations as such one Lorentz tensor.

I didn't make the second statement, so the implication isn't valid. The only statement that matters, for the present discussion,  is that the spatial part of the Minkowski metric doesn't become anisotropic under Lorentz boosts.  

The radius of the sphere isn't invariant under Lorentz transformations, because it isn't a Lorentz scalar. So it doesn't mean anything, once more. But the quadratic form of the spatial part, if it's isotropic in the old coordinates, is isotropic in the new. And this is utterly trivial: x^2 + y^2 + z^2 - (ct)^2 = (x')^2 + (y')^2 + (z')^2 - (ct')^2; the coefficients of the spatial part on both sides are all equal. The metric is, always diag(1,1,1,-1) (or diag(1,-1,-1,-1) depending on convention) in any frame. Squeezing would mean that a Lorentz boost changes the three spatial 1's to  values, different from 1, but whose product is equal to unity-and a direct calculation shows this isn't the case.  

Arturo,

I am not knowledgeable about how you wish to use invariants in biology.  I wish I knew more.  Pressed for time like most people/parents.  It would be a nice service to write a short introduction to these ideas for the outsiders.  I'm prone to writing exhaustive mini-tomes.  Trying to learn to take my own advice.  

C

    Dear Stam,

    Minkowski metric obviously is kept invariant under the Lorentz group SO(3,1). That is the starting point for defining this Lie group. Obvious!

    But that is not the same that to say that when you move in one inertial system of velocity v respect to another, you see that the coordinates in the direction of the velocity are transformed by Lorentz transformations while the orthogonal components remain equal. This creates a distortion of the space as the photo of Christian has shown. This is all and please forget to employ Lorentz invariants as the only physical relativistic observable, please. Every tensor can be observed, for instance the electromagnetic field is a 2 tensor that can be measured and it is not at all one Lorentz invariant. In a rest system you would observe just electric field and in another in motion the magnetic field appears too as new components.

Dear Arturo, 

I have understood that you have alluded to the fact that it is often stated that Einstein conceived his interpretation of the Lorentz transformation by wondering what a person would see if he/she moved at the speed of light (Probably in relation to the three-dimensional physical space where Maxwell's theory can be formulated assuming isotropy of space).

I notice that a photograph can not always compensate for the mathematical contraction of Lorentz as suggested by the title of the article "Invisibility of the lorentz contraction by J. Terrell"

Indeed, if two material points move with the same constant velocity and on the same straight line in the observational reference frame of a camera, it is necessary to distinguish the situation where these objects move away of the camera and the situation where they get closer to the camera.

Indeed, the object furthest away from the camera will always emit its ray of light before that which is closer to the camera (so that the two rays meet on the camera):

1/ If these objects are moving away from the camera then the second object will move closer to the position where the first one emitted a ray of light before emitting its own ray of light. So these two objects will appear closer in the picture.

2/ If these objects are moving closer to the camera then the second object will move away from the position where the first one emitted a ray of light before emitting its own ray of light. So these two objects will appear more distant in the picture.

Thus the apparent diameter of an object depends not only on its speed, but also on the direction of this velocity yet the contraction of Lorentz does not depend on the direction of motion.

No distortion, once more: Any Lorentz boost preserves the Minkowski metric-not just its determinant. So, while x'≠x, what matters is that x^2+y^2+z^2 gets transformed into (x')^+y^2+z^2 and NOT to a^2 (x')^2 + y^2 +z^2 for a boost along x. So a sphere remains a sphere for all observers. Its radius seems to have changed, but this is an illusion, since the time it would have taken to measure it has also changed, in just the amount necessary to cancel the first illusion (which is the expression, in words, of the fact that all observers, in their respective frames, measure the same value for 

x^2 + y^2 + z^2 -(ct)^2 ). But what matters here is the statement that, if one observer describes a sphere, i.e. an object for which the spatial part of the metric is (1,1,1), then all observers, in their frames do, as well. SO(3) invariance isn't broken by a Lorentz boost to a subgroup thereof. The spatial and temporal components get mixed up, that's all. 

And for the umpteenth time: ONLY Lorentz invariant quantities-i.e. that transform as scalars under Lorentz transformations-are physical. Electric and magnetic fields, that aren't Lorentz invariant quantities,  are used to compute Lorentz invariants and make sense only in terms of the latter. The end result matters, not the intermediate results. The statement that electric and magnetic fields are measured is misleading, that's all-it implies that the Lorentz invariant quantities deduced from them can be easily constructed, once the fields are known. 

Hello,

It was wrong then and now. See my list, of facts undisputed: https://www.researchgate.net/post/What_are_the_worse_yet_enduring_misconceptions_about_mass_and_energy_in_special_relativity?_tpcectx=profile_questions

Cheers, Ed Gerck 

The problem of the apparent shape of a sphere moving with relativistic velocity was discussed by Lampa (1924) and later by Penrose (1959) and Terrel (1959). According to the commonly accepted opinion, based on Lampa-Penrose-Terrel, the relativistic moving sphere will be visible as a sphere. This fact was reinforced by the proofs provided in the answers of this thread.

Based on the notion of "Invisibility of the Lorentz contraction" it is not likely that the results based on the relativistic polygon rendering (presented in the paper which I have suggested few days ago) will be accepted. This paper still should be examined to find any potential problems leading to the ellipsoidal shapes presented in Fig. 3.

After accepting the Lampa-Penrose-Terrel theorem we can ask the subsequent question: What is the equation describing geodesics on the sphere moving with relativistic velocity? Only after answering this question we can reach the complete understanding of the effects of a relativistic motion of the sphere.

Some useful additional observation and references are provided in the paper by Y. Pierseaux "Einstein Spherical Waves versus Poincaré's Elipsoidal Waves" published in the Annales de la Fondation Louis de Broglie

http://aflb.ensmp.fr/AFLB-303/aflb303m378.pdf

Hello,

The problem of the apparent shape of ANY object moving at ANY apparent speed, relativistic or not, must be answered as "no change with speed" under penalty of invalidating physics. Acceleration, not speed, is the cause  of change. More details, and references, at https://www.researchgate.net/post/What_are_the_worse_yet_enduring_misconceptions_about_mass_and_energy_in_special_relativity   [accessed Jul 28, 2017].

Cheers,  Ed Gerck 

     The Lorentz coordinate transformations are

                      t'=γ (t-vx/c2)

                      X’=γ (x-vt)

                      Y’= y

                      Z’=z

    choosing a direction unitary vector of the relative velocity n=v/v, the Lorentz transformations can be written is the following form

                    t'=γ (t- nrv/c2)

                    r'=r+(γ-1) (nr)n-γtvn

    where it is forward to see that the coordinates present a spatial anisotropy respect to the direction of motion. This is what explains the figure of Christian where a photo appears deformed. In other more compact form the coordinates transform as Lorentz vectors

                xi= gijxj

      (I have introduced latin letters in the last indixes for simplicity, but i,j=0,1,2,3. This has obviously a Lorentz scalar associted

xixi=gij xixj

     which is a Lorentz invariant due to the invariance of the Lorentz metric within the Lorentz group SO(3,1).

Hi to all! in the meantime I was able to partially pubish my idea on a very good Journal (IF 13,8):

Article Einstein and the physics of the mind: Comment on "Physics of...