
It had to happen...: here is the loedel diagram for the rolling wheel and the pole flying by.
Thanks for scrutinizing. I only guessed position of events of chord and spoke per ground at t=0.866
Is this thread the continuation of the rolling wheel II thread? Why not continue the other thread?
Edit: spoke and chord per road corrected as per JT's post #306. Thanks, JT.
Edit 2: wheel t'=0.433
Last edited by VeeDee; 10042014 at 11:33 AM.
Thanks for that. So the rotational movement is totally irrelevant to the shape. Is this because the time dilation counteracts the length contraction differences?
According to the road frame, the velocities of points on the wheel are greater near the top than they are near the bottom. So, we would expect the wheel to be more lengthcontracted near the top than it is near the bottom. But that does not necessarily mean the shape must be a pear shape. It is possible for the atoms of the wheel to be more concentrated near the top, even if the shape is an ellipse. Using the correct equations of relativity, that turns out to be exactly what we find. In the figure below, the axle frame is depicted on the left, and the road frame is depicted on the right. Note the way all the lines curve upward and end up closer together near the top. This makes the wheel much more dense near the top than it is near the bottom:
Also, in post #257 I provided a pretty simple proof that the shape must be the same whether it is rotating or not. The proof is based on the idea that no part of the circular disk is ever located outside of the geometric circle represented by (r')² = (x')² + (y')² where r' is the radius of the circular disk, as measured by the axle frame. The proof is also based on the idea that there is never any space between the edge of the circular disk and the geometric circle represented by (r')² = (x')² + (y')² where r' is the radius of the circular disk, as measured by the axle frame. So, according to the road frame, the wheel must be the same shape as that geometric ellipse. Otherwise there would be a contradiction, a true paradox, a violation of causality, and all sort of other nasty things.
So is the counteracting effect due to the time dilation?
SYA: Well, can you maybe explain why you would say such a thing? That might help get to the bottom of this.
Sure. You said: SYA: Yes, I must have explained 50 times to cinci that the rotating cord atoms follow types of cycloids,...
Then I said: Besides, you don't use those velocities in Gron's analysis.
Then you said: Gron's analysis uses v, because the LTs use v.
Then I said: So all your talk about cycloid velocities is just hot air.
The you said: Well, can you maybe explain why you would say such a thing?
So if you want to get to the bottom of this, explain why you brought up cycloid velocities when Gron's analysis doesn't use them. We're all ears.
Only because you do not understand what Gron's analysis means, that's all. Since you have never understood why his theory is valid, we have all been trying to explain it to you in other ways, such that the light bulb might light up. Cycloids, was one such attempt. Anyone who understands relativity should recognize that your cord atoms all travel different cycloid paths, ie are of different speed and direction in the axle instant of time t'. This is to say, they all travel different paths thru spacetime, and they are not even parallel to one another. As such, there's no chance whatever that the Fitzgerald length contraction can apply to that cord, in the same way it applies to the inertial rod (the LTs apply though). No relativist expects the LCF to apply to your defined cord, but you do. The reason you do not understand, we all know. It's because you do not understand how length contraction and time dilation apply "in unison as a single mechanism". There are many who have this difficulty. One cannot grasp the complete meaning of RoS until they understand that. If you would allow JT to go thru a few simple all inertial scenarios, front to end, you would likely know in a month what you haven't come to learn in 30+ yr. However, you argue eternally instead, and forever change the subject when confused. You should take the bull by the horns!
I recommend you study spacetime diagrams, because that'll expedite your learning curve. It did for me
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SinceYouAsked
SYA: Only because you do not understand what Gron's analysis means, that's all.
cinci: In other words, you really don't understand the cycloid velocities and had no real reason for bringing them up...AKA hot air.
The rest of your comments are ridiculous in light of the fact that I have analyzed the wheel and got Gron's results. So if I did that without understanding anything as you claim, I must be a physics savant. Thanks for the compliment.
Ahhh, no. IOW, every relativist here understands what you have never, the Gron analysis. Anyone can tinker with equations, but understanding them is another thing. The Fitzgerald length contraction formula (LCF) applies to an axisofmotion that the atoms of the body are always on, and the atoms must all always move at the same unwavering speed. Since your cord atoms are on differing cycloid paths, and on differing circumferences of the rotating disk, they cannot be on the same axisofrotation or travel at like speed. Anyone who understands relativity should know the LCF cannot effectively apply to cinci's defined cord. So given you last post's response, answer these questions, just so everyone knows ...
1) Do you think the atoms of your y=y' cord at t'=0 possess the same speed, in any inertial system?
2) Do you think the atoms of your y=y' cord at t'=0 travel the same direction, in any inertial system?
3) Do you think your cord's atoms are all on the same axisofmotion?
4) Do you think your cord's atoms ever change speed as they go?
5) Do you think your cord's atoms ever change direction as they go?
Given your answers to the above 5 questions, please explain to everyone here how it is that you believe your LCF should work for your rotating cord, as it does for your defined inertial rod. Then, we'll know where you stand in the savant category.
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Last edited by SinceYouAsked; 10042014 at 04:14 AM.
Where would the cycloid velocities even be used? I suppose we could use them to approximate the rest lengths of small bits of the wheel. For example, take a really small bit of wheel near the top, where the cycloid velocity is 0.989c, and that bit should be contracted from its rest length by 1/7. So if the small bit you look at is 1 mm long in the road frame, its rest length is 7 mm. Is that what cinci wants us to do? Who knows?
SYA, your last post is you trying to recover from your nonsensical statements and previous errors about cycloids. Sorry, it's too late for that.
Even JT can't understand why you bring them up.
JT Where would the cycloid velocities even be used? I suppose we could use them to approximate the rest lengths of small bits of the wheel. For example, take a really small bit of wheel near the top, where the cycloid velocity is 0.989c, and that bit should be contracted from its rest length by 1/7. So if the small bit you look at is 1 mm long in the road frame, its rest length is 7 mm. Is that what cinci wants us to do? Who knows?
cinci: I'm not now nor was I ever the one who suggested using cycloid velocities. That nonsense is all SYA. Ask him what he thought he was doing. Now he's trying to weasel out of it and he has you fooled.
Not to mention, each atom of cinci's horizontal cord in the axle POV has a different length wrt x, even though the atoms are properround before relativistic effects alter that. He doesn't seem to realize that there are many more atoms in the disk cord than in his superposed inertial rod (endtoend wrt x). That's true per the axle POV as well. Also, he doesn't seem to realize how the rotating disk has higher atomic density nearer the top. The rolling disk is 1.0 wide (at y=y'=0.866) per ground because there are a lot more atoms wrt x at that height per ground, than the inertial rod has per ground. There are always the required number of (differently) contracted atoms at the y=0.866 height to make the rolling disk 1.0 wide, per ground ... and those atoms are not even the same atoms of the horizontal cord defined in the axle system at t'=0. For the rod, they are the same atoms.
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Hmm, I'm just not seeing it. If I paint a small smiley face on the rim of the wheel and say said smiley face is on the part of the rim in contact with the ground as the wheel rolls past the hedgehog. Isn't the smiley face stationary wrt the hedgehog at that point so it won't be distorted in any way? What I am trying to say is that I see whey the wheel will be contracted at the top but not at the bottom.
You are right. Atoms higher up on the wheel are more contracted, generally speaking, because they move faster. The atom at ground contact is virtually stationary wrt the ground, and as such uncontracted. Nonetheless, the wheel maintains a standard contracted elliptical shape. It can do so because the radial elements of the disk dynamically curve, per the ground POV. They are curved in such a way that there are much fewer atoms low on the rolling disk than (higher or) high on the rolling disk, generally speaking. By the Gron analysis, the atoms have no choice in that respect, because it's the nature of the relative POV that drives it. Said dynamic curving of radial elements per ground induce no forces unto the disk that don't exist per the axle POV. In fact, if they did not curved to recipe, then forces would be required per ground that don't exist per axle, which no valid theory would predict.
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So the very small element of the rim in contact with the ground is still the same shape as the nonrolling wheel? Or does the face get fatter?
cinci: Well since everybody is interested in cycloids and cyloid velocities, I'm pretty sure neither SYA nor JT or even VeeDee knows how to calculate cycloid velocities. Let's make that a question, do you? And since you "always" answer my questions, "I await with baited breath".
Let me answer the way you like answering questions:
Why do you need somebody who can calculate cycloid velocities? To transform the wheel from axle to road you don't need to calculate cycloid velocities.
Because we know the dimensions and shape of the wheel, chord and spokes in the axle frame and transform from there.
We can do it for the pole flying by (JT showed it gives correct results, but you apparently never understood it) . And we can do it for the rotating wheel. The problem is you still don't know how to do it.
The smiley face is "essentially undistorted", yes. Technically speaking though, given a single atom at rest is defined by (its field, say) a tiny spherical region of spacetime, only that which would be considered the ground contact point (of a rotating atom) is totally at rest with the ground and totally uncontracted. But that's a 0 dimensional point, that has no size. The other points of that atom's spherical field are rotating, because all atoms of the disk are always rotating ... that motion caused by the rotation does not vanish when considering the instant of ground contact, just as velocity doesn't vanish when we consider the instant of a body's position plot. However, for the sake of simplicity in trying to understand the case of rotating bodies at relativistic rate, one can say the atom at ground contact is essentially uncontracted. And yes, it would virtually measure the same as if the wheel were nonrotating at rest. The disk atom immediately above that (ground contact) atom is "almost at rest", but moves a little faster and thus a wee bit contracted. Next atom immediately up, same thing wrt the atom immediately below it, and so forth.
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At height y=0 there is only one event of the wheel the take into consideration: the contact ground point. That event will be he same event popping up in the road reference frame
But at height y= 0,0000000000000000000000000000000...1 the wheel has some length, i.o.w. There is more than one event of the wheel to take into consideration. From axle frame to road frame 'Relativity of Simultaneity of events' (RoS) will yield a different set of simultaneous wheel events in the road frame, which total length will be shorter than the sum of simultaneous events in the 'axle frame'.
As SYA has pointed out the ground contact point has zero dimension..., hence there is in fact no material at all that will not be contracted. But it's not uncommon in relativity to give some 'body' to an event. So, indeed, your ground contact smiley face will not be contracted.
VeeDee: Why do you need somebody who can calculate cycloid velocities?
cinci: Ask SYA. He's the one who thinks it's a big deal.
JT: Why don't you show us how to do it? Instead of asking us to do it for you? Einstein provides you with the equations in OEMB 1905. I await with baited breath your presentation.
cinci: Presentations for you are a waste of time. I presented a solution for the rolling wheel years ago, compared answers with you, and you still say I don't understand it.
Besides, I want to know how SYA thinks it has anything to do with Gron's rolling wheel solution. Don't you?
You are in no position to complain about wasted time, lol.
SYA already told you that Gron does not use the cycloid velocities because he does not have to. Simply put.
Gron is transforming axle coordinates to road coordinates using the LT equations. Is it possible that you still do not realise that the v in the LT equations represents the relative velocity between two inertial frames? The cycloids are not inertial frames.
I brought up cycloids because of a specific point, ie how the LTs apply to axes of motion and uniform velocity. That's all the LTs work for, in their own right, alone. Each atom on your cord of the rolling disk travels a (cycloid) path thru spacetime that differs with all the paths of all other atoms of your same defined cord, and their velocities all differ in both speed and direction. Not only that, the speed and direction of any specific disk atom on your rotating cord changes with time, and it's not even a uniform change. Those points I make are all that matters, and no analysis of the velocity of a rotating disk atom at a specific location is required, as anyone who knows anything about cycloids knows what I say to be true by mere inspection of a cycloid plot, and especially if it is animated (which it does not need to be). As such, the LCF cannot apply, because it requires uniform motion on a single axis of motion. Geesh.
Gron's analysis is effective because he runs the LTs between axle and ground, both inertial systems. He uses ω and sine/cosine functions to predict which atom is at the axle point x',y',t' and the LTs say where it must be in the ground system X,Y,T. He then focuses his soln on T=0, because he needs only the solns that are simultaneous in the ground system to determine its shape and size. Why does it work? Because an event such as a specific (say 1 red) disk atom at location x',y',t', must exist at X,Y,T per the LTs. You simply don't understand this, because you disbelieve that the LTs can transform the location of an entity that moves in the axle system, to the ground system in which it also moves. Put simply, you do not understand spacetime events (ie LT coordinate inputs and/or outputs).
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Last edited by SinceYouAsked; 10052014 at 09:30 PM.
Indeed. It follows a cycloid thru space over time. Would you figure it more correct to say it travels a helix thru spacetime, or that it exists as a helix in spacetime?
I would add ... Interestingly, we do no perceive spacetime (ie 4space), but rather only 3space over duration. The 4space must exist per SR, deducable by the theory, and validated by the measure of the individual relativistic effects. So far as perception goes, we can perceive only cycloids. Per the theory, it must travel the helix (or exist as such) in the 4space perspective that we are not (casually) privileged to, as the theory requires it.
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SinceYouAked
JT: SYA already told you that Gron does not use the cycloid velocities because he does not have to. Simply put.
cinci: I agree they are of no use in this context, so bringing them up was nonsense. Agreed?
JT: Gron is transforming axle coordinates to road coordinates using the LT equations. Is it possible that you still do not realise that the v in the LT equations represents the relative velocity between two inertial frames?
cinci: It's statements like this that show doing a "presentation", as you put it, is a waste of time. I did the wheel analysis.
JT: The cycloids are not inertial frames.
cinci: Neither are rolling wheels.
Your LCF calculation of the chord is of no use either, but you go on doing it for ages.
For the newcomers, here is how Cinci handles the wheel chord contraction.
The error is: his rest length calculation and final contraction calculation cannot be correct because the chord atoms are rotating, not moving linearly.Relativistic Rolling Wheel II
Here is the way I analyze the 60 degree chord with length contraction and the LTs. Find an error if you can and if you can't, then admit it.
Length of chord in axle frame = .5
Speed of chord relative to axle frame is Vch = .866*sin60 = .75
Rest length of chord is measured in (x", t").
Lch"(rest) = Lch'/(1  .75^2)^.5 = .755929
Or, Lch"(rest) = (Lch'  Vcht')/(1  Vch^2)^.5 = (.5  .75*0)/(1  .75^2)^.5 = .755929
Vch(relative to road) = (.75 + .866)/(1 + .75*.866) = .979695
Lch(road) = Lch"rest(1  Vch(relative to road)^2 )^.5 = .755929*(1  .979695)^2)^.5 = .151559
Cinci doesn't understand this, he doesn't want to understand this, he probably cannot understand this, and unortunately he will never understand this.
Either way seems fine to me. I still like to think of time flowing normally up the vertical axis, (but I am relatively new to Minkowski and Loedel diagrams).
If there were a movie camera at rest in the axle frame, it could film the rotating wheel. Put a paint spot on the wheel to make it easier to identify the rotations. Now develop the movie film, cut each frame into an individual picture, and stack them up in the correct temporal order.
And the same thing could be done with a camera at rest in the road frame, or any other inertial frame for that matter. So I was wrong to correct you on the shape, as it could be anything from a helix to a cycloid, or anything in between.
He was making a point about your pole method. Your pole looks a lot like the chord in the axle frame where everything lines up simultaneously for a zeroduration instant of time. But in the road frame, the pole only lines up with the chord over a nonzeroduration of time. During that nonzeroduration of time, points on the chord move along their cycloid paths, whereas the pole only moves horizontally. So the pole does not look like the chord at all in that frame.
Correct. And that is why we don't use the wheel's own frame. We use the axle frame, where we know exactly where each and every wheel atom is located at any given time.
I'm glad you brought this question up JT. When I free up a little more here, and after I think it through further, I'll start a new thread in relation to this. Basically regarding the relativistic effects, and the differences in our perception of 3space (x,y,z) versus time (ct). I would say it this way, personally ...
The rolling diskatom travels a cycloid thru space over time. It must travel a helix thru 4space, or at least exist as a helix in 4space. The helix is only deducable, and not directly perceptable, because of the difference in the way we perceive time vs 3space. By the theory, the helix must exist in the 4space (a theoretical prediction) for the (measurable) relativistic effects to arise as predicted. Basically, I'm interested in a thread regarding the nature of time itself, but strictly in regards to the requirements of SR and Minkowski spacetime, as opposed to an independent conjecture of time.
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SinceYouAsked
Last edited by SinceYouAsked; 10062014 at 11:32 PM.
VeeDee: The error is: his rest length calculation and final contraction calculation cannot be correct because the chord atoms are rotating, not moving linearly.
Cinci doesn't understand this, he doesn't want to understand this, he probably cannot understand this, and unortunately he will never understand this.
cinci: The calculation is correct for a rod which matches the chord's xdirection velocity and length. No it doesn't match Gron's analysis and it is not correct for the wheel. All of which I have acknowledged.
I understand Gron's analysis and did it years ago using length contraction and the time LT. You apparently can't understand how that is done but your inability to understand it doesn't make it wrong.
Apparently you think repeating over and over that I don't understand relativity improves your selfimage. So repeat it as often as you like if it makes you feel good; it makes me laugh so both of us get some good out of it.
SYA: But the rod's rest length calculation would be correct no matter if the xdirection related "velocity and length" were the same for the two bodies or not. So one must ask, what was your point here, if any?
cinci: The original point was to test the idea of building a wheel at speed. You need to work on your imagination and curiosity, not to mention your memory. This question has been answered for you numerous times.
No it wasn't. For the past year here, and for many years before that, every time you realized you found yourself backed in a corner regarding the operation of LTs and basic relativistic principles and effects, you then change the subject to (something the dialog was nothing about ...) regarding the addition of real material during an impossible roll up, or wonderous wheel building procedures for magic wheels at speed. Now then, back to the real problems here ...
The dialog was about these below. Cincirob does not understand why ...
1. LTs can be used to determine the length of Gron's disk, that moves in both systems. (Cinci thinks it must be at rest in 1 of the 2 systems).
2. The composition of velocities is not required to determine the length of Gron's rolling disk. (Cinci thinks Gron flawed because he didn't use it).
3. The Fitzgerald LCF does not relate the length of Gron's disk between axle and ground systems. (Cinci thinks it should work as his inertial rod).
OK, so each time cincirob realized he did not understand what everyone else was helping him with, he changed the subject to these unrelated matters ...
a. Addition of real material during an roll up to (an impossible) relativistic rate.
b. (wonderous) wheel building procedures for (magic) wheels at relativistic speed.
c. Determining the rest length of a rotating disk cord in it's own noninertial POV as stationary, which requires a noneuclidean spacetime system.
And yet, you just did that again here. Geesh ...
As many times before, I'd encourage you to stick to 1 > 3 until you master them first, learn about spacetime events, then switch gears to a > c later. Don't understand 1 > 3, then no chance of ever addressing or understanding a > c properly.
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Sounds like it would be an interesting thread.
I do have one issue with what you say there. The shape can be described as a helix, for lack of a better word. But it could also be considered to be a skewed helix. There is more than one way to draw a Minkowski diagram. If we draw the vertical axis as ct and the horizontal axis as x, (i.e., road frame coordinates), then the helix is skewed to a cycloidlike three dimensional shape. Since all inertial frames are equally valid, this is just as valid of a shape as the perfect helix that we get from drawing the vertical axis as ct' and the horizontal axis as x', (i.e., axle frame coordinates).
SYA: No it wasn't.
cinci: Yes, it was. It started years ago. When I realized it wouldn't work I began to wonder why. Since you have no curiosity, you didn't.
But that rod has nothing to do with the rotating set of chord atoms we are interested in.
You apply LCF to your coincident rod, but that would only be correct if the rod moves lineraly at that speed over time. There is no set of rod, nor chord atoms moving linearly at your rim xdirection velocity.
The chord atoms do not move lineraly, hence your coincident rod LCF calculation, rest length etc is wrong.
For the 'pole and firecracker' flying buy the LCF calc is correct, because it moves linearly.
How many time do we have to repeat this?
Never read a book about SR?
Don't you want to learn something about SR instead of fantasizing about it, or worse inventing your own SR bubble?
No it doesn't match Gron's analysis and it is not correct for the wheel. All of which I have acknowledged.
I understand Gron's analysis and did it years ago using length contraction and the time LT. You apparently can't understand how that is done but your inability to understand it doesn't make it wrong.
Apparently you think repeating over and over that I don't understand relativity improves your selfimage. So repeat it as often as you like if it makes you feel good; it makes me laugh so both of us get some good out of it.
However, before those "years ago", you had just as much difficulty with the SR basics as you do today here.
As stated many times, no one will help you with those more difficult issues, until you first learn some basic tenants of SR. For otherwise, the discussions could never get anywhere productive. For example, your rolling wheel threads here are a perfect example in that respect.
So no matter how many times you switch the subject from your misunderstandings of relativistic effects, and divert the discussions to wonder wheel building procedures or rest lengths of noninertial POVs, I will repeat the above. Everyone knows.
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Agreed. The skewed helix is just another form of a helix. A perfect helix is per the frame of the axle, given the stationary POV and the symmetry of the disk and uniformity of the rotation. I just refer to both "helix and skewed helix" as "helix", just to save on wording. Similarly, I refer to "cycloids and curtate cycloids" as "cycloids", just to expedite posting.
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That makes sense. I guess the most general thing we can say is that the shape of a worldline of a point on the wheel is some kind of helix, skewed or not.
We are lucky that the disk does not change its z' coordinate. If we had to include z' coordinates, we would not be able to talk about the shape of a worldline of a point on the wheel at all. It would have too many dimensions to make any sense.
Agreed, it would make things a little more difficult. However, it should be the same shape (helix) no matter what in the 4 dimensional continuum. One would just have to add coordinates for z' (and z) as well, to define it's plot. The LTs are designed for x aligned with the propagation axis, which simplifies the coordinates and thus the math. Don't define an axis aligned with propagation, things become much more complicated mathematically, yet the worldtube of the disk atom still exists in the 4d continuum just the same, either way. And, it has a shape independent of one's election of coordinates, a helix. That's my take on it.
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Imagine the entire wheel is very thick along the z' axis, with a small hole through the entire thickness of the wheel, (somewhere near the rim of the wheel). Let an ant walk through the hole, like a tunnel for the ant. Consider the worldline of a point on the ant, as plotted with x' and y' as the horizontal axes, and ct' as the vertical axis. That worldline would be a perfect helix, but it does not reflect the ant's walking motion. It is the same worldline that we would get if the ant did not walk at all. What is the shape of the worldline that includes the z' axis as well as the other three axes? It would have to be some kind of four dimensional shape, I would think.
True, because the zaxis is essentially not presented.
Agreed.
If z/z' is also presented, the shape is 4 dimensional, yes. That's hard the wrap one's head around. The helix, is a 4dimensional helix, not a 3 dimensional helix. This is why I referred to a worldtube prior. Instead of the worldline (of a point) that is a 1 dimensional line thru 4space, the ant occupies a 3d volume of space in any instant, so it exists as a 4 dimensional worldtube in 4 dimensional spacetime (4space). The worldtube is defined by the propagation of a 3d volume thru time, versus a 0dimensional point thru time (worldline). The 4d helix simply exists in the continuum, but since we perceive the passage of time, we perceive only a 3d volume in any instant of ever changing time ... ie we perceive only cross sectional 3d slices of the 4d continuum in an everchanging NOW.
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That makes perfect sense. So, by limiting the number of spacial dimensions to two, (x' and y' but no z'), we have the advantage of being able to work with worldlines instead of worldtubes. This gives us three dimensional shapes that we can wrap our heads around, like the helix.
But if you had to draw a Minkowski diagram for a point on the ant, how would you do it? I think I would replace y' with z' for one diagram, then replace x' with z' for another diagram, then look at all three diagrams at the same time. Is that how you would do it? Just curious...
Yes. It's like imagining the realm of a flat lander, a person who perceives only 2space + time person on a flat sheet of paper. He exists only on the surface of the sheet. That's all the dimensions he knows. Draw a wall across the middle of the page, and tell him he must pass. He cannot, because he'd have to go around, but the wall extends to opposite ends of the sheet of paper. Now assume there is a 4th dimension called height, that flat lander cannot perceive, but we can. We might say ... just go over the wall ... but that has no meaning at all to a flat lander who has no concept of height, so he could not. However, in real life, we envision a 3d sphere, but flatlander would consider it a 2d circular area encompassed by a circle. Try to tell flatlander what a sphere is, he cannot really know, because he cannot perceive the extra height dimension. He has to try and extrapolate that concept in the abstract ... as we do the 4d tesseract ... link > Tesseract  Wikipedia, the free encyclopedia
The problem stems from all axes being orthogonal to one another. We can call ct a spatial axis, but soon as we do, we now have 4 orthogonal space axes. I drew my 3space figure (to lay to rest all cincirob's arguments), and that easily conveyed 2space + time. No problem there because we are used to having 3 orthognal axes (length width height), and so it has meaning. However, add ct as a 4th space dimension, then we are like the flatlander who cannot wrap his head around the height of a wall, because height is beyond his perception. He must extrapolate that notion in the abstract, geometrically ... not easy.
Following your idea here, that would be fine. Putting those figures together in a single higher perspective figure would require what was done for the tesseract, and the tesseract doesn't present the full scope of 4d either though. I wonder if there are any genius around that can envision basic geometric objects in 4 space?
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Good point. I'd say that if numerous geniuses all come up with the same solns independently, then that would be worth something. Also, if it matched independent computer analyses, or was at least consistent with them, that may be good as well.
I have heard (or read somewhere before) that there exists individuals who presumedly have the ability to envision higher dimensional sacetime, but I would venture as in the sense a tesseract is envisioned (not a full 4d perception). Being that there exists a consensus on the tesseract, I would venture folks might extrapolate other shapes wrt 4d. In thought, one would figure we may be able to envision it, if one were sharp enough.
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Last edited by SinceYouAsked; 10092014 at 11:39 PM.
I've been trying, but so far, I cannot envision four orthogonal axes. But I did come up with this idea in the process:
If we keep the x', y', and z' axes orthogonal, and we let time 'flow' one moment at a time, then we have a three dimensional animation which can be viewed from any angle. This has all of the features of a four dimensional world tube, except the worldtube somehow shows all times at once, whereas our animation shows the times in sequence.
We can modify our three dimensional animation to show all times at once, but it requires allowing multiple instances of atoms to exist in the same space. That is unrealistic, but it is not too difficult to imagine. For example, in the axle frame, we could show the wheel at t'=0.000 with the ant tunnel at (say) the 12:00 o'clock position, and we can also show (superimposed) the wheel at t'=0.605 with the ant tunnel at the 1:00 o'clock position. The ant would be located at two different z' coordinates at those two different times, and both ants would be shown, (even though there is really only one ant). Theoretically, we could superimpose the continuous rotation of the wheel this way, and the ant would appear as a blurred helix in this spacetime. Note that a point on the wheel would only appear as a blurred circle, not a blurred helix like the ant.
If we do the same thing from the road frame, the point on the wheel is a blurred elongated cycloid, and the ant is a blurred elongated cycloid that changes z position. This method makes it easy to imagine worldtubes, but it is arguably not as mathematically useful as a four dimensional Minkowski diagram would be.
This should probably be in a thread of its own, as it's worthy of one. I'll need to think on that a wee bit further. We do depict a bodily progression upon an x vs y 2space figure (time implied), and it seems that you are basically adding in the 3rd spatial dimension, so an x vs y vs z (with time implied). No? Animating a 3space over duration, seems to be the same as direct experience. You are suggesting something different though, but I'm not quite sure what that is yet.
Imagine a flatlander in 2space. He exists on an x vs y plot. He cannot perceive a zaxis, because his entire perceptible cosmos is within only x & y. Yet, we know a zaxis exists, all 3 axes orthogonal to each other. Here's the important thing IMO ... the flatlander cannot imagine how a 3rd axis could be orthogonal to the other 2. Similarly, if we try to imagine a 4th spatial dimension (none are ct), we cannot envision how it is orthogonal to the other 3. The extra space must exist, "but also be inperceptible". For instance, the curled spaces of string theory are inperceptible.
Thank You,
SinceYouAsked
If you stick to the wheel you don't need 3D space + Time, because to keep it simple the wheel doesn't change its 2D shape in the z direction.
If you really want to consider 3D shapes changing shape ove time, then you will have a hard time showing the 4D shape on a 2D sheet of paper.
For the cube/ tesseract shown in box 4 here:
http://upload.wikimedia.org/wikipedi...levels.svg.png
the z dimension is plotted at a 45 degrees, and the 4th dimension at another 45 degrees.
This looks neat because the cube doesn't change shape over time.
But say the cube changes shape from cube to a sphere, then along that 4th dimension (45 degree direction) you have to show all those different shapes, and they will be superimposed if you choose small increments of time.
Then enters special relativity. For the events per another frame you have to '3D cut' that full 4D shape under a specific angle....
And then you plot that weird 3D shape on a 2D sheet of paper. Obviously you need quite a few 2D plots to look around that 3D shape.
Is this what you want to achieve?
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