The case for True Length = Rest Length

Branching from another thread, 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.

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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
\((.5c)^2+(.86c)^2=c^2\)

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...and indeed, SR calculates that your wife's watch would be ticking at 86% of yours as she speeds away.

[LIMIT OF 3 PICTURES PER POST, SO OP CONTINUED AS FIRST POST IN THREAD]

 

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[...CONTINUED]

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.

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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...

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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^)

 

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I've not read the whole of this, and that will have to wait until at least tomorrow, but you've got this wrong.

If an object has mass you cannot perform a Lorentz transformation such that the speed of the object with respect to the observer is c. What you seem to be saying is that an inertial object can travel at c wrt some observer. One look at the Lorentz transformation equations...
\( x' = \frac{1}{\sqrt{1- \frac{v^2}{c^2}}} \left( x - v t\right) \)

.. etc. Straight away it's obvious that if \(v \to c\) then \(x' \to \infty\) which makes no sense.

The punchline to this is that an object can be inertial or moving at c, but not both. Saying the contrary is like saying the object is both massless and massive.

 

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When you say "speed of the object" you're actually just referring to what I call the spatial-component of the object's constant velocity. My explanations break no laws if you keep this in mind. A bit unconventional, maybe, but I hadn't really considered that this part of the post was novel. What I do believe is novel, though, is my carrying through of this way of thinking about SR to explain length contraction. I believe you'll find the concept is sound if you give it some thought.

 

I think he is just trying to say the the length of a Lorentz 4-vector is invariant under Lorentz transformations, which of course it is by definition:

\(s^2=\eta_{\mu\nu}x^{\mu}x^{\nu}=\eta_{\mu\nu}x'^\m x'^\nu\)

which for 4-velocity is always \(\sqrt{s^2}=c\)

 

RJ: you use a car above. Simplify matters by taking one atom of that car. Then simplify it further by taking one electron of that atom. Now remind yourself that an electron (and a positron) can be created from a photon via pair production, and an electron has spin angular momentum and magnetic dipole moment, and remind yourself of the Einstein-de Haas effect. Take the leap away from "the electron is a point particle" and look to the evidence. There's something going round and round, and that something is light. It isn't travelling through time at c when the electron is at rest with respect to you, it's travelling through space at c. However when you move the electron and follow the track of one point on the circumference of a "ring of light", you find that it describes a helical path. Do this with all points on the circumference, and the ring of light is now a cylinder of light. If you look at this from the side it's similar to the simple inference of time dilation. The Lorentz factor comes straight out of Pythagoras' theorem - the hypotenuse is the light path wherein c=1, the base is your speed as a fraction of c, and the height is the Lorentz factor. There's a reciprocal that distinguishes time dilation from length contraction.

The electron isn't actually a ring of light, it's more like a sphere. But when it's moving fast this is still smeared out in the direction of motion. Scale this up to a person travelling at .5c, and they are too. Hence they see everything they pass looking shorter. But I wouldn't say length contraction was an illusion. Observer effect captures it better. The bottom line is that when you move fast towards a star it looks flatter to you because you've changed, not the star. You know this because it starts looking flat as soon as you move, not 8 minutes later.

 

I disagree. Neither I nor the star has changed. Nothing has changed but perspective. In the Rubik's Cube analogy above, consider two of them facing each other, and then turning one of them to the side...the "apparent face-width contraction" is RECIPROCAL. Don't you agree that an illusion is what is demanded if both cubes conclude that the other cube is shorter?

 

Something has definitely changed. Time dilation is real, and it goes hand-in-hand with length contraction. What's changed is you.

No, sorry, I don't agree that an illusion is demanded if both cubes conclude that the other cube is shorter. If you assert that one cube is moving fast past another, then it isn't a cube any more, it's smeared out into a rectangular cuboid. But as far as its concerned it's still a cube, so the other cube looks shorter. And motion is relative. When you assert that the other cube is the one that's moving, then the other cube is the cuboid. The reciprocality is there because motion is relative, and it isn't just an illusion.

 

Well I work under a presumption of objective reality; that's just how I think, and how I analyze Physics as I learn it. To me, saying that what is "actually going on" depends on "who you consider to be moving" does not sit well. I completely understand what Relativity says about this, but I'm concerning myself here with what Relativity DOESN'T say...and that is, what is really going on, as well as WHY THE HELL does Nature appear that way. Also, when you say objects are smeared out, such as the electron or the rectangular cuboid, that seems to contradict the notion that length contraction occurs in the direction of travel, doesn't it?

 

Er, Lorentz invariance of the laws of physics? The physical laws we know about all naturally predict length contraction. For example, as I said in the other thread, around 1889 Heaviside showed that the electric field surrounding a moving charge would be squashed in the direction of motion. That's what Maxwell's equations predict. They're incapable of predicting anything else about the electromagnetic field surrounding a charge. Since matter is largely held together by electrostatic forces, what effect would you expect this to have on moving matter?

The point of relativity is that you can deduce this sort of effect without having to know any details of composition of a material object or the interactions holding it together. You only need to know that it is governed by physical laws that are Lorentz invariant. This is analogous to the way you don't need to know the details of what a 30 cm ruler is made of to deduce that it will still be 30 cm long if you rotate it 45 degrees on your desk. You just need to know that the physical laws governing the structure of the ruler are invariant under rotations.

...except you've drawn the proper time on the vertical axis of your diagram. This isn't a useful representation: the proper time is only a locally relevant parameter, so your diagram wouldn't work so well as soon as you put two or more trajectories on it. For example, two events coincide (same place, same time) if their spatial and temporal coordinates are the same in a given reference frame, but not necessarily if their proper times are the same. So you could have situations in which the paths of two observers appeared to cross on your diagram, but didn't cross in reality. Worse, proper times are only really defined along the worldlines of point particles. How would you represent something like an electromagnetic field, which has no proper time associated with it, on your diagram?

If you put the reference frame's time on the vertical axis, you find that it is possible to come up with a geometric view of the invariant quantities in relativity, but you get a sign change in the pythagorean relation. For example, the proper time accumulated along an inertial world line is

\( c^2 \Delta \tau^2 \,=\, c^2 \Delta t^2 \,-\, \Delta x^2 \,. \)​

There are many other relativistically invariant relations like this one. For example, you can pack the energy and momentum of a particle into a space-time vector. The "norm" \((E/c)^2 - p^2\) is invariant and turns out to be proportional to the particle's (rest) mass squared:

\((E/c)^2 \,-\, p^2 \,=\, (mc)^2 \,.\)​

Similarly, the density \(\rho\) and current \(j\) of a fluid are frame-dependent, but the quantity \((c\rho)^2 - j^2\) is invariant and is related to the (local) rest density \(\rho_0\) by

\((c\rho)^2 \,-\, j^2 \,=\, (c\rho_0)^2 \,.\)​

In electromagnetism, the electric and magnetic fields are frame-dependent but the combination \(| \bar{E} |^2 - c^2 | \bar{B} |^2\) is invariant.

You've drawn the cars extended across time instead of space. You've just illustrated time dilation again, not length contraction.

 

What you're saying "Physics is described by these equations, and these equations predict length contraction, therefore length contraction is explained." Mathematical prediction and an explanation are not the same thing to me. Others are at peace knowing the behavior while I want to know the cause.

The auto graphs are imperfect; I was trying to do this over lunch today. But I think you're confused...the graphs are doing double-duty here: 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.

Well, I would represent the constant spacetime velocity of a photon with a horizontal line of length c; photons have no proper time, as you said. This isn't a traditional spacetime diagram, and may require some imagination, particularly in the ones with 2 vehicles because it is then that the "double-duty" aspect kind of breaks things.

If you understand the Rubik's Cube analogy you'll see what I'm trying to get at. As the cube's face is gradually twisted away from the width dimension in which we are measuring it, it's apparent face 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.

 

There's really no objective distinction between the two. All explanations in physics show how to deduce consequences from certain postulates. Length contraction follows if you accept Maxwell's theory (among other things) as a postulate for instance, or if physics is Lorentz invariant in general. The only way you can really avoid postulates is to engage in circular logic.

Ultimately the "cause" of relativistic effects like length contraction and time dilation (ie. due to Lorentz symmetry in physics) is entirely analogous to the "cause" of how the x, y, and z components of spatial vectors depend on an observer's orientation (ie. due to rotational symmetry in physics). There's little conceptual difference between the two. In fact, the former is a generalisation of the latter. The only way you can go deeper than this is to find an explanation for why the laws of physics have the symmetry properties they do. I've never seen an "explanation" of Lorentz invariance that was more than just a synonym for it.

Your relation only works if you define the "time-component" of the velocity as something like \(c\frac{\mathrm{d}\tau}{\mathrm{d}t}\) where \(\tau\) represents proper time. In other words, the vertical component of your velocity vector is the "proper time" component, and the vertical axis of your diagram points in the "proper time" direction.

I actually experimented with diagrams like these, representing proper time on the vertical axis, a few years ago while I was still learning relativity. This isn't the first time I'm seeing them.

I see the time dilation just fine. But not length contraction. You've simply shown a contraction in the time component and asserted that this somehow illustrates length contraction.

If you're suggesting that space and time intervals are components of space-time that "rotate" as we change frames, then what exactly did you think the Lorentz transformations represented? The only distinction is that the Lorentz transformations are hyperbolic rotations which preserve space-time intervals. It's possible to represent them as:

\( \begin{align} t' \,&=\, \cosh(\phi) t \,-\, \sinh(\phi) x \\ x' \,&=\, -\sinh(\phi) t \,+\, \cosh(\phi) x \;. \end{align} \)​

The parameter \(\phi\) is called the rapidity. It's the Lorentz transformation's analogue of a rotation angle.

 

I would say length contraction is not fundamental, but Lorentz transforms are. Also relativity of simultaneity is vital.

So I would work out that the consequences of the Lorentz transformations were and only then ask if they represent reality. Since they do, it's unfair to call anything derived an illusion -- they are simply fairly-held points of view.

Let A and B be time-like, inertial world-lines such that for a given inertial observer they both have the same velocity \(\vec{u}\). Then that observer may write: \(\vec{x}_A = \vec{u} t + \vec{x}_{0,A}\) or for a more space-time feel to it, every event on A satisfies: \(\begin{pmatrix} c t_A \\ \vec{x}_A \end{pmatrix} = \begin{pmatrix} 0 \\ \vec{x}_{0,A} \end{pmatrix} + \begin{pmatrix} c \\ \vec{u} \end{pmatrix} t\). Likewise for B.

Now we may Lorentz transform these events into the events seen by the observer who is moving at velocity \(\vec{v}\) with respect to the first observer.
\( \begin{pmatrix} c t'_A \\ \vec{x}'_A \end{pmatrix} = \begin{pmatrix} \gamma & - \frac{\gamma}{c} \vec{v}^{T} \\ - \frac{\gamma}{c} \vec{v} & \bf{I} + ( \gamma - 1 ) \frac{\vec{v} \vec{v}^{T}}{v^2} \end{pmatrix} \begin{pmatrix} c t_A \\ \vec{x}_A \end{pmatrix} = \begin{pmatrix} c \gamma ( t_A - \frac{1}{c^2} \vec{v} \cdot \vec{x}_A ) \\ \vec{x}_A + ( \gamma - 1 ) \frac{\vec{v} \cdot \vec{x}_A}{v^2}\vec{v} - \gamma \vec{v} t_A \end{pmatrix}\). Likewise for B.

The first observer sees the simultaneously-measured separation between any two world lines as:
\(\vec{\sigma}(0) = \left. \vec{x}_A - \vec{x}_B \right| _ { t_A = t_B } = \vec{x}_{0,A} - \vec{x}_{0,B}\)

But if \(t_A = t_B + \Delta_t\) then \(\vec{x}_A - \vec{x}_B = \vec{x}_A - \vec{x}_B - \vec{u} \Delta_t = \vec{\sigma}(0) - \vec{u} \Delta_t\), which illustrates that simultaneity is important to nailing down the separation between A and B as a well-defined value.

The second observer sees the simultaneously-measured separation between any two world lines as:
\(\vec{\sigma}(\vec{v}) = \left. \vec{x}'_A - \vec{x}'_B \right| _ { t'_A = t'_B } = \left. ( \vec{x}_A - \vec{x}_B ) + ( \gamma - 1 ) \frac{\vec{v} \cdot ( \vec{x}_A - \vec{x}_B )}{v^2}\vec{v} - \gamma \vec{v} ( t_A - t_B) \right| _ { (t_A - t_B) = \frac{1}{c^2} \vec{v} \cdot ( \vec{x}_A - \vec{x}_B ) } \)

This last condition is saying \(\Delta_t = \frac{ \vec{v} \cdot \vec{\sigma}(0) }{c^2 + \vec{v} \cdot \vec{u} } \) which gives us \(\vec{\sigma}(\vec{v}) = ( \vec{\sigma}(0) - \vec{u} \frac{ \vec{v} \cdot \vec{\sigma}(0) }{c^2 + \vec{v} \cdot \vec{u} } ) + ( \gamma - 1 ) \frac{\vec{v} \cdot ( \vec{\sigma}(0) - \vec{u} \frac{ \vec{v} \cdot \vec{\sigma}(0) }{c^2 + \vec{v} \cdot \vec{u} } ) }{v^2}\vec{v} - \gamma \vec{v} \frac{ \vec{v} \cdot \vec{\sigma}(0) }{c^2 + \vec{v} \cdot \vec{u} } \)
If I haven't blown the algebra, this works out to:

\(\vec{\sigma}(\vec{v}) = \vec{\sigma}(0) + \frac{ \vec{v} \cdot \vec{\sigma}(0) }{c^2 + \vec{v} \cdot \vec{u} } ( \frac{c \sqrt{c^2-v^2} - c^2}{v^2} \vec{v} - \vec{u} )\)

Sanity checks:
\(\vec{v} = 0 \Rightarrow \vec{\sigma}(\vec{v}) = \vec{\sigma}(0) \)
\(\vec{\sigma}(0) = 0 \Rightarrow \vec{\sigma}(\vec{v}) = 0 \)
\(\vec{u} = 0 \Rightarrow \vec{\sigma}(\vec{v}) = \vec{\sigma}(0) + \frac{ \vec{v} \cdot \vec{\sigma}(0) }{v^2} ( \sqrt{1-\frac{v^2}{c^2}} - 1 ) \vec{v} \Rightarrow \sigma^2(\vec{v}) = \sigma^2(0) ( 1 - \frac{v^2 \cos^2 \theta}{c^2} ) \leq \sigma^2(0) \)

 

I disagree with this. QM is DESCRIBED by equations, but its EXPLANATION lies in its various interpretations. The "explanation" step is the very one that bridges the mathematical world to the physical one for consumption by humans. (Please note: previous comment notwithstanding, I consider many mathematicians to be human...)

We all understand that parallax can make length appear contracted - it's an illusion that we're all familiar with. I've just pointed out that there is a mathematically analogous explanation to spatial parallax that explains "why" length might appear contracted as an object's constant spacetime velocity obtains a greater spatial-component. Your comments bringing the analysis back into the mathematical world, as if this is somehow dismissing my argument, are redundant because they were the basis for my conclusions!

Forget the graphs. I assume you're aware that the length contraction factor is equal to the time dilation factor...I pointed out that BOTH of them are equal to the time-component of the constant spacetime velocity. I assumed this was well established, but possibly not well known, and did so for the sake of others reading the thread. If you don't believe me, take a pencil and measure its length, L. Now take two other pencils on each side of it, keeping them parallel to each other, and angle the middle one such that the distance between the two pencils is .5L, and you will find that the middle pencil's extent of traversal of the other two is .86L. If you think this is condescendingly trivial I would agree with you, which is why I'm confused by your above comment.

Not exactly. I'm suggesting that the apparent width of a Rubik's Cube face, as it is rotated from our "width" dimension to our "depth" dimension, is exactly what we calculate that same width to be if we were to "rotate" that Rubik's Cube through the space and time graphs above, starting at rest (where the constant spacetime velocity is vertical) and rotating it to the right (where the Rubik's Cube is now moving at a relative spatial velocity of c to us).

 

Hi rpenner! I don't mean to ignore the amount of work you put into your post, but my 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.

 

I don't know , but it looks like an illusion to me . Seeing in 3d helps . Understanding 3D and seeing it when looking at 2D yeah Man

 

It is well accepted that objects DO have an absolute proper length.
It does not follow that length contraction is an illusion, because "length" means "moving length", not just "proper length".

Using the Rubik's cube analogy, "proper length" is the length of an edge of the blue face, while "length" is the east-west distance covered by a horizontal edge of the blue face.

Taking that analogy to 1+1 space-time, "east-west" is space-like, "north-south" is time-like, and "length" simply means the space-like separation between the worldlines of the two ends of the edge.

You could argue that its an illusion if you want, but I just see a different paradigm. I don't know if your paradigm is more useful or not.


But!

Perhaps there is some corollary in the question of whether "relativistic mass" is a useful paradigm or not.
You seem to be arguing that relativistic length is an unuseful concept just as some argue that relativistic mass is unuseful; that we should only talk of invariant length and invariant mass.

Care to explore? What are the similarities and difference between the concepts of length contraction and relativistic mass?

 

As a preface, I've had a few glasses of wine (good Malbec from Argentina), and may or may not have imbibed other intoxicating substances.

But you haven't done anything original here. The analogy of "velocity through time" is something I learned probably 10 years ago, in Brian Greene's first book about string theory. And I'm sure he learned it from somewhere else, too. I would be willing to put money on the fact that Einstein himself knew of the analogy.

And you certainly haven't explained the cause of the Lorentz transformations. To do so would first require one to understand why (for example) there are four dimensions, or why the signature happens to be (+---). A good, non-anthropic, first principles proof (that probably involves math) of only one time dimension doesn't even exist, hence the theories with two time dimensions. The fact that one dimension is different that the others is fundamental to both the special and general theories of relativity.

A priori, there's no reason that one can think of as to _why_ the Lorentz transformations should be. Just like there's no reason _why_ Maxwells Laws should be, or _why_ the Standard Model should be. But this is why physics is fascinating. These are numbers given to us by God (or, the great lagrangian in the sky, if you like). You can try to understand where the symmetries come from, but that's something I'm confident won't be resolved in these fora. You need some underlying principle that will dictate that the universe must be this way. Theorists today are happy to declare victory if their theories dictate that the universe _can possibly_ be this way.

 

Yes, and chasing this analogy, if we equate the angle away from the east-west direction that the horizontal edge of the blue face possesses with the "angle" of its constant spacetime velocity in my graphs above, then what you are calculating as length here is exactly what one would calculate as its "moving length" in SR. My point is that IF you consider parallax-induced length to be an illusion then you must (should? could?) also consider Lorentz-contracted length to be an illusion.

No. I'm not saying SR contraction is unuseful any more than I'm saying parallax-induced contraction is unuseful. The other day I had to fit an ottoman through a narrower door; I turned the ottoman at an angle and that solved my problem, but that doesn't mean that I actually reduced its length.

I hadn't given relativistic mass any thought to be honest. Maybe there's another, purely spatial, analogy that shows relativistic mass to be another illusion?

 

Hey bro it's all good, I prefer Bourbon myself.

The whole "velocity through time" thing wasn't really my point and I certainly don't take credit for it. That was just a foundation for my true analogy which lies with the Rubik's Cube. I'm clearly doing a piss-poor job of explaining it, though...

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