I wanted to discuss Lorentzian length contraction (and time dilation, for that matter). Books on the subject do a fine job describing it but I've generally found that they lack an adequate explanation of it. What follows is my personal explanation. Special Relativity, in my opinion, is best explained in the following way: there is but one speed in the universe, c, at which all objects travel for a given (inertial) observer. In SR, though, "travel" occurs through spacetime, rather than space only, and one must consider the space- and time-vector component of such travel when making measurements. Time Dilation. In the picture below, the car is sitting in your driveway. It's spacial travel component, relative to you, is null; in other words, it's "travelling" through time along with you at a speed of c and there is no time dilation. Now your wife takes the car out to go shopping, and tears off down the road at a speed of .5c. Since we postulate that her "spacetime" speed is constant at c, and we know her "space-component" speed is .5c, we calculate that her "time-component" speed to be .86c because (.5c)^2+(.86c)^2=c^2 ...and indeed, SR calculates that your wife's watch would be ticking at 86% of yours as she speeds away. Length Contraction. In this description we're only concerned with treating dimensions as temporal or spatial but Lorentzian length contraction has a purely spacial analogy: Hold a the blue face of a Rubik's Cube squarely in front of your face and measure it with the ruler also squarely facing you. Now, turn the Rubik's Cube face partially away from you without moving the ruler and...it's length will APPEAR to contract. Turn it such that the blue face is completely to the side and its width appears to be zero. In fact, if we consider in this analogy the blue face to be the c invariant, the width dimension to be temporal, and the depth dimension to be spatial, SR makes the same predictions as shown below... Considering SR in this light, one could make the case that objects DO have an absolute length, that being their maximally-measured inertial length, and that any Lorentzian contraction is in fact an illusion. Thanks for your time and feedback. *8^)
Your diagrams are confused because they have the car's spatial length be parallel to the time axis--i.e. the front of the car seems to be at a later time than the back! If you're going to have a diagram with 1 space and 1 time dimension, better to imagine a 1-dimensional car parallel to the horizontal axis whose "back" is at the left and whose "front" is at the right. Alternately you could have a 3D diagram with two space dimensions, but either way you need a diagram showing the "world-tube" (analogue of a world-line for an object that's extended in space) of the car where a horizontal cross-section (space at a particular moment in time) shows the complete spatial extent of the car at that moment in time, both front and back. It would then be more obvious that disagreements about length have to do with the relativity of simultaneity and the fact that different frames slice spacetime into spatial cross-sections at different angles, which means they disagree about what a single cross-section of the car's world-tube looks like. And hopefully you agree that there is no absolute truth about simultaneity (and likewise no absolute truth about velocity, so there can be no objective fact about their "time-component speed" and whether it's zero or nonzero)
Yes, the graphs require some imagination. The spatial dimension isn't there to "show length" but rather "describe the constant-velocity components". They are also performing the double-duty of giving a visual aid which would have been more appropriately done in another graph. I tried to show that considering a Lorentz-contracted length to be valid is equivalent to considering the length-contracted width of a Rubik's Cube face turned at an angle. If this analogy is valid, it is always your prerogative to consider ALL Rubik's Cube apparent face widths to be on equal footing but I don't believe that would be popular sentiment.
But the analogy isn't valid as it depends entirely on the fact that you have misrepresented what "a car" looks like on a spacetime diagram, drawing it exactly the same as you would in an ordinary spatial diagram, and misrepresented its "length" as just somehow viewing a single ordinary car from an angle in space. If you had more accurate graphs, a better analogy would be if we had some solid 3D objects like cylinders, and different observers were disagreeing about the "width of a 2-dimensional horizontal cross-section" of the cylinders, but the reason they disagree is that they use different coordinate systems that define "horizontal" differently. Arguing for a true "width of a 2-dimensional horizontal cross-section" would require believing in a "true" definition of "horizontal", just like arguing for a "true" length in relativity would require believing in a true definition of simultaneity, a fact which is covered up by your distorted diagram.
JesseM, the car's length in the diagram is primarily representative of it's velocity "direction", where direction in this case is broken up into either spatial or temporal components. I was exploiting the fact, perhaps our of laziness, that the actual MEASURED length of the vehicle contracts at the same proportionality as it's temporal velocity component does. The validity of the analogy should stand independent from my sloppy use of graphs, unless you're telling me that you're actually unable to understand what I meant by them. Is that the case?
Yes, I have no idea what you meant since the whole concept of "viewing something at an angle" and seeing it foreshortened visually, which is the basis for both your graphs and the Rubik's Cube analogy, seems to have no real connection to length contraction.
rjbeery, I am curious, why did you put "True Length = Rest Length" in your title when you have no discussion of "True Length" at all. It seems like a very mis-named thread. In any case, we have a very long thread already currently running on this topic. I would recommend you go through it and see if you think you have anything that has not already been discussed in excruciating detail. https://www.physicsforums.com/showthread.php?t=469311
No problem. I'll see if I can come up with a "better" graph. In the meantime, if you happen to have an epiphany of imagination and see what exactly it is I'm trying to communicate I'd love to continue the conversation. If you read the thread all the way through, my closing statement references objects' "true length". Also, I've never seen anyone make the case above for Lorentzian length contraction as being an illusion (specifically for the reasons mentioned), including in the other thread that you linked to. If you don't mind, I felt my points are connected to the other thread in only a single way, which is that we both mention rest length, and that it would be a disservice to myself and the other author to conflate our competing ideas into a single conversation.
No it doesn't. Specifically which argument do you believe is novel? I bet I can find it in that other thread.
DaleSpam, with respect, are you a moderator? I had assumed so but you seem a bit argumentative for a Mod. When I say ...I am equating "absolute length" with "true length". This should be clear when I refer to other measured lengths as "an illusion". I have to feel that you're being a bit disingenuous if you're protesting that I didn't LITERALLY use the phrase "true length" when it should be apparent that I referred to it nonetheless. Here are my two points, summarized. If you can find both of these in the other thread I will remove this one. 1) There is a direct connection between the time-component of the constant spacetime velocity of an object and its time dilation and length contraction factors. 2) As a measured length, such as that of the face of a Rubik's Cube, is gradually twisted away from the width dimension in which we are measuring it, it's apparent width is altered in the same proportion as the car's apparent length is altered as it's constant spacetime velocity is "twisted" from being purely temporal to having a spatial component.
Can you explain what it means to have a "time component" speed of something something c? What does it mean to travel through time at some "speed" (which is measured in meters/second)?
If you can represent what you mean on a normal spacetime diagram drawn accurately, please do so. But if you can't, then consider the possibility that there isn't any well-defined idea that you are "trying to communicate", that in fact you just have a vague analogy that you have convinced yourself is meaningful even though maybe it isn't.
Can I explain it? Yes and no. It's a good question. The problem is that we're blending SR's concept of spacetime with traditional definitions based on a strict separation of space and time. Velocity is defined in terms of distance (or space) / time, as you said. In my explanations above, you must simply consider the "constant spacetime velocity" to be the physical manifestation of the Lorentzian invariant quantity under SR, while what you would traditionally consider "velocity" now becomes the "spatial component" of that constant spacetime velocity.
The auto graphs are clearly imperfect; I was trying to do this over lunch today. But I think you're confused...normal spacetime diagrams represent world-lines, etc. Here, the graphs are doing double-duty: 1) The car length is a representation of the direction of an object's constant spacetime velocity as it relates to it's temporal and spatial components. It says nothing of time lines, proper times, etc. 2) It also serves as a visual aid to show the connection between the time-component of the constant spacetime velocity and its time dilation and length contraction factors. On the graphs with 2 automobiles, the 2nd auto doesn't really apply to the graph itself, it's rather an attempt to "show" this connection visually.
I have a comment to make but I have already made it in post #5 in the thread the DaleSpam linked to in post #7 of this thread: And I have the same questions for you that I presented to Greg in that post.
rjbeery, here is a quick copy of one of the posts. Does this represent the idea you are offering here? Was Greg (with his discussion below) trying to make the same point you are?
The term "absolute" is well defined. It means that the quantity in question is frame invariant. Everyone agrees that the rest length is absolute. EDIT: this may not be correct, see below The term "true" on the other hand is not well defined. While you are certainly free to arbitrarily define "true" to mean "absolute" the choice is completely arbitrariy and not without controversy. That is the whole point of the other thread and by making this personal definition you are simply doing the same thing that has been discussed at length in the other thread. This point was made multiple times by multiple different people in the other thread. It appears that I am not the only one who thinks this thread is redundant with the other one.
Are you defining "spacetime velocity" the same as Greene does on p. 392 of The Elegant Universe? If so can you explain what you mean by "direction of an object's constant spacetime velocity as it relates to it's temporal and spatial components" in mathematical terms like this? Are you imagining a sort of graph where we plot [tex]\sqrt{(d\vec{x}/dt)^2}[/tex] (which is just the magnitude of the velocity vector, i.e. speed) on one axis and [tex]d\tau/dt[/tex] on the other, such that the length of the vector for any object always adds up to 1? And is the "direction" you're talking about in this abstract space of speed vs. time dilation, rather than direction in ordinary spacetime? If so how does this have anything to do with length contraction? What "connection" would that be? Just that they both happen to involve a gamma-factor when you relate their value to the object's speed? I don't see how that would imply we should view the rest length as "true" or how this helps make sense of an analogy involving visual foreshortening.
That's exactly the same impression I had. rjbeery, the repeat visitors to this forum understand spacetime diagrams quite well. You really can't convey the story you have in mind if your spacetime diagram is not at least qualitatively correct. It really doesn't work to provide an incorrect sketch and assume people will know what you meant (that is, if you did mean for it to be different than what you sketched). Please don't think I intend anything critical here. Spactime diagrams can be tough to interpret for some of the newer visitors sometimes, even when done correctly.
Could you please give me some links in the other thread? While I was unable to find it myself, I would be curious to see how others are framing my argument. JesseM and bobc2, yes Greene's description of a constant spacetime velocity is equivalent what I was attempting to outline. I had assumed that this description of SR was well-known and that a rigorous graphical and mathematic proof was not necessary. Also, there is nothing particularly ground-breaking in this way of looking at SR. My novelty lies in my Rubik's Cube analogy, so if you could please reread it and critique it such that I can make it clearer it would be most appreciated. My ultimate point is that to the extent that my spatial-parallax analogy (i.e. width-to-depth dimensional perspective) applies to to the constant spacetime velocity concept (i.e. temporal-to-spatial dimensional perspective) we are able to say that a rest length is just as valid as a squarely-measured length (i.e. they are both "true lengths"), and a Lorentz-contracted length is just as illusory as a parallax-affected one. If you would like to argue that the analogy does not hold, that's OK, but the math is equivalent which I find to be compelling.