Are the transformations just observed ones or real ones?

[Noyhcat]

A few little corrections:

The clock must be made to tick at a slower rate to compensate for the combined effects of speed and gravitation as predicited by GR. See: https://en.wikipedia.org/wiki/Error...sitioning_System#Calculation_of_time_dilation

Right! Thanks for the great link. Net GAIN of 38640ns. I was trying to give a real world example of how time dilation is "real" vs "observed" per OP's original question, but I may just have confused matters by including another layer of complication by bringing GR into it.

"Relative to the satellite" doesn't mean much: the satellite isn't even nearly in rest in any inertial frame (and I did not copy your last sentence which I could not parse).
[addendum: and the clock on the wall uses the ECI frame]

I've only started to get into GR (evinced by my complete omition of its existence in my post), and I get the idea that there is nothing inertial about an object that is actively being accelerated on. I concede your Yuiop's points and will be doing some more reading.

Though, does every reference frame need to be inertial? If the clock on the lab wall and the satellite are moving relative to the ECI frame, allbeit at different speeds, I get that, but can we not speak of things from the satellite's reference frame as well? Shouldn't a person who is "standing" at the center of the Earth be able to state the laws of physics just as a person who is on board the satellite should?

If your point is "stop mixing IRF's with non inertial ones" I get that too. :)

 

[yuiop]

Though, does every reference frame need to be inertial?

No, they do not need to be inertial, as for example in the Rindler metric which considers the reference frame of an accelerating observer. I suspect (although it is hard to be sure) that the OP is interested in the tangible physical differences between measurements made between two purely inertial observers where all the measurements are symmetrical. It is easy to show that there are tangible physical differences when non inertial reference frames are considered because the measurements are not symmetrical.

If the clock on the lab wall and the satellite are moving relative to the ECI frame, allbeit at different speeds, I get that, but can we not speak of things from the satellite's reference frame as well? Shouldn't a person who is "standing" at the center of the Earth be able to state the laws of physics just as a person who is on board the satellite should?

I think we can, by stating the laws of physics in an invariant way that all observers can agree on. When we consider the results purely due to a Lorentz boost with no proper acceleration involved, there are no physical quantities that vary in a invariant way. This I think is the crux of the matter that the OP is asking about. (I hope I stated that correctly as I am not very good with the formal language of relativity.)

We can rule out GR (but not non inertial motion) by modifying your example and placing the reference clock on an absurdly high tower that has the same altitude as the satellite. Now the orbiting clock will show less elapsed time than the reference clock on the tower, each time it passes. I think someone also gave the more practical example (that also excludes GR) of how the half life of particles is considerably extended when they are circulating at high speed in a cyclotron, but again that is a non inertial example.

 

[harrylin]

In addition:

[..] Shouldn't a person who is "standing" at the center of the Earth be able to state the laws of physics just as a person who is on board the satellite should? [..]

If you want to keep things conceptually simple and straightforward as in classical physics and SR, then you (and those persons) should stick to using Newtonian ("Galilean") reference systems. The ECI frame is approximately such a system (only approximately due its orbit around the Sun).

 

[Noyhcat]

I think I'm going to stick to trains for a bit longer. :)

 

[harrylin]

I think I'm going to stick to trains for a bit longer. :)

The trains (as long as they go at constant speed in a straight line) are great.

 

[phyti]

You didn't get it. You define time as what you see, i.e, as the information you get from photons. Time is related to motion, photons do travel, so without time photons would not even travel. And the EM process is just photons, you again define time as the motion of photons which is incorrect.

'Time' does not cause anything. Photon motion can define a unit of 'time'.
Refer to drawing.
Light is emitted from a source in a direction p, perpendicular to x, the direction of motion, and reflects from a mirror a distance d=1, to a detector/counter. For the clock to function, the photon path must have an x and p component. The x component compensates for the motion of the clock at speed v. The p component becomes the active part of the clock. Since the photon speed is constant, its path in any direction generates a circular arc for the 90º between the p axis and x axis. This means the relative photon speed along p = c*sqrt(1-(v/c)^2) = c/γ, i.e. the clock ticks slower, the faster it moves past an observer.
The clock moves in a 1-dimensional space, while/(simultaneously) the photon moves in a 2-dimensional space. The clock is counting spatial increments of (2γd) which are labeled in the traditional manner as ‘time’.

With vt the x component and pt the p component, the relation can be rephrased as
1. (vt)^2 + (pt)^2 = (ct)^2, or
2. (object motion)^2 + (light motion)^2 = (light motion)^2, or
3. (object motion)^2 + (object time)^2 = (light motion)^2
Line 3 is the misconception, equating ‘object time’ to ‘object motion’, that leads to the idea of ‘moving thru time’.

‘Time’ is a relation between events, a scalar or number (thus no direction) that is always cumulative. A clock never runs backward reducing ‘time’. It can be likened to a ships log, or a diary, or any method of historical record keeping.

 

[Dale]

The term "length contraction" has two different meanings; one meaning relates to a reduction in a moving object's or system's equilibrium length according to a system in which that object or system was in rest before. This was also how Einstein used it in 1905: "let a constant velocity v be imparted in the direction of the increasing x of the other stationary system".

You are taking the Einstein quote out of context. The full quote, in context, is:

... a uniform motion of parallel translation with velocity v along the axis of x in the direction of increasing x is then imparted to the rod. We now inquire as to the length of the moving rod, and imagine its length to be ascertained by the following two operations:—

(a)The observer moves together with the given measuring-rod and the rod to be measured, and measures the length of the rod directly by superposing the measuring-rod, in just the same way as if all three were at rest.
(b)By means of stationary clocks set up in the stationary system and synchronizing in accordance with § 1, the observer ascertains at what points of the stationary system the two ends of the rod to be measured are located at a definite time. The distance between these two points, measured by the measuring-rod already employed, which in this case is at rest, is also a length which may be designated “the length of the rod.”

So in Einstein's 1905 the imparting of velocity v is merely setting up the initial conditions that the rod is moving in the "stationary" system. The actual length contraction comparison (a vs b) is clearly between frames, not before and after acceleration in a single frame.

 

[harrylin]

You are taking the Einstein quote out of context. The full quote, in context, is: [..]
So in Einstein's 1905 the imparting of velocity v is merely setting up the initial conditions that the rod is moving in the "stationary" system. The actual length contraction comparison (a vs b) is clearly between frames, not before and after acceleration in a single frame.

You took my side remark ("This was also how") to be an issue. As a reminder, this concerns your insistence that:

It is not a change in an observed phenomenon [..] Again, length contraction isn't about changes in length, it is about disagreement between frames.

I referred to a century old paper to illustrate that we are telling you nothing new - I could also have cited from Bell and others incl. a paper by myself not so long ago. However, as your reading of Einstein appears to be different from mine and this may also be instructive for more people, I now spent some time to clarify Einstein's development with a fuller context in italics and with some added precisions by me in non-italics. Einstein's symbol l may be confusing, so I replaced it by L₀.

Let there be given a stationary rigid rod; and let its length be L₀ as measured by a measuring-rod which is also stationary.
[ Definition: L₀ = length as measured in rest in "stationary system" S. ]
We now imagine the axis of the rod lying along the axis of x of the stationary system of co-ordinates, and that a uniform motion of parallel translation with velocity v along the axis of x in the direction of increasing x is then imparted to the rod. We now inquire as to the length of the moving rod [..]
In accordance with the principle of relativity [..]"the length of the rod in the moving system"- [let's label that L'₀] must be equal to the length L₀ of the stationary rod.
[According to POR: L'₀ = L₀]
[..] "the length of the (moving) rod in the stationary system" [let's label that L_t] [..] we shall find that it differs from L₀.
Current kinematics tacitly assumes [..] that a moving rigid body at the epoch t may in geometrical respects be perfectly represented by the same body at rest in a definite position.

He finds thus that in S, L_t ≠ L₀.
And a note about "kinematics": classical mechanics makes its assumption for the equilibrium length of the object under the condition of negligible plastic deformation; and the same specifications apply to the ruler for the measurement in motion. SR doesn't change those specifications.

Einstein elaborated next (emphasis mine):

Physical Meaning of the Equations Obtained in Respect to Moving Rigid Bodies and Moving Clocks

[..] The equation of the surface of this sphere moving relatively to the system K with velocity v [..]
Thus, whereas the Y and Z dimensions of the sphere (and therefore of every rigid body of no matter what form) do not appear modified by the motion, the X dimension appears shortened in the ratio 1/sqrt{1-v^2/c^2}, i.e. the greater the value of v, the greater the shortening. [..]
Equation of shape in S: Δx = Δx₀/γ
A rigid body which, measured in a state of rest, has the form of a sphere, therefore has in a state of motion -viewed from the stationary system- the form of an ellipsoid'

Once more, I illustrated that with my calculation example and yuiop illustrated it as follows (emphasis mine):

[..] Ehrenfest paradox [..] the length contraction of the outside edges of a rotating object causes real stresses that would eventually tear the the object apart if the radius was not permitted to alter as the rotational speed varied. The next best demonstration is Bell's rockets paradox, where a string of fixed length (in one reference frame) breaks due to length contraction[..].

BTW, I thought that it was understood -notwithstanding Einstein's limited scope in 1905- that Ehrenfest's disc can't be rigid and that Bell's string isn't supposed to be rigid - but see next!

Stevendaryl added as further clarification (emphasis mine):

Yeah, there are two "length contraction" effects, one having to do with the changes in the measured equilibrium length of an object that is set in motion, and the second having to do with a comparison of distances in two different inertial coordinate systems.

There are similarly two "time dilation" effects: the changes in the measured rate of a clock that is set in motion, and the second having to do with a comparison of elapsed times in two different inertial coordinate systems.

Of course, these pairs of effects are closely related:

• From the assumption that clocks and rods undergo time dilation and length contraction when set into motion, one can show that a coordinate system based on those moving clocks and rods will be related to the original coordinate system through the Lorentz transformations.
• From the assumption that the forces governing rates of clocks and lengths of objects are Lorentz-invariant, one can derive that they must undergo time dilation and length contraction.

But then the following amazing remarks appeared in a parallel thread despite all the preceding:

[..] There is no such thing as length contraction in one frame.

In reaction I suggested to digest the information by myself, yuiop and Stevendaryl in this thread, but apparently that did not happen:

Fixed it for you [AT added rigid]. This the key element that people often forget, when assuming "length contraction" in that historical sense. And it is a good reason to avoid that historical usage [..] This leads to confusion [..] generally in Bell-Spacehip-Paradox threads.

[my correction deleted by Dalespam]

[..] I don't want any historical apologists cluttering up the thread.

[my correction deleted by Dalespam]

The above is self-explaining, so I will leave it at that.

 

[Dale]

I referred to a century old paper

Your reference was out of context. Now, your explanation here is also very selectively edited to keep it out of context.

Your first [..] hides his comment

and imagine its length to be ascertained by the following two operations:—

(a)The observer moves together with the given measuring-rod and the rod to be measured, and measures the length of the rod directly by superposing the measuring-rod, in just the same way as if all three were at rest.
(b)By means of stationary clocks set up in the stationary system and synchronizing in accordance with § 1, the observer ascertains at what points of the stationary system the two ends of the rod to be measured are located at a definite time. The distance between these two points, measured by the measuring-rod already employed, which in this case is at rest, is also a length which may be designated “the length of the rod.”

Which clearly identifies the comparison being done between frames.

Your second [..] hides the comment

the length to be discovered by the operation (a)—we will call it

Your third [..] hides his statement

The length to be discovered by the operation (b) we will call

And your fifth [..] hides

that the lengths determined by these two operations are precisely equal

All of which completely cement the fact that all of Einstein's previous comments are concerning the comparison between two frames, the operations a and b he defined.

Your selective reading of Einstein's paper is truly amazing. It makes me no longer certain that there even is a historical case to be made for the pre- vs. post-acceleration interpretation of length contraction. Certainly, it isn't to be found in Einstein's 1905 paper, and if your reading of his paper led you to that conclusion then I suspect that your reading of other sources also led you to a similar erroneous conclusion.

Furthermore, English is always ambiguous, and more so German translated into English. So the best place to look for an unambiguous definition is in the mathematical derivation. All the mathematical derivations of length contraction which I have seen (including Einstein's 1905 derivation in section 3) are a comparison of the length in two frames, not a comparison of the length before and after acceleration in a single frame. That further casts doubt on the idea that length contraction has multiple historical definitions.

Can you find even a single example where length contraction is mathematically derived using pre- and post-acceleration lengths in a single frame rather than deriving length contraction as a comparison between two frames?

 

[kaplan]

Can you find even a single example where length contraction is mathematically derived using pre- and post-acceleration lengths in a single frame rather than deriving length contraction as a comparison between two frames?

The length of an object is an experimentally measurable quantity, and it depends only of the object's current state and how you do the measurement, not the object's past history.

In other words it doesn't make the slightest bit of difference whether an object got to its current state of motion "after acceleration in a single frame" (whatever that even means) versus always being in that state of motion, or something else.

Einstein might never have said that explicitly, but if so it's because he considered it obvious. If you want a related example that Einstein did discuss, take the twin paradox.

 

[universal_101]

The length of an object is an experimentally measurable quantity, and it depends only of the object's current state and how you do the measurement, not the object's past history.

In other words it doesn't make the slightest bit of difference whether an object got to its current state of motion "after acceleration in a single frame" (whatever that even means) versus always being in that state of motion, or something else.

Einstein might never have said that explicitly, but if so it's because he considered it obvious. If you want a related example that Einstein did discuss, take the twin paradox.

Exactly, Twin Paradox is all about the comparison of state of NO motion and after some acceleration the state of Relative Motion. In daily scientific practice, accelerators are also very good example and they are almost always accelerating the particles, but we do apply all the relativistic corrections nonetheless. And these corrections are applied because there is a relative motion between lab frame particles and tunneled ones, neglecting the acceleration needed to achieve the state of relative motion. So, if the two states are not related (i.e. no motion and relative motion), why do we compare them for the time dilation!

 

[Dale]

The length of an object is an experimentally measurable quantity, and it depends only of the object's current state and how you do the measurement, not the object's past history.

Correct. That is another reason why length contraction is correctly understood as a disagreement between frames, not a change over time.

 

[Dale]

And these corrections are applied because there is a relative motion between lab frame particles and tunneled ones, neglecting the acceleration needed to achieve the state of relative motion.

Here you seem to understand. I am not sure where you are missing the connection in the case of current.

 

[stevendaryl]

Correct. That is another reason why length contraction is correctly understood as a disagreement between frames, not a change over time.

I just don't understand why you say that. It's BOTH a disagreement between frames AND a change over time. If you take a stiff rod of length $L$, initially at rest in some frame $F$, and give it a really hard shove on one end so that it is moving at speed $v$ relative to $F$, the rod will contract. A compression wave will propagate through the rod, and when it reaches the far end, that end will start moving forward. Between the times that you push one end and the time the compression wave reaches the other end, the rod is shrinking. That's because the rear end is moving forward, but the front end is not. After the whole rod is moving, it's length will fluctuate. After compressing to a minimum length of $L (1-\frac{v}{c})$, it will then expand and contract, until it reaches an equilibrium length of $L \sqrt{1-\frac{v^2}{c^2}}$. That is all from the point of view of a single reference frame. If instead, you had pulled the front end, then initially the rod would stretch to a maximum length of $L (1+\frac{v}{c})$, and then it would contract, and expand, etc., until eventually it settled down to the same final length of $L \sqrt{1-\frac{v^2}{c^2}}$.

Now, I certainly agree with you that there is no easy way to prove what the final length would be without using multiple frames, but the facts as outlined above are all about what happens according to a single frame.

 

[kaplan]

Correct. That is another reason why length contraction is correctly understood as a disagreement between frames, not a change over time.

If an object undergoes linear acceleration, its length - as measured in a fixed inertial frame - will change over time. That follows immediately from the Lorentz transformations.

 

[Dale]

If an object undergoes linear acceleration, its length - as measured in a fixed inertial frame - will change over time.

Not necessarily. For example, consider a modification of Bell's spaceship scenario where the ships are connected by an elastic band. In the launch frame the length of the band does not change over time during the linear acceleration.

 

[Dale]

I just don't understand why you say that. It's BOTH a disagreement between frames AND a change over time.

Can you provide a source which derives length contraction as a change over time? If not, then I contend that it is not a change over time.

If an object is undergoing rigid motion then that fact can be used together with length contraction (between frames) to determine the length of the object in a given inertial frame over time, but that does not mean that the over-time comparison is length contraction.

 

[Samshorn]

[Length contraction is] BOTH a disagreement between frames AND a change over time.

Right, and of course Lorentz's theorem of corresponding states is, in a sense, the whole basis for the physical significance of the systems of coordinates related by Lorentz transformations. The equilibrium configuration of a solid object, originally at rest in one standard system of inertial coordinates, when set into motion and allowed to reach equilibrium in another system of standard inertial coordinates, is found to be spatially contracted in terms of the original coordinates. Of course, in terms of the second system of coordinates the object was spatially contracted in its original state, and after the acceleration (and stablization) it exhibits its rest length in terms of the second system.

Needless to say, the fact that the the spatial contraction of a solid equilibrium configuration after accelerating from one frame to another agrees exactly with the Lorentz transformation between those frames is not merely a coincidence. It is the basis for the physical significance of the Lorentz transformations. If the equilibrium configurations of solid objects didn't physically contract when set in motion, then the Lorentz transformations would not have any physical significance. Note that, if length contraction didn't imply that the spatial extents of solid objects change as their states of motion change, then it obviously couldn't account for the Michelson-Morley experiment. Relative to a single inertial coordinate system, the arms of the interferometer must change their lengths (in different directions) as the apparatus is re-oriented.

Now, I certainly agree with you that there is no easy way to prove what the final length would be without using multiple frames...

Sure there is (well, depending on what you consider "easy"). Lorentz did it in 1904. That is his Theorem of Corresponding States. Given the correct laws of mechanics and electrodynamics expressed in terms of any single system of inertial coordinates, the spatial contraction of any given equilibrium configuration when set in motion can be determined. This leads unavoidably to the conclusion that the equilibrium configuration contracts spatially in the direction of motion by the factor sqrt(1-v^2). Of course, this all assumes no plastic deformation of the object, representing a permanent change in the equilibrium configuration. As Einstein said, "This conclusion is based on the physical assumption that the length of a measuring rod does not undergo any permanent changes if it is set in motion and then brought to rest again".

It's important to be clear about this, because confusion on this point has served as the launching pad for many neo-Lorentzian crackpots. For example, some individuals have made careers out of writing articles for philosophical magazines advocating the Lorentzian interpretation of special relativity. Their basic mis-understanding is the same as the one expressed by some participants in this thread, namely, they mistakenly think if the laws of physics, expressed in terms of one system of coordinates S1, predict that physical phenomena will behave in a way (contracting, slowing, etc) that ensures they will satisfy the same formal laws in terms of a relatively moving system of coordinates S2, then (so they think) this proves that the S1 coordinates are the "true" coordinates and S2 are just mathematical artifacts. The obvious flaw in this reasoning is that it applies equally well to S2 as the true coordinates and S1 as mathematical artifacts. Lorentz himself credited Einstein with pointing out this "remarkable reciprocity", which reveals Lorentz invariance as a fundamental symmetry of nature, and makes it meaningless to argue for the primacy of S1 or S2 - at least in terms of the local physics. Neo-Lorentzians habitually conflate the possibility of a Lorentzian interpretation with its necessity or physical meaningfulness.

The same applies to Euclidean geometry (as a physical description, not an abstract axiomatic system). It would make no sense to claim that the Pythagorean theorem applies only to the coordinates of a single un-moved rod in terms of two relatively tilted coordinate systems, and to deny that it applies to re-oriented rods in a single coordinate system. We can obviously define any families of coordinate systems we choose, and there can be all kinds of funky differences between the descriptions of solid objects depending on which specific coordinate system we select, but that has no physical significance. The physically significant coordinate systems are the ones that correspond to the operationally defined metrical behavior of equilibrium configurations of solid bodies.

 

[A.T.]

If an object undergoes linear acceleration, its length - as measured in a fixed inertial frame - will change over time. That follows immediately from the Lorentz transformations.

No, it doesn't immediately follow from the Lorentz transformations. You have to additionally assume that the proper-length of the object stays constant (or doesn't increase too much) over time.

 

[Dale]

The equilibrium configuration of a solid object, originally at rest in one standard system of inertial coordinates, when set into motion and allowed to reach equilibrium in another system of standard inertial coordinates, is found to be spatially contracted in terms of the original coordinates.

While this is true, that doesn't mean that it is the phenomenon that is referred to by the term "length contraction". As I mentioned before, all of the derivations I have seen specifically refer to the between-frames comparison.

Sure there is (well, depending on what you consider "easy"). Lorentz did it in 1904. That is his Theorem of Corresponding States.

I would be interested in this. Do you have a link? This type of derivation would make it clear that Lorentz referred to the over-time comparison as "length contraction".

 

[Samshorn]

All of the derivations [of length contraction] I have seen specifically refer to the between-frames comparison.

I would say the derivations you've seen actually do cover both aspects of length contraction, provided you keep in mind the physical meaning of "frames", i.e., of standard inertial coordinate systems. Remember, they are the coordinate systems in terms of which the laws of physics take the same simple homogeneous and isotropic form. This means that the equilibrium configurations of a solid object (for example) will be the same, for any state of motion of the object, when expressed in terms of the co-moving system of inertial coordinates. With this understanding, the fact that those systems are related by Lorentz transformations implies length contraction, which entails the equivalence between (1) the contraction of a solid object (at equilibrium in terms of a given reference frame) when the object's state of motion is changed, and (2) the difference in spatial extent of an unaccelerated object in terms of two different frames of reference. The whole point of special relativity - the reason it works and has physical significance - is that these two things are perfectly equivalent, or rather, they are the same thing, looked at in two different ways.

While this is true, that doesn't mean that it is the phenomenon that is referred to by the term "length contraction".

Yes it does. More precisely, it means that the comparative differences in spatial lengths corresponding to processes (1) and (2) are simply two manifestations of the same attribute (Lorentz invariance) of physical phenomena. They are the same thing, and that thing is called length contraction.

I really think it would help if you thought about the corresponding facts of Euclidean geometry. In terms of a Cartesian system of coordinates, the x projection of a rod of length L is Lcos(q) where q is the angle between the rod and the x axis. So if q=0 we have x=L, but if q>0 the x projection is less. This can be called “x contraction”. Your claim is that the term “x contraction” applies only if we hold the rod fixed and compare the x projections of two different coordinate systems, but not if we hold the coordinate system fixed and consider the x projections for two different orientations of the rod. Hopefully you can see that, given that the coordinate systems are ultimately defined in terms of equilibrium configurations of rods, these two situations are equivalent.

I would be interested in this. Do you have a link? This type of derivation would make it clear that Lorentz referred to the over-time comparison as "length contraction".

Do you have access to a copy of "The Principle of Relativity", in the Dover edition? If not, I'd highly recommend it. It contains the original papers of Lorentz, Einstein, and Minkowski on relativity. In particular, Lorentz's 1904 paper contains his theorem of corresponding states, by which he provides a physical justification for the Fitzgerald contraction effect that he had previously just postulated ad hoc. By 1904 he had realized that precisely this contraction was to be expected as a consequence of the form of the physical laws (expressed in terms of a single coordinate system):

"If we suppose each particle of a solid body to be in equilibrium under the action of the attractions and repulsions exerted by its neighbors, and if we take for granted that there is but one configuration of equilibrium, we may draw the conclusion that the system S', if the velocity v is imparted to it, will of itself change into the system S. In other terms, the translation will produce the deformation [by the factor sqrt(1-v^2) in the direction of v]."

But I hope you don't think this is just an antiquarian derivation. Again, every derivation you've ever seen entails essentially the same concepts. Given the laws of physics, expressed in terms of any single coordinate system, if we determine the result of accelerating a measuring rod to some state of motion, we can obviously determine the amount of spatial contraction it undergoes, simply by applying the laws of physics in that single coordinate system. The physical significance of the Lorentz transformations is that they relate the coordinate systems in terms of which the laws of physics take the same simple form, and hence the equilibrium configurations of objects are the same.

 

[Dale]

I would say the derivations you've seen actually do cover both aspects of length contraction, provided you keep in mind the physical meaning of "frames", i.e., of standard inertial coordinate systems.

No, they don't. The derivation of length contraction (between frames) only requires the principle of relativity and the invariance of c. As such, it is always valid, regardless of the scenario and regardless of the specific theory of matter. The over-time interpretation requires a theory of matter with "rigid" objects, and further it is only applicable in scenarios where the objects undergo purely rigid acceleration.

The derivations of length contraction that I have seen have been based on the Lorentz transform without requiring the additional assumptions.

Remember, they are the coordinate systems in terms of which the laws of physics take the same simple homogeneous and isotropic form. This means that the equilibrium configurations of a solid object (for example) will be the same, for any state of motion of the object, when expressed in terms of the co-moving system of inertial coordinates.

You are here assuming solid objects in equilibrium configuration, and also the laws that produce such objects. It is therefore of limited applicability, and none of the derivations I have seen introduce such needless complexity and limitations. As I said, the additional assumptions required for over-time length contraction are NOT implied by the usual derivations of between-frames length contraction.

"If we suppose each particle of a solid body to be in equilibrium under the action of the attractions and repulsions exerted by its neighbors, and if we take for granted that there is but one configuration of equilibrium, we may draw the conclusion that the system S', if the velocity v is imparted to it, will of itself change into the system S. In other terms, the translation will produce the deformation [by the factor sqrt(1-v^2) in the direction of v]."

This seems to be a postulate of Lorentz's theory (which is consistent of my understanding of his theory) rather than a derivation, but since postulates aren't derived it seems completely valid to me. It does make it clear that the over-time interpretation of length contraction was a correct historical usage of the term. I will make sure to modify my comments accordingly in the future.

 

[Samshorn]

The derivation of length contraction (between frames) only requires the principle of relativity and the invariance of c.

Well, relativity along with memorylessness, isotropy, and homogeneity of inertia. And the point is, those are precisely the principles that imply that the two aspects of length contraction are equivalent -as they must be if the Lorentz transformation is to have any physical significance.

As such, it is always valid, regardless of the scenario and regardless of the specific theory of matter. The over-time interpretation requires a theory of matter with "rigid" objects, and further it is only applicable in scenarios where the objects undergo purely rigid acceleration.

No, not at all. As you said, the derivations require the principle of relativity, which asserts that the laws of physics take the same form when expressed in terms of any system of inertial coordinates. This signifies that - by definition - the equilibrium configuration of a solid body (governed by the laws of physics) will be the same, regardless of its state of motion, when described in terms of a co-moving system of inertial coordinates. For example, the arms of Michelson's interferometer, when their orientation changes, undergo changes in their spatial lengths in terms of the single reference frame of the Earth. It doesn't matter what kind of acceleration or process created that object in that state of motion, nor does this require any "theory of matter", it requires only the existence of stable and persistent entities, such as the "measuring rods and clocks" on which Einstein based his theory.

The derivations of length contraction that I have seen have been based on the Lorentz transform without requiring the additional assumptions.

No, you're overlooking the fact that the physical significance of the Lorentz transformations is entirely due to the fact that inertial coordinate systems (i.e., coordinate systems in terms of which the laws of physics take the same simple form) are related by those transformations. You could dream up any weird kind of coordinate transformations you like, and "derive" the "length contraction" implied by those coordinates, but it would have no physical significance. Length contraction is not an algebraic fact derived from an arbitrarily selected transformation, it's a physical fact due to the Lorentz invariance of physical phenomena, as exemplified by Einstein's (ideal) measuring rods and clocks.

Again, the analogy to Euclidean geometry makes this perfectly clear. As explained previously, the x-contraction is purely a function of the angle between the rod and the x axis, regardless of whether we rotated the rod or the coordinates (because ultimately the coordinates are defined in terms of the persistent properties of ideal measuring rods). Granted, a real physical object could be deformed by rotation, just as a real physical clock could be broken by acceleration, but this doesn't invalidate the concept of ideal measuring rods and clocks. It's understood that we don't rotate the arms of Michelson's interferometer fast enough to deform them. This is why Einstein noted that "This conclusion is based on the physical assumption that the length of a measuring rod does not undergo any permanent changes if it is set in motion and then brought to rest again".

This seems to be a postulate of Lorentz's theory (which is consistent of my understanding of his theory) rather than a derivation...

Well, he originally just postulated it, but by 1904 he derived it (more or less) from the laws of electrodynamics (combined with the "molecular force hypothesis" that says whatever forces hold the elementary particles together transform the same way that electromagnetic forces do) in terms of a single reference system. This is called the Theorem of Corresponding States. Note that Heaviside and Searle had already derived in the 1800s the fact that the spheres of equi-potential of an electric charge are contracted into ellipsoids by the factor sqrt(1-v^2) in the direction of motion, so physical length contraction is not a surprising phenomenon.

 

[stevendaryl]

Can you provide a source which derives length contraction as a change over time? If not, then I contend that it is not a change over time.

*IF* a rod has an equilibrium length $L$ when at rest, then Lorentz invariance of the forces within the rod would imply that it would have that length in any inertial reference frame. So, if a rod starts off at rest in frame F, and then is gently accelerated to speed v, and then allowed to return to its equilibrium length, then its length will be$L/\gamma$ in frame F.

Now, not everything has an equilibrium length. A lump of chewing gum doesn't. But if it has an equilibrium length, then its length will necessarily decrease if you set it gently in motion.

 

[stevendaryl]

No, they don't. The derivation of length contraction (between frames) only requires the principle of relativity and the invariance of c. As such, it is always valid, regardless of the scenario and regardless of the specific theory of matter. The over-time interpretation requires a theory of matter with "rigid" objects, and further it is only applicable in scenarios where the objects undergo purely rigid acceleration.

Right. In the cases in which an object has an equilibrium length, then it will undergo physical length contraction if it is accelerated (as measured in its original rest frame). But that really is the normal case when people are talking about rockets and measuring rods and so forth. If rods didn't have equilibrium lengths, then they would be pretty useless for measuring.

When people talk about Rindler coordinates for a rocket undergoing constant proper acceleration, the usual assumption is that the rocket's length remains constant in Rindler coordinates. If that were not the case, then that would be kind of weird.

 

[Dale]

Well, relativity along with memorylessness, isotropy, and homogeneity of inertia. And the point is, those are precisely the principles that imply that the two aspects of length contraction are equivalent

No, they are not equivalent. If they were equivalent then both over-time and between-frames length contraction would necessarily apply in the same situations. Over-time length contraction does not apply when objects undergo non-rigid acceleration whereas between-frames length contraction does. Therefore they cannot be equivalent. It is a proof by contradiction and none of your response changes that simple and obvious fact.

Over-time length contraction is not equivalent to between-frames length contraction because it requires additional assumptions which can be violated in specific scenarios. Please re-read your own explanations to see the clear fact that you are making those additional assumptions.

Please do not post more of the same, they are not equivalent as you should be well aware by now.

 

[Dale]

Right. In the cases in which an object has an equilibrium length, then it will undergo physical length contraction if it is accelerated (as measured in its original rest frame). But that really is the normal case when people are talking about rockets and measuring rods and so forth.

Yes, in the case where you additionally assume a constant equilibrium length then the between-frames length contraction together with those additional assumptions imply the over-time length contraction. While that is the normal case when talking about rockets etc. it is not the case when talking about currents or Bell's spaceship or many other scenarios. This is because the additional assumptions required for the over-time length contraction are not met. Therefore, from a pedagogical standpoint it is not good to confound those two distinct concepts, which is why the modern usage is between-frames as can be clearly seen from the derivations.

 

[DrGreg]

This is, of course, an argument over words and not an argument over physics. The argument can be avoided by just doing the maths and not attempting to describe it with words.

I would say that the primary meaning of length-contraction is the between-frames version. I would say that the over-time version is a consequence of, or corollary to, length-contraction, as it applies only in special cases where objects behave as if they are "rigid", whereas the between-frames version always applies.

I think this matters because people learning the subject can get confused. They think that "acceleration causes contraction", and then get confused in cases such as Bell's spaceship paradox, or electrons moving in an uncharged wire, where things accelerate but don't move closer together.

They can even get confused when A and B start out initially at rest relative to each other, then B accelerates and then coasts at constant velocity relative to A. They understand that B contracts relative to A's frame, because "acceleration causes contraction", but they don't understand that A contracts relative to B's frame. Either they don't accept it at all, or they think A's contraction is an illusion and B's contraction is "real". And all because of a misunderstanding of what "length-contraction" really is.

 

[kaplan]

No, it doesn't immediately follow from the Lorentz transformations. You have to additionally assume that the proper-length of the object stays constant (or doesn't increase too much) over time.

Yes, you're right. But it will stay constant, at least if (a) it's reasonably rigid, or I accelerate it slowly enough, and (b) I measure the length long after the acceleration is over.

All of which is to say that I don't understand what DaleSpam is getting at.

 

[Dale]

Yes, you're right. But it will stay constant, at least if (a) it's reasonably rigid, or I accelerate it slowly enough, and (b) I measure the length long after the acceleration is over.

All of which is to say that I don't understand what DaleSpam is getting at.

What I am getting at is that (a) and (b) are additional assumptions which are not always valid. What is known as length contraction in modern terms does not include those additional assumptions and is therefore more general.

 

[Samshorn]

This is, of course, an argument over words and not an argument over physics.

I agree that some of the disputation is due to inapt wording (on my part), but I think there may be some actual physics lurking in this discussion. What I've been talking about is the physical meaning of the Lorentz transformation in terms of the equilibrium configurations of physical entities, exemplified by ideal measuring rods and clocks. Others here are focused on non-equilibrium configurations of physical entities, and they want to be sure no one confuses these with equilibrium configurations. That's fine, and I applaud their efforts to make that important distinction. But that distinction doesn't contradict what I've been saying - see below.

I would say that the primary meaning of length-contraction is the between-frames version. I would say that the over-time version is a consequence of, or corollary to, length-contraction, as it applies only in special cases where objects behave as if they are "rigid", whereas the between-frames version always applies.

The point I've been laboring (apparently without success!) to convey is that "frames" are ultimately defined in terms of ideal measuring rods and clocks, so when we talk about evaluating length "between frames" we are actually talking about evaluating two different sets of ideal measuring rods (and clocks) in two different states of motion - but of course these two sets of ideal measuring rods and clocks must be 'intrinsically identical'. For example, a one meter rod at rest in S must be 'intrinsically identical' to the one meter rod at rest in S'. But this has meaning only if we know what 'intrinsically identical' means - and therein lies the rub.

One way of establishing the intrinsic identicality of relatively moving rods would be to create two sets of measuring rods in a single reference frame and compare them side-by-side to make sure they are equal, and then accelerate (arbitrarily slowly) one set of rods to some state of motion. But this implies that a "between-frame" comparison is nothing but a disguised over-time comparison of ideal rods. This is unacceptable to anyone who insists that the between-frame comparison is not equivalent to an over-time comparison of ideal rods. That person needs to contend that each ideal measuring rod can serve as a measuring rod only for one specific frame, and can never be put into another frame for comparison with any other rod. But then how do we physically correlate ideal measuring rods in different states of motion?

Conceptually we can define a certain molecular structure, consisting of a certain number of (presumed) elementary particles arranged in a certain way, maintained by the electromagnetic interactions and whatever other interactions (e.g., the strong nuclear force) are involved in maintaining the equilibrium structure, and then we could create a similar arrangement of elementary particles at rest (on average) in a different frame and allow it to reach equilibrium, and declare that these two structures are intrinsically identical. But of course the simplest way of doing this would be by the first method (i.e., by creating them side by side and then accelerating one arbitrarily slowly), and in any case it is obviously equivalent to the first method. So, again, the frame-comparison is equivalent to the over-time comparison of ideal measuring rods in equilibrium.

Note well that this does not imply that an over-time comparison between ideal measuring rods (or any other physical configurations in equilibrium) is equivalent to a comparison of configurations that are not in equilibrium. That should go without saying. By the way, I really think the Euclidean analogy for x-contraction of a rod makes this perfectly clear. No one disputes that Lcos(q) is the same regardless of whether q was due to re-orienting the frame or re-orienting the rod.

 

[stevendaryl]

Yes, in the case where you additionally assume a constant equilibrium length then the between-frames length contraction together with those additional assumptions imply the over-time length contraction. While that is the normal case when talking about rockets etc. it is not the case when talking about currents or Bell's spaceship or many other scenarios. This is because the additional assumptions required for the over-time length contraction are not met. Therefore, from a pedagogical standpoint it is not good to confound those two distinct concepts, which is why the modern usage is between-frames as can be clearly seen from the derivations.

I thought one of Bell's points in his thought experiment is that the intuition that objects undergo physical contraction when they are accelerated gives the correct answer immediately. The normal behavior of a string that is accelerating is for it to be contracted (relative to its initial rest frame). To prevent contraction, you have to apply stress to the string (you have to stretch it relative to its equilibrium length). So with large enough acceleration, the string will break.

I agree that equilibrium length is not always an appropriate concept, but then, neither is the concept of the "rest frame of an object". If the object is not rigid, then the pieces can have nonzero velocity relative to each other, and it doesn't make sense to talk about a single rest frame for the object.

 

[stevendaryl]

I think this matters because people learning the subject can get confused. They think that "acceleration causes contraction", and then get confused in cases such as Bell's spaceship paradox, or electrons moving in an uncharged wire, where things accelerate but don't move closer together.

As I said to Dale, Bell's spaceship thought experiment is a case where the intuition of physical contraction gives you the RIGHT answer. The normal case for an accelerating string is that its length becomes shorter with time. To prevent this, you must exert additional stress on the ends of the string (you must stretch it relative to its equilibrium, velocity-dependent length). Eventually, that stress will cause the string to break.

 

[Samshorn]

As I said to Dale, Bell's spaceship thought experiment is a case where the intuition of physical contraction gives you the RIGHT answer.

Right, that was the whole point of Bell's paper. He used that spaceship example to show that many people who've been taught special relativity in the standard way are unclear about the fact that length contraction implies that coherent physical objects actually do tend to spatially contract when put into motion (just as the x component of a rod's length actually does tend to get smaller when it is re-oriented away from the x axis).

 

[Dale]

This is going nowhere.

The two length-contraction concepts are distinct with the over-time definition requiring additional assumptions than the between-frames definition. The over-time derivation and definition of length contraction does have a legitimate historical basis. The modern usage and derivation is the between-frames definition of length contraction.