The victorious truther

I'd like to point out that in lieu of modern spacetime formalisms and mathematical knowledge it is also the case that even in a classical context that gravity should be treated as a force could be considered dubious. Versions of Newton-Cartan gravity exist which is basically a geometrical realization of classical gravitation or, if you are avoidant of such terminology, is a version of newtonian gravitation/Classical physics in which what an inertial reference frame is or what path they follow is found to be dependent on the local/global mass distributions. So if you placed an object at rest or give it some initial velocity it wouldn't remain at rest or follow merely a straight line but a curved one in the presence of some mass distribution while all local experiments would convince you as to it being an inertial reference. 

Do note that together with the curvature based formulation of General Relativity there is also a version of General Relativity known as tele-parallel gravity that would replicate most if not all of General Relativities predictions but seems to appear to be a force field. This is done by assuming that the curvature is absent from determining the interaction and rather its the torsion (how much your world line bends) in spacetime that determines how the gravitational interaction takes place. So, is gravity then a force once again?

As was showcased in the first paragraph in the classical situation perhaps this is all but arbitrary and what you are really looking for has less to do with force. Perhaps, it has more to do with what you consider to be inertial or non-inertial and how much you're willing to apply occam's razor. 

I've been, as of late, debating with rather fringe or crackpot physicists on ResearchGate and in the process of such an exchange i've found myself forced to be rather particular about my terminology. It was a debate regarding Special Relativity and the person in question made the obvious mistake of assuming that such a theory couldn't be extended to accelerating reference frames. I came to clarify to him that in Special Relativity you could only deal with reference frames that accelerate as a result of dynamical influences or forces as General Relativity would come into the case for accelerated motions without forces. In hindsight, it is relatively true this is the case in so much as General Relativity accounts for everything that Special Relativity could account for as well as much more. However, I came to later realize in some respect that, in a similar manner to Classical mechanics, a fictitious forces would arise in any non-inertial reference frame I extend Special Relativity to so therefore it should equally as much handle it as well.

Then it occurred to me; the definition of an inertial reference frame or the general intuition many have had through out our physics history, from Galileo to Newton and then to Einstein's statement of it in his 1905 paper, all seemed to implicitly assume that for constant velocity motion the fact of it being an inertial frame came whole sale. In an opposite manner when discussing accelerating frames these are usually thrown into the bin of non-inertial frames as well. However, there are cases that do not relegate this sort of strict intuitive dichotomy in that you can view accelerated inertial frames from a non-inertial reference frame (linearly or in rotational motion) and such that the people outside the non-inertial frame will claim that their laws of physics remained valid despite this relative acceleration you perceived. 

Its as if the definition of an inertial frame that includes not just the statement of the invariance of some collection of laws of physics but also of a constant velocity has exceptions that do not warrant such a strict adherence to only constant relative velocities. Even Newton proclaimed in a strangely familiar manner in Corollary 6 of his Principia that,

Quote

If bodies, any how moved among themselves, are urged in the direction of parallel lines by equal accelerative forces, they will all continue to move among themselves, after the same manner as if they had been urged by no such forces. 

Then the question arises as to why we prefer to distinguish two different frames of reference where one is part of a homogeneous influence of forces (non-inertial) and the other is inertial (perceived from the other frame to be accelerated) when the physics conspires to not have us distinguish them. It seems then that you can extend this argument to any magnitude of homogeneous forces or any dependency on direction/magnitude, so long as its homogeneous, such that any general kinematical motion could be further relegated also to being inertial. It seems to be fairly redundant and i'm curious if Newton or others ever attempted to specify why inertial reference frames were so relegated usually to only constant velocities if they even noticed these sorts of equivalent situations.

For Newton's own gravitational theory this relativization and treatment of 'gravitationally' accelerated motion can be performed creating a Classical geometrical version of Newtonian gravitation called Newton-Cartan gravity. Entirely from arguments based on the above corollary along with assumptions regarding the WEP and SEP (Weak and strong gravitational equivalence principles). You don't have to see it as the inclusion then of a warped spacetime but merely as a redefinition and generality of inertial frames that responds to corollary 6 by making all such equivalent frames a part of the definition of being inertial.  

This is a curiosity to my Classical intuitions as they have always then betrayed me into thinking about constant velocity inertial motion as a strict given when these exceptions screamed for Occam's razor to be applied to them. Perhaps an approach that emphasizes the relativity of inertial frames in many cases, as is the case here, would be more amenable in certain teaching settings for a gentler transition from Special Relativity into General Relativity which wouldn't be too distinct in core concepts when going from Newtonian gravitation to Newton-Cartan theory. 

The key point being that kinematical motions are somewhat of a byproduct of being inertial or non-inertial and may be sometimes necessary but not sufficient to make a distinction between them. 

What are your thoughts? 

I'm fairly unfamiliar with data processing and data analysis in astrophysics so i'm unsure how to address certain claims as to the veracity of astronomical measurements. The paper by a crank physicist named Pierre-Marie Robitaille got me thinking. What is a respectable signal to noise ratio in astrophysics and, in particular, why a ratio of less than 2:1 for the Wmap data could be then considered an accurate recording of the CMB? What could be considered a standard in astrophysics for a signal to noise ratio that would indicate an accurate reading? If you could supply some extra reading material at the undergraduate level i'd appreciate it. 

802260431_PierrePaper.pdf

So i've been arguing with someone on ResearchGate regarding Special Relativity and other caveats related to his own pet Aether theory. He is, as many of these non-mainstream critics of Special Relativity are, a big fan of specific interferometer results from the late nineteenth or early twentieth century as he supports his theory on the back of a re-analysis of previous results. He claims that any vacuum interferometer will give a null result and that interferometers with a medium for the light beams would then give a non-null result. Something that was analyzed in this paper here: https://arxiv.org/pdf/physics/0205070.pdf. I'm curious about what your perspectives on it are given the derivations from their initial assumptions seemed to check out. 

If you could find a resource(s) for Lorentz Violating experiments conducted in a medium i'd heavily appreciate that. I had found a few that I thought had a relation to this such as https://arxiv.org/abs/0706.2031v1. 

His own paper is given here: https://www.researchgate.net/publication/350770907_MM-Cahill

The key idea is that this analysis combines both Lorentz contraction and the idea of light having a different speed in a chosen medium of 

\[ V = \frac{c}{n} .\]

You basically then perform the same analysis to determine the time difference between a path in the direction of motion and the path of the interferometer arm perpendicular to that. Which comes out to be 

Compared to it without the Lorentz contraction taken into account 

I got the second equation wrong when I expanded it out.  It should look like that. 🙁

14 hours ago, Markus Hanke said:

Is your density distribution a continuous function, or are you dealing with a collection of point masses?
At first glance your system of equations looks ok, though I’m not immediately sure whether (3) is actually needed at all, as (1)+(2) should already imply (3) - I haven’t explicitly checked though. As for numerical methods, that’s not my area of expertise, so I can’t offer any suggestions; I’d say it would be very difficult to implement that in code without constraining the form of the density and velocity functions in some way. 

How will the user of the software input the functions? Will there be an interface where the user types the functions symbolically (in which case you need to implement a suitable parser), or do they go directly into the source code, or what did you have in mind?

Well, the density distribution could be discontinuous and the idea here is that there are so many innumerable masses i'm thinking you could model it as an evolving non-interacting continuum of sorts (aside from gravitational interactions). Sort of like how we have equations to deal with individual H2O molecules and perhaps also a handful of them but as you increase the number of them then at some point you could fairly well approximate it by throwing out the assumption of a discretized fluid. Then model the fluid continuously with the Naviar-Stokes equations. I'm hoping the same can be done with a density distribution of sorts. 

Further, at the moment i'm just thinking of directly doing it in the source code. 

I'm most importantly wondering if this system of PDE's correctly describe exactly what i'm trying to model. Simulation of matter moving around while not interacting with itself aside from being gravitationally attractive. No mass accumulation or other non-gravitational forces present. No mass sinks or sources. 

\[ \begin{bmatrix} 4 \\ 5 \end{bmatrix} + \]

Dear forum, 

So I was curious about methods through which to simulate gravitational interactions and while I know there are numerous methods through which to numerically deal with multiple individual bodies I decided to find one to deal with general mass distributions. The model in question having ignore their interactions through non-gravitational forces or add in any accumulation of mass (formation of planetesimals) but still account for the change in the density distribution as well as the distribution of velocities. 

Firstly the primary equation in question that would be used is gauss law of gravitation to calculate the current gravitational potential distribution from the density distribution. Then i'd need an equation to encapsulate the conservation of mass to help with designating what the next distribution of matter would be. Finally, i'll need one more to determine the next velocity distribution and then repeat the whole process. I sort of guessed at this and found this set of PDE's that would suit my interest; gathered from continuum mechanics. 

1) \( \nabla^{2} \phi = 4 \pi G \rho \)

2) \( \frac{\partial \vec{u}}{\partial t} + \vec{u} \cdot \nabla \vec{u} = - \nabla \phi \)

3) \( \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{u}) =0 \)

The first is gauss law for gravity then the second is basically newtons second law but with a total derivative with the mass canceled out because of the weak equivalence principle (also because of mass conservation it takes on that form). The third and final equation is basically just a statement of mass conservation. 

I attempted to go on through and write these out full sale to then later discretize them or apply some other finite difference scheme to put into some programmable format using scipy, python, numpy, etc. These forms are given below, 

1) \( \frac{\partial^{2} \phi}{\partial x^{2}} + \frac{\partial^{2} \phi}{\partial y^{2}} + \frac{\partial^{2} \phi}{\partial z^{2}} = 4 \pi G \rho \)

2) 

\[ \begin{bmatrix} \frac{\partial u_{x}}{\partial t} \\ \frac{\partial u_{y}}{\partial t} \\ \frac{\partial u_{z}}{\partial t} \end{bmatrix} + \begin{bmatrix} u_{x} \frac{\partial u_{x}}{\partial x} + u_{y} \frac{\partial u_{y}}{\partial x} + u_{z} \frac{\partial u_{z}}{\partial x} \\  u_{x} \frac{\partial u_{x}}{\partial y} + u_{y} \frac{\partial u_{y}}{\partial y} + u_{z} \frac{\partial u_{z}}{\partial y} \\  u_{x} \frac{\partial u_{x}}{\partial z} + u_{y} \frac{\partial u_{y}}{\partial z}  + u_{z} \frac{\partial u_{z}}{\partial z}  \end{bmatrix} + \begin{bmatrix} \frac{\partial \phi}{\partial x} \\ \frac{\partial \phi}{\partial y} \\ \frac{\partial \phi}{\partial z}  \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \\ 0 \end{bmatrix} \]

3) \( \frac{\partial \rho}{\partial t} + u_{x} \frac{\partial \rho}{\partial x} + u_{y} \frac{\partial \rho}{\partial y} + u_{z} \frac{\partial \rho}{\partial z} + \rho \frac{\partial u_{x}}{\partial x} + \rho \frac{\partial u_{y}}{\partial y} + \rho \frac{\partial u_{z}}{\partial z}= 0  \)

Any assistance in terms of implementing these or stable numerical methods would be highly appreciated. 

On 11/9/2020 at 4:27 PM, joigus said:

What brings about the decay of the muon is the W^- weak boson.

Are all decays in QM or QF mediated by the weak force? 

On 11/7/2020 at 3:22 AM, joigus said:

I'm not aware that muons are aware

Aware in the same sense that you sitting in a chair makes you and the chair 'aware' of each other. I should have used a word like 'interaction'. In Quantum Mechanics or Quantum Field theory what theoretical entities are mean't to bring about the decay of a muon? 

1 hour ago, swansont said:

1 hour ago, The victorious truther said:

how does a muon keep track of time internally?

Why does it have to? You are anthropomorphizing this.

In special relativity we assume that certain objects have their own respective proper time and i'm assuming this translates over to QM as well as QF. Is it arbitrary when it decays then and how long afterwards? If not, then how does it know when it should be more probable to decay versus when it should if it can't keep track of how long it has been existent? Does it not have its own sense of proper time in Quantum Field theory or Quantum Mechanics?

I'd be anthropomorphizing this if I said the muon must have a top hat and a pocket watch. 

On 8/9/2020 at 7:15 PM, MigL said:

By the same token, the victorious truther is proposing that muons decay, not according to their subjective time, but according to 'external' time/change. Does that mean a muon in an otherwise empty universe will not decay as there are no 'external' changes ???
( his proposal is easily falsified by the fact that time dilation extends a muon' decay time, but, is purely dependent on the muon's subjective speed )

The muon is not in the absence of changes or in this case if we assume its monadic properties are retained across time (via a physical instrument) it would be wrong to say it doesn't undergo or is absent from any change at all. Relational change (relative velocity), that is, not a change in its monadic properties (though how we would know its monadic properties didn't change is a tricky one). I'm not entirely saying that its relational changes cause it to decay but they do influence it (relative speed to other frames of reference). All of this is in the end is taking for granted the muons own clock and its proper time of which i'm not sure we've investigated. . . how does a muon keep track of time internally?

On 8/11/2020 at 2:54 AM, studiot said:

On 8/10/2020 at 12:41 PM, The victorious truther said:

As far as that interpretation goes but this depends on whether you identify spacetime as distinct from matter and its configurations (classical substantivalism), that it cannot exist in the absence of matter with its accompanying configurations (relationism), or that matter is fully identified with a specific spacetime region so an empty spacetime is basically a spacetime devoid of quantum field looking spacetime configurations (super-substantivalism).

You can even speculate whether it would mean anything for there to be spacetime without matter in GR without resorting to having to construct a new theory but newly analyze it again. In that arxiv article, while I cannot fully vouch for its veracity, presents the intriguing possibility that even in a universe thinly distributed with matter you could still get back a spacetime that was geometrically Minkowskian. We are assuming that spacetime could take on background inherent geometries independent of the mass distribution (as there isn't any supposed matter) in the empty spacetime. 

 

This may be your thread, but your latest post moves the goalposts.

Before you can discuss time in universes we are not in, you have to prove that the time in these universes is the same as time in out universe.

Difficult since we have not yet arrived at a definition of time in our universe.

Sorry. . . I was just playing a bit loose there with the terminology and positions from a sort of naive perspective. 

5 hours ago, John2020 said:

Prediction: Mass (m) will accelerate due to the inertial centrifugal force (no reaction will appear at the rest of the system), creating an accelerating change in CoM that will eventually accelerate the system (M+m) as a whole.

There is no centrifugal force when seen from your inertial frame of reference.  

On 10/18/2020 at 7:23 AM, John2020 said:

Because I addressed the rotational energies coming from the torque (nut) and counter torque (upon the screw), where both have opposing direction of rotation (conservation of angular momentum), thus they cancel each other.

At some point the whole thing is at rest then at a later time it's moving at a constant linear and angular velocity with respect to its center of mass. Thusly, there wasn't actually any conservation of angular momentum or energy because work was done (in this case both linear as well as rotational work). Energy was not in fact conserved if this bolt went from not moving to moving. 

26 minutes ago, John2020 said:

I am preparing a drawing to explain all these and we will make the analysis for three cases: rigid rope, semi-rigid and non-rigid rope.

If you are truly to do this correctly you CANNOT be ignorant to what frame of reference you are in whether inertial or in the rotating frame of reference. Further, fully analyze it from the inertial frame of reference then move into the non-inertial one with mathematics to back up your conclusions. 

On 10/18/2020 at 10:48 AM, John2020 said:

Then it has to be utilized in a rotating frame. Creation of an Euler force.

If it doesn't work when seen from an inertial frame of reference then it doesn't work at all.

On 10/18/2020 at 10:56 AM, John2020 said:

In an accelerating spinning chair will appear centrfugal and Euler forces. They should be possible to be  utilized.

The only force that can be utilized is the one that is actually accelerating you. Fictitious force only arise and are present in non-inertial frames of reference because mathematically we attempt to treat an accelerating frame of reference as if it actually is at rest so we have to make up other forces to give rise to the phenomenon we observe while remaining at rest in that non-inertial frame of reference. Those forces which do not disappear after we switch frames of reference from say non-inertial to inertial (centripetal force for example) are the only real forces that you can do anything with. 

On 9/25/2020 at 8:20 AM, michel123456 said:

Because he uses the laws of Relativity exactly in the same way. So B observes A aging less, and A observes B aging less. They are observing THE SAME THING.

If you make a spacetime diagram of this situation in which one twin remained inertial while the other had a device to measure that they flipped frames of reference (remember it's relativity of inertial frames of reference not relativity of NON-INERTIAL frames of reference) then with a instantaneous turn around the non-inertial twin would notice the other twin actually jump ahead in time in a way the other twin (inertial one) would not see. If you plot the diagram you'll notice the time loss as a missing triangular section if I recall.  

On 9/25/2020 at 10:27 AM, michel123456 said:

But A is also at rest. He sees (observes, measures) the Earth length contracted & the distance to the Earth is also contracted. A & B must be considered on the same ground. The situation is symmetric, reversible, how to say. Motion is relative: there is no Earth at rest & A travelling.

No, both WORLD LINES cannot be treated entirely as the same as the moving twin CHANGES frames of reference half way through his journey (his proper time). So it isn't symmetric just as if a car drives away and comes back we know it was the car that accelerated (changed inertial frames of reference) as it was not the person given they both had accelerometers. If they both remained inertial then they would never meet back up again and if they were both non-inertial then their accelerometers would have noticed the inclusion of some force fictitiously into each of their respective frames of reference but the person on the ground never noticed such inertial forces arise so the only conclusion is that the car was moving, ergo it WASN'T symmetric. Why is this so hard to understand when you literally can go into a parking lot and forget about special relativity to then show this to be the case. Remember that special relativity has to simplify down to classical physics in the low speed limit. 

Also, acceleration is not relative in the sense that velocity is or inertial frames of reference are just as in CLASSICAL PHYSICS or in GALILEAN SPACETIME. In special relativity (as in classical physics) it isn't relative whether you are ACCELERATING or are ROTATING. We can disagree the magnitude of said quantities but whether you are or are not accelerating is not frame dependent but can rather frame independent. 

@michel123456 Do you understand that the postulates of special relativity are:

1. All inertial frames of reference are to be treated as equivalent in that the laws of physics behave the same in any inertial frame of reference.

2. The speed of light as measured from any arbitrary inertial frame of reference in free space is constant. 

This would mean that if you had a ANY non-inertial motion then by definition we could detect it and know that we are in a non-inertial reference frame because these postulates wouldn't apply. Further, this would also mean the situation of one non-inertial observer compared to an inertial one (the twin paradox) would mean by these postulates that their world lines COULD NOT be treated symmetrically or in other terms equivalently as if the non-inertial reference was inertial. AGAIN, the postulates are not, 

\( 1^{*} \). All frames of reference have the laws of physics behave the same within them. 

 \( 2^{*} \). The speed of light as measured from any frame of reference in free space is constant. 

You seem to be under the delusion that the starred ones are what define special relativity when in relativity it's the un-starred ones. 

6 hours ago, joigus said:

And @The victorious truther, you misquoted me.

I must supremely apologize then. I must have missed some important context involved when I abruptly popped in here. 

6 hours ago, Eise said:

As space and time are relative, but spacetime is not, this is no problem at all.

Stated differently there is no relativity principle for non-inertial frames of reference in classical or relativistic physics. 

9 hours ago, joigus said:

I offer this thought, in all humility - if Special Relativity theory can generate the above 6 long, complicated pages of debate among highly intelligent people,  without their being able to reach any agreement about what it actually means, as appears to be the case -  could this be because the theory isn't right?

A theory such as special relativity or general relativity can be a successful well understood mathematical/theoretical and applied theory of how the world works. Yet we could still disagree on philosophical interpretations of the relationship of our naive realism and previous theories with their accompanying previous ontologies to this specific theory (as well as fully interpreting the theory itself). Take the discussion of proper mass and relativistic mass and ask the legitimate question of whether one/both/neither of these newer concepts in special relativity would match up intuitively/ontologically (philosophically) to the classical physics concept of inertia (mass). Is relativistic mass the relativistic counterpart to inertia or is this a false comparison? This is a legitimate question but it doesn't really come to much difficulty when it comes to actually applying said theory if you had a case to make that we applied it, expected a certain result but got the wrong one, but under a different interpretation the experimental results could be construed to be correct then i'd agree there is something contentious. . . i'll await for you to actually propose this. 

In the history and philosophy of physics the discussion of what forces are as well as whether we should even accept action-at-a-distance ones wasn't settled with Newtons Principia but this philosophical side discussion didn't stop us from applying successfully the mathematical results there in for generations. 

7 hours ago, michel123456 said:

1.I hope I have that much

You just said your biased. . . 

7 hours ago, michel123456 said:

2. Again, Relativity is OK to me as long it is considered as a Theory that describes (between others) a phenomena comparable to perspective: a kind of effect caused by the different states of motion of different observers, a Theory that explains how one observer can relate his observation to another observer.

Which is technically what the GALILEAN transformation already did in CLASSICAL physics but I do not see you screaming at the top of lungs about that and how it must be a perspective thing. Classical physics did the same thing of relating certain quantities in one frame of reference to that of another but they were no less real with respect to those frames of reference. 

7 hours ago, michel123456 said:

Taking the example of length contraction, it seems evident (to all of us I hope) that once the traveler stops, the resting observer does not observe length contraction anymore,  the phenomena has vanished.

Be careful with your language here. . . novice. . . when you say STOP it's implied you mean non-inertially accelerate or change reference frames so that they entered the specific one in question. Obviously if we go to classical physics in which a person who was moving with respect to our frame of reference then suddenly de-accelerate to enter our frame of reference. . . they aren't moving any more. . . cause they stopped with respect to our frame of reference. . . so if you ask whether the non-moving observer is moving then clearly no. . . because they aren't moving. To be length contracted or appear as such from other frames of reference the object in question would have to be moving. 

7 hours ago, michel123456 said:

In fact, in his own FOR, the traveler was never contracted at all: length contraction is an observational effect that appears from the FOR of reference (at rest). Exactly as the forearm presented in perspective: it does not change length. Is this doubting of Relativity?

Velocity is a real effect. . . but it's frame dependent on whether it arises and you won't see the same thing from every frame of reference nor could you claim that one particular velocity was more real or not or even declare that he is actually at rest when in other frames of reference he is moving. If this was purely a perceptual effect then we would think this wouldn't be accompanied by clear dynamical and kinematical issues involved with taking measurements. . . course you could mathematically reproduce this understanding with a classical theory of literal length contraction (as well as a counterpart one with no length contraction but finite speed of perceptual effects) and compare. . . to you know. . . show who is right. 

26 minutes ago, swansont said:

My mistake. The questions are still focusing on irrelevant details, given the fundamental misunderstanding of relativity.

That's fine. . . simple mistakes pail in comparison to those regarding our fundamentals. 

\[ \begin{bmatrix}a & b\\c & d\end{bmatrix}\]

7 minutes ago, swansont said:

The picture depicts the object (die) moving, and pictured at different times. Basically like snapshots put into the same picture.

In the description is declares that it's a row of dice moving. Not a single (die) pictured at different times while it was moving if I recall correctly. 

7 hours ago, michel123456 said:

Quest1. Why is the distance d2 larger than the distance d1? Shouldn't it be contracted?

How about you go through the mathematics and create an image plane with the relativistic object having its light happen to reach some focus then figure it out. It's difficult and perhaps not actually correct to analyze some image analysis as has been given as we do not know what the units were nor whether your naive perceptual decision on what is longer is truly correct. Should the image of a cube appear larger than it should in the way you claim or is this not what special relativity would predict optically? Remember, no "I think it would look like this," just do the mathematics or perhaps wait for some one competent in that respective to show it in a simplified situation.  

7 hours ago, michel123456 said:

Quest2. Why is the red distance smaller than the orange one? I am counting 4 intervals for the red one (4 squares) and 8 or more for the orange one. If the cube is moving at constant velocity, the 2 lines should be equal.

A simple imaging plane and a focus will construe the image like that as objects farther to the left will appear more scrunched together i'm assuming. . . we're talking about what you would see optically. In fact. . . just you wait as i'll perform the some mathematical investigation into this but derived via some simple vector mechanics with objects being focused unto an imaging plane with a focus. Then we can compare the images showing that those objects farther away parallel wise to the imaging plane do seem to have their distance scrunched up even if we happen to possess an equally dispersed series of lines in reality. 

7 hours ago, michel123456 said:

And I have to admit that I had in mind picture B. I was wrong again as it seems. But I still don't know why: in this picture B everything looks fine.

Glad you admit your fault and you must please understand that you should not use your intuition as if it's a judicial gavel of physics as even in CLASSICAL PHYSICS with optical imaging i'm willing to bet we could both make similar mistakes in thinking if we didn't actually do the proper mathematical preliminaries to derive how things would actually play out. I'll attempt to get back to you as soon as possible for the classical then the special relativistic case with all proper mathematical foundations. I'll be using equally spaced constant velocity rods then for both the classical as well as the relativistic to compare. 

1 hour ago, michel123456 said:

Nicely presented. Except that intuition would say that the guy who covered the long path through the corners would be older than the one going straight away, while the result of the twin paradox is the contrary.

You would think this if you didn't consult the spacetime interval in which \( s^{2} = (ct)^{2} - x^{2} = (c\tau)^{2}\). NOTE that the traveler happens to have a longer path through spacetime and together with the objectively long path they traveled in space some of that distance in SPACETIME that they happened to traverse has some of that temporal component eaten up by the distance they traveled. 

The spacetime interval for the person at rest is: 

\[ \tau = t_{person at rest} \]

For the twin who (because he was non-inertial) was objectively traveling a longer path through spacetime they started and ended the journey at the same spot yielding in terms of the rest frame time \( \tau \) or it took \( \tau \) time for them to leave then come back: 

\[  (c\tau)^{2} = (ct_{traveler})^{2} + x^{2} \]

They traveled objectively some certain distance \( x \) and using your knowledge of pythagorean theorem you notice that the length of the one side (the time by the traveler) couldn't exceed the length of the two others. Objectively given they traveled away there was no way the time of the traveler could in fact exceed or even equal that of the time given by the clock at rest if it must abide by special relativity and likewise the spacetime interval. 

\[ t_{traveler} \neq \tau \]

@joigus Did I do this right? 

14 minutes ago, md65536 said:

Michel has established that he's spent 20 years denying relativity. There are many, many hijacked threads over the decades, any that have certain keywords (in this case, "time, direction"), turned into 6+ pages of failed attempts to get him personally to accept something about SR. He's stated in this thread that he's not interested in relativity, and of all the hundreds and hundreds of answers to his repeated questions, not a single one of them he acknowledges as an answer to his questions. The only time I've seen any calculation or attempt to work through a problem, is when he's twisting his beliefs and made-up definitions into a nonsensical answer that confirms that relativity is wrong. Trying to understand relativity is destructive to his goals, and completely avoided. So the answer is yes, anything that shows relativity working will be ignored. If you're enjoying explaining it, that's good. If you're hoping Michel will learn something, ... I wouldn't.

Always bound to come across somebody like this on any forum. . .

12 hours ago, michel123456 said:

The analogy with length contraction is pure coincidence, I guess.

Or not. Are the laws of physics a single thing that enties everything ( the way we see things on a daily basis), or are the laws of perspective totally independent of Relativity? (as it seems to be the case at first sight).

And respectively, can Relativity ignore the fact that objects appear smaller as their distance from the observer increase? And not taking count of this effect when representing the distortion of objects that move at near to c velocity?

The arm does look smaller or larger depending on your perspective. . . but if you actually made a measurement and sent a light beam (or a radio signal or even used a measuring rod that was at rest with respect to your arm) you would happen to find the length had remained unchanged in BOTH situations. So. . . no length contraction. . . bad analogy. 

When you talk about the laws of perspective being independent of Relativity note the difference between what a camera or perhaps even a human being could potentially see and what kind of raw measurements we could make in which these sort of perceptual effects wouldn't come into it. You get the SAME PERCEPTUAL effects in CLASSICAL PHYSICS and we rightly so do not designate them as actual length changes but this is because of the specific collection of dynamical/kinematical laws we are assuming to then analyze this. The same is in special relativity in which length contraction is usually treated as the sort of frame dependent observation that is consistent with Lorentzian transformations while what you would see is (c) instead of (b) so you CANNOT go off of pure visual observations so to speak to find this contraction but you would measurably notice it in special relativity,

Not only that. . . are you just going to ignore the meat of my previous posts. . . anything to say. . . the laws of optics are not ignored in special relativity rather they are amended yielding the above image (c) rather than the classical optical image (e). If you desire to chalk it up to CLASSICAL OPTICS then please be my guest and explain how this can be the case that in special relativity we seem to measure/record lengths as being shorter when in reality they are not supposed to be but we measure them as. 

Further, there isn't entirely something wrong with the philosophical question of whether it's the dynamical laws that give rise to or are fundamental to the kinematical ones or vice versa as other philosophers in spacetime philosophy have claimed that dynamical symmetries must be symmetries of spacetime. Dissenters have argued that we should flip the arrow of explanation from the kinematically explaining dynamical laws (spacetime structure -> dynamical laws) to seeing them as rather fundamental (dynamical laws -> spacetime structure). The question then of whether the Lorentz length contraction is more real in one perspective or the other one isn't really a question that would get a wrong answer in this situation as if we emphasized dynamical considerations then the objects from other perspectives do contract seemingly (dynamical laws require us to measure them as shortened) or it's the spacetime structure that results in our. . . wait for it. . . length measurements to result in being shorter. In either situation it wouldn't be any less real or entirely more perceptual.