The 'mechanism' of length contraction

[analyst5]

Let's suppose that we have a rod which is 2 meters long in its rest frame. Its rest length can be defined as a set of points which do not occupy the same place, measured in a frame which is at rest with the rod as a whole.

Now if we travel relative to the rod, it gets length contracted, I understand this, but does this mean that some of its points somehow 'disappear'. By this I mean does length contraction imply the loss of some points which combined together make the original length of the rod? If the rod gets length contracted by 90% does it lose some of its parts, or are the spacetime points of the rod that exist measured in the rest frame still there? I really don't understand how can two points that are one next to another get contracted. I hope someone can explain the mechanism of this relativistic effect to me, and how to all the points of the initial rod get preserved in moving frames, if they do?

 

[WannabeNewton]

The rod clearly cannot "lose" some or any of its parts i.e. the constituent particles of the rod cannot "disappear". This obviously violates Baryon number conservation. Even if we were in a high-energy regime where particle number is not conserved, the creation and destruction of particles would be consistent amongst all Lorentz frames.

Now your second question is a much harder and deeper one to answer in a satisfactory manner. Prior to Einstein, the fluid defined by the ether picked out a mean rest frame with respect to which electromagnetism played out. It was calculated that a rod moving relative to the ether would contract in length and the explanation given for this, along with a valid calculation based upon Maxwell's equations, was that the electrostatic interactions between the constituent particles of the rod get modified due to motion (relative to the ether) and hence the length of the rod contracts since the spacings between particles are of course determined by the electrostatic interactions between them. This was effectively a constructive explanation of length contraction relative to the ether.

Special relativity does not offer such a constructive explanation for the symmetrical effect of length contraction between inertial frames that results from its basic postulates (I have italicized these terms because they characterize the crucial differences between length contraction in the ether theory and one form of length contraction in SR-note there is still an analogue of the former type of length contraction in SR wherein a rod initially at rest in an inertial frame once accelerated will begin length contracting in that same frame).

Here are some articles for you to read on this:

http://www.euregiogymnasium.ch/alumni/images/pdf/aeneas_wiener-lorentz_contraction.pdf
http://www.lophisc.org/wp-content/uploads/Frisch.pdf
https://webspace.utexas.edu/aam829/1/m/Relativity_files/SHPMPFitzGerald.pdf

However it is not hard to understand the deductive reasoning behind length contraction within the framework of SR. It is simply due to the fact that the notion of the length of a (Born) rigid ruler (an inherently space-like notion in causal terms) depends on a notion of simultaneity. More precisely, each point of the rigid ruler is described by a world-line in space-time thus ascribing to the ruler a "world-tube". The length of the ruler would then be defined as the distance between simultaneous events on the extremities of the world-tube. So clearly the notion of length is simultaneity dependent (be it radar simultaneity or slow-clock transport or whatever you like) and this in turn implies, due to the relativity of simultaneity (which follows deductively from the fundamental postulates of SR), that observers in different frames in general ascribe different lengths to the rigid ruler.

Geometrically speaking the length of the ruler, relative to a given frame, is given by the intersection of said world-tube with the simultaneity hypersurfaces of this frame and as you know these simultaneity hypersurfaces tilt at different angles from one frame to another relative to those of a given frame, this being an inherent property of SR. As a result we get length contraction. This does not mean that the particles making up the ruler "disappear" when going from one frame to another nor does it imply that the ruler loses its parts in such a transition (as already stated this is obviously impossible)-rather it is simply that length itself is a simultaneity dependent notion and as such is affected by the relativity of simultaneity.

EDIT: In case the lack of a constructive explanation (in this case a dynamical explanation in terms of Maxwell's equations) disturbs you, note that this kind of deductive reasoning is quite pervasive in physics. In statistical mechanics we start with the postulate that in thermal equilibrium, all possible accessible microstates of a system are equally likely and deduce the macroscopic properties (pressure, temperature, chemical potential etc.) through derivatives of the partition function without any fundamental dynamical explanation of the microscopic physics (of course the situation in statistical mechanics is a bit different in that such an explanation is readily available to us through Newton's 2nd law whereas in SR the situation is more subtle).

 

[PeterDonis]

Its rest length can be defined as a set of points which do not occupy the same place, measured in a frame which is at rest with the rod as a whole.

No, that's not how the rod's rest length is defined; it's how the rod itself is defined. More precisely, the rod is a set of *worldlines* of distinct points, i.e., a set of non-intersecting timelike curves which occupy a continuous "world tube" in spacetime. These curves exist in all frames, obviously, since they are curves in spacetime.

The rest length of the rod is defined as the length along a particular spacelike slice through the set of worldlines that defines the rod, namely, the slice which is orthogonal to every worldline in the set.

Now if we travel relative to the rod, it gets length contracted, I understand this, but does this mean that some of its points somehow 'disappear'.

No, it means that, if we travel relative to the rod, the rod's length in our rest frame is defined by a *different* spacelike slice through the set of worldlines that defines the rod--namely, the slice that is orthogonal to *our* worldline, rather than to the rod worldlines. This length is shorter than the rod's rest length, so we say the rod is "length contracted" in our rest frame. But all the worldlines are still there, so all the rod's points are still there (since each point of the rod corresponds to a distinct worldline).

I hope someone can explain the mechanism of this relativistic effect to me, and how to all the points of the initial rod get preserved in moving frames, if they do?

On the spacetime viewpoint, there is no "mechanism"; length contraction is just geometry, i.e., it's a consequence of taking a different spacelike slice through the rod. Nothing about the rod itself changes; all that changes is which spacelike slice we use to define its "length" relative to us.

 

[A.T.]

but does this mean that some of its points somehow 'disappear'.

How many points can you fit into 2m? And when some of them disappear, how many do you have?

 

[bahamagreen]

Rather than disappear, don't they get squashed closer?

If that is what "geometrically" means, does it mean that what we might locally believe are spherical (protons, neutrons, s-orbitals, equal potential radius from point charge, etc...) are thought to be existentially flattened in the direction of travel for length contracted objects?

If so, how is this understood with respect to the physical laws holding good in all IRFs? If the shape of the s-orbital, for example, is observed to be a serverly contracted to an oblate spheroid, yet the observed c is constant in all directions from the center of that spheroid, doesn't that envoke a change to the laws describing the shape of the orbital, and likewise other flattened things that need accounting for... if c got "contracted" as well it might all work out fine in proportion, but with c invarient, don't the laws have to be changed when the local spherical things are observed at speed to be contracted along one axis?

 

[PeterDonis]

Rather than disappear, don't they get squashed closer?

Not in any invariant sense, no. See below.

If that is what "geometrically" means, does it mean that what we might locally believe are spherical (protons, neutrons, s-orbitals, equal potential radius from point charge, etc...) are thought to be existentially flattened in the direction of travel for length contracted objects?

No, because the geometry in question is spacetime geometry, not spatial geometry. When we call an object "spherical" we mean that a spacelike slice of it in our rest frame is spherical; but that's obviously frame-dependent, i.e., it depends on how you take a spacelike slice of the object.

The *spacetime* geometry of the object, OTOH, is invariant, and there are invariant ways of determining whether or not the individual atoms in the object are being "squashed closer". The general method is called the "kinematic decomposition", and is described on Wikipedia here:

http://en.wikipedia.org/wiki/Congru...atical_decomposition_of_a_timelike_congruence

The expansion scalar is the particular piece of the kinematic decomposition that tells whether the object is being "squashed" in an invariant sense; in the case we have been discussing (a rigid rod moving inertially), the expansion scalar is zero, indicating that the rod is not being squashed or stretched at all.

If so, how is this understood with respect to the physical laws holding good in all IRFs?

Physical laws must always be expressible in terms of invariants, like the expansion scalar, not frame-dependent quantities like the spatial geometry of an object. That's why they hold good in all IRFs.

If the shape of the s-orbital, for example, is observed to be a serverly contracted to an oblate spheroid, yet the observed c is constant in all directions from the center of that spheroid, doesn't that envoke a change to the laws describing the shape of the orbital

Atomic orbitals are a bad choice of example here, because the theory that is used to derive their shapes is non-relativistic; AFAIK there is no relativistic version of it.

But I think what you're really focusing on doesn't depend on any particular feature of atomic orbitals, so let's use a simpler example: a light source and a "spherical" detector, where "spherical" means "in a frame in which the light source and the detector are both at rest, the detector is perfectly spherical, and the light source is at its exact center". See further development of this example below.

if c got "contracted" as well it might all work out fine in proportion, but with c invarient, don't the laws have to be changed when the local spherical things are observed at speed to be contracted along one axis?

No. If we look at an invariant description of the light source and detector in my example above, what we will have is a set of worldlines describing a certain geometric object in spacetime (not space!). What invariant properties do these worldlines have?

Well, we know one property at the outset: light rays emitted at some instant by the light source in all directions will strike the detector simultaneously in the frame in which the source and the detector are all at rest (call this frame O). As I've just stated the property, it doesn't sound invariant, but we can remedy that easily: we simply pick out the events at which the light rays strike each individual piece of the detector, and call that set of events set D. Set D is then a geometric object in spacetime, most easily described as the intersection of two other geometric objects: the future light cone of the emission event at the light source (call this event E), and a particular spacelike slice which is orthogonal to the light source's worldline (and all the detector worldlines too, of course). And if we look at the spatial geometry of set D (i.e., its geometry as seen in the spacelike slice that picks it out), it will be spherical.

Now, suppose we are moving relative to the light source and detector. What does the detector "look like" in our rest frame? Well, first of all, we realize that that question requires the use of a *different* spacelike slice, one that's orthogonal to our worldline, not the light source/detector worldlines. And the intersection of this slice with the family of worldlines describing the detector will give us a *different* set of points in spacetime, set D'. And the spatial geometry of *this* set of points, in the spacelike slice we are using now, will *not* be spherical; it will be an ellipsoid with its shorter axis in the direction of motion.

But also, notice that set D' does *not* describe a set of events at which light rays from the source strike the detector! It can't, because that set of events is set D, *not* set D'. In other words, in the moving frame, the detector is ellipsoidal, not spherical, but also light rays from the source do not all strike the detector at the same time. *That* is how c can be the same in the moving frame even though the detector is length contracted.

This is a good illustration of the fact that length contraction is not a fundamental concept in relativity; i.e., you can't use length contraction, by itself, to analyze a scenario, or you will make mistakes. Whenever there is length contraction present, you also have to take into account relativity of simultaneity (and possibly time dilation as well) in order to do a correct analysis. This is a big reason why I prefer the spacetime approach: the correct analysis is just geometry, but *spacetime* geometry, not spatial geometry.

 

[Chestermiller]

Rather than disappear, don't they get squashed closer?

From the perspective of the traveling frame, all the individual material points of the rod would still be visible, although they appear to be closer together. Also, the people in the rest frame would claim that the people in the traveling frame are not seeing the material points of the rod all at the same time, although the people in the traveling frame claim that they are.

If that is what "geometrically" means, does it mean that what we might locally believe are spherical (protons, neutrons, s-orbitals, equal potential radius from point charge, etc...) are thought to be existentially flattened in the direction of travel for length contracted objects?

Yes. This is what would be reckoned from the moving frame.

If so, how is this understood with respect to the physical laws holding good in all IRFs? If the shape of the s-orbital, for example, is observed to be a serverly contracted to an oblate spheroid, yet the observed c is constant in all directions from the center of that spheroid, doesn't that envoke a change to the laws describing the shape of the orbital, and likewise other flattened things that need accounting for.

Yes. We've already seen how conventional mechanics is modified to be consistent with SR for particle momentum, etc. I don't know how this is accounted for in the theory that you have described, but it undoubtedly has been worked out. Another area where modifications would have to be made is in deformational mechanics, particularly solid mechanics, where conventional deformational kinematics evaluates strains based on the configurations of bodies at constant time (as reckoned from their rest frame). As reckoned from another inertial frame, the same bodies would be distorted (even without developing stress), and calculation of strains and stresses would be problematic.

Chet

 

[Dale]

does this mean that some of its points somehow 'disappear'

I am not sure what you mean. There are an infinite number of points between any two points. Points are not countable.

 

[jtbell]

Consider a one-meter stick and a two-meter stick. There is a one-to-one correspondence between points on the two sticks. You can match every point on the one-meter stick with a point on the two-meter stick, and you can match every point on the two-meter stick with a point on the one-meter stick.

 

[Fredrik]

Now if we travel relative to the rod, it gets length contracted, I understand this, but does this mean that some of its points somehow 'disappear'.

Short answer: No, they just move closer together. (In the coordinate system in which the rod is originally at rest, the rear will have a higher acceleration than the front).

If I were to write a longer answer, I would emphasize the same things as PeterDonis. Let's say that Alice is comoving with the rod before the boost, and Bob is comoving with the rod after the bost. I think it's very important to understand that after the boost, what Alice thinks of as the length of the rod is a number that she associates with a certain line segment in spacetime, and what Bob thinks of as the length of the rod is a number that he associates with a different line segment in spacetime. They're not measuring the same thing.

 

[WannabeNewton]

Rather than disappear, don't they get squashed closer?

Read the links I posted. This topic enshrines an age-old debate amongst philosophers of physics.

 

[PeterDonis]

(In the coordinate system in which the rod is originally at rest, the rear will have a higher acceleration than the front).

Let's say that Alice is comoving with the rod before the boost, and Bob is comoving with the rod after the boost.

Just to clarify, the example I have had in mind does not involve any acceleration of the rod; the rod moves inertially the whole time. Only the observer changes his state of motion--i.e., he starts out comoving with Alice (and the rod), and ends up comoving with Bob (who is moving relative to the rod).

In other words, I was interpreting the OP's words "if we travel relative to the rod" to mean that *we* accelerate, not that the rod does.

 

[Fredrik]

Just to clarify, the example I have had in mind does not involve any acceleration of the rod; the rod moves inertially the whole time. Only the observer changes his state of motion--i.e., he starts out comoving with Alice (and the rod), and ends up comoving with Bob (who is moving relative to the rod).

In other words, I was interpreting the OP's words "if we travel relative to the rod" to mean that *we* accelerate, not that the rod does.

I guess I should clarify something too then. (It's probably already clear to you, but perhaps not to other readers, including the OP). My first paragraph is about the scenario where the rod is accelerated to a new velocity gently enough to allow internal forces to keep distances between "this atom and the next" approximately the same in all the comoving inertial coordinate systems (a different coordinate system at each event on "this" atom's world line).

My second paragraph (the one where I said I agree with you) was about what happens if we just describe the same unaccelerated rod using two different coordinate systems.

I didn't see the OP's words "if we travel relative to the rod". If I had, I might have skipped the first paragraph. I thought that it would be more tempting to ask if "points disappear" when the rod accelerates, so I assumed that he was talking about that.

 

[bhobba]

Interesting posts here, Peterdonis etc is spot on - SR is about geometry.

In SR nothing happens physically to the rod - its proper length is unchanged - all you are doing is a hyperbolic rotation in space-time so its apparent length is different.

It's the exact analogue to getting a long stick through a door. You rotate it and it fits through. You haven't done anything physically to the length of the rod, but the geometry of the situation means it now fits through. Same with GR - you haven't done anything to the length of the rod, but you have geometrically rotated it so it now fits through smaller openings.

Thanks
Bill

 

[analyst5]

I am not sure what you mean. There are an infinite number of points between any two points. Points are not countable.

What about two adjacent points, so point basically doesn't have a length and cannot contract?
Does this imply that any worldtube is made of an infinite number of wordlines, since there are infinite points that 'make it up'

 

[Chestermiller]

What about two adjacent points, so point basically doesn't have a length and cannot contract?

The differential segment of rod between the two adjacent points length contracts.

Does this imply that any worldtube is made of an infinite number of wordlines, since there are infinite points that 'make it up'

Yes. You can look at it that way. In Materials Science (solid and fluid mechanics), we talk about bodies being represented as continuua, and being comprised of an infinite array of differentially separated material points. When a body deforms, the material points can get closer together or farther apart, and we use this to quantify the local strains that the body experiences.

Chet

 

[Dale]

What about two adjacent points,

There is no such thing as two adjacent points. Between any two points there are an uncountably infinite number of points.

Does this imply that any worldtube is made of an infinite number of wordlines, since there are infinite points that 'make it up'

Yes, and it is an uncountably infinite number of worldlines.

 

[analyst5]

No, it means that, if we travel relative to the rod, the rod's length in our rest frame is defined by a *different* spacelike slice through the set of worldlines that defines the rod--namely, the slice that is orthogonal to *our* worldline, rather than to the rod worldlines. This length is shorter than the rod's rest length, so we say the rod is "length contracted" in our rest frame. But all the worldlines are still there, so all the rod's points are still there (since each point of the rod corresponds to a distinct worldline).

On the spacetime viewpoint, there is no "mechanism"; length contraction is just geometry, i.e., it's a consequence of taking a different spacelike slice through the rod. Nothing about the rod itself changes; all that changes is which spacelike slice we use to define its "length" relative to us.

But why does it get contracted in the sense of different simultaneity use? Imagine that we take a space-like slice while traveling with 0.5 c relative to the rod, we still take each worldline, just their different time stages from the initial rest frame and it this case it can be stated that the distance between the end and the front will remain the same, the rod just won't be made of the same 'parts'. Does the gamma factor regarding simultaneity have to do something with this, because it seems to me that simultaneity itself isn't sufficient to explain why a different spacelike slice has a new length that is shorter than the original one.

 

[Fredrik]

But why does it get contracted in the sense of different simultaneity use? Imagine that we take a space-like slice while traveling with 0.5 c relative to the rod, we still take each worldline, just their different time stages from the initial rest frame and it this case it can be stated that the distance between the end and the front will remain the same, the rod just won't be made of the same 'parts'. Does the gamma factor regarding simultaneity have to do something with this, because it seems to me that simultaneity itself isn't sufficient to explain why a different spacelike slice has a new length that is shorter than the original one.

Do you see that "the rod" is a line segment in spacetime to the observer who's comoving with the rod, and a different line segment to the observer who's moving at 0.5c relative to the rod? If you draw a spacetime diagram, the latter line segment will be longer (in the diagram). You're right that the gamma factor (more accurately: the Lorentz transformation) is involved in explaining why the coordinate distance along that line segment is actually shorter.

 

[Nugatory]

it seems to me that simultaneity itself isn't sufficient to explain why a different spacelike slice has a new length that is shorter than the original one.

We're talking about two different space-like slices so there's no reason why they must have the same length, any more than two different people must have the same weight or height or shoe size.

The relativity of simultaneity comes in because it's the reason why we're talking about two different space-like slices. The length of an object is, by definition, the distance between where one end of the object is and where the other end is at the same time. Relativity of simultaneity means that "where the other end is at the same time" is different for the different observers; for one the length is the distance between point A and point B and for the other it is the distance between point A and point C. There's nothing surprising about these being different.

 

[analyst5]

Do you see that "the rod" is a line segment in spacetime to the observer who's comoving with the rod, and a different line segment to the observer who's moving at 0.5c relative to the rod? If you draw a spacetime diagram, the latter line segment will be longer (in the diagram). You're right that the gamma factor (more accurately: the Lorentz transformation) is involved in explaining why the coordinate distance along that line segment is actually shorter.

I understand that from the space-time diagram, so why is the rod shorter if its line segment is in fact longer for a moving frame?

 

[analyst5]

We're talking about two different space-like slices so there's no reason why they must have the same length, any more than two different people must have the same weight or height or shoe size.

The relativity of simultaneity comes in because it's the reason why we're talking about two different space-like slices. The length of an object is, by definition, the distance between where one end of the object is and where the other end is at the same time. Relativity of simultaneity means that "where the other end is at the same time" is different for the different observers; for one the length is the distance between point A and point B and for the other it is the distance between point A and point C. There's nothing surprising about these being different.

In defining length we use the worltubes of the back and the end of the rod, so if we use different spacelike slices the distance in space must remain constant for each spacelike space unless we use the gamma factor, my question again is how relative simultaneity (which is most certainly affected with the gamma factor) comes into play when defining length, since from what you're saying I still can't grasp why different simultaneity surfaces, or spacelike slices cannot have a constant distance between its endpoints, and I know this can't be true...

 

[Nugatory]

I understand that from the space-time diagram, so why is the rod shorter if its line segment is in fact longer for a moving frame?

A line segment between the positions of the ends of the rod at the same time in the frame in which the rod is moving is shorter than a line segment between the positions of the ends of the rod at the same time in the frame in which the rod is at rest. That's "shorter", not "longer".

It doesn't look that way in the diagram because the diagram is drawn in pixels on the screen of your display device, and distances on the physical surface of that screen obey the Pythagorean theorem: $s=\sqrt{\Delta{x}^2+\Delta{y}^2}$. However, the lengths of space-like line segments in space-time obey the Minkowski geometry instead: $s=\sqrt{\Delta{x}^2-\Delta{t}^2}$.

 

[Chestermiller]

But why does it get contracted in the sense of different simultaneity use? Imagine that we take a space-like slice while traveling with 0.5 c relative to the rod, we still take each worldline, just their different time stages from the initial rest frame and it this case it can be stated that the distance between the end and the front will remain the same, the rod just won't be made of the same 'parts'. Does the gamma factor regarding simultaneity have to do something with this, because it seems to me that simultaneity itself isn't sufficient to explain why a different spacelike slice has a new length that is shorter than the original one.

As reckoned from our rest frame, both ends of the rod are moving. So, to measure its length, we need to record the locations of the two ends of the rod as a functions of time, and subtract the two measurements (at constant times, as reckoned from the synchronized clocks in our frame of reverence). However, to the people in the rod's frame of reference, when our clocks at the ends of the rod read the same time, their clocks at the ends of the rod read different times; so we are not seeing each ends of the rod at the same time. When our clocks read the same time, we are seeing an older version of one end of the rod and a newer version of the other end of the rod.

Chet

 

[analyst5]

A line segment between the positions of the ends of the rod at the same time in the frame in which the rod is moving is shorter than a line segment between the positions of the ends of the rod at the same time in the frame in which the rod is at rest. That's "shorter", not "longer".

It doesn't look that way in the diagram because the diagram is drawn in pixels on the screen of your display device, and distances on the physical surface of that screen obey the Pythagorean theorem: $s=\sqrt{\Delta{x}^2+\Delta{y}^2}$. However, the lengths of space-like line segments in space-time obey the Minkowski geometry instead: $s=\sqrt{\Delta{x}^2-\Delta{t}^2}$.

So when using spacetime diagrams to draw lines of simultaneity (each angle basically represents speed) with each line that is not parallel with the x-axis we pick points and distances that represent the state relative to the origin of the coordinate system, where v=0.
Let' me be clearer with an image:

http://http://upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Simultaneity_Lines.svg/602px-Simultaneity_Lines.svg.png

So the line of the observer with the velocity of 0.25c basically represents the set of points in spacetime view of the observer with zero velocity, who is at origin. His line doesn't get 'contracted', just represents the worldlines from the perspective of the origin observer and what points are simultaneous to him, but the line doesn't represent length contraction in any way?

 

[Fredrik]

so why is the rod shorter if its line segment is in fact longer for a moving frame?

Consider my ugly diagram.

The two vertical lines (one of them coinciding with the t axis) are the world lines of the endpoints of the rod. Let $L_0$ be the rest length of the rid, i.e. the length of the rod in the (t,x) coordinate system, and let $L$ be the length of the rod in the (t',x') coordinate system. Note that at every point on the t' axis, we have $x=vt$, but on the x' axis, we have $t=vx$. In particular, at the point on the x' axis where $x_B=L_0$, we have $t_B=vL_0$.
\begin{align}
&(t_A,x_A)=(0,L_0)\\
&(t'_B,x'_B)=(0,L)\\
&(t_B,x_B)=(vL_0,L_0)
\end{align} The last two coordinate pairs are related by a Lorentz transformation.
\begin{align}
\begin{pmatrix}0\\ L\end{pmatrix} =\gamma\begin{pmatrix} 1 & -v\\ -v & 1\end{pmatrix}\begin{pmatrix}vL_0\\ L_0\end{pmatrix} =\gamma L_0\begin{pmatrix}v-v\\ -v^2+1\end{pmatrix} =\gamma L_0\begin{pmatrix}0\\ 1/\gamma^2\end{pmatrix} =\begin{pmatrix}0\\ L_0/\gamma\end{pmatrix}.
\end{align} This implies that $L=L_0/\gamma$.

Edit: I think it helps the intuition a bit to be aware of how the scale on the t' and x' axes is determined by invariant hyperbolas. See. e.g. Schutz, in particular figure 1.11 on page 15. The x' coordinate at a point where an invariant hyperbola intersects the x' axis is the same as the x coordinate at the point where the same invariant hyperbola intersects the x axis.

 

[A.T.]

I understand that from the space-time diagram, so why is the rod shorter if its line segment is in fact longer for a moving frame?

Seeing length contraction directly in a Minkowski space-time diagram is tricky. The geometrical relationships are somewhat more obvious in a space-propertime diagram:

http://www.adamtoons.de/physics/relativity.swf

Here the contracted length is just a projection of the rotated proper-length onto the space-axis.

 

[Nugatory]

In defining length we use the worldtubes of the back and the end of the rod, so if we use different spacelike slices the distance in space must remain constant for each spacelike space unless we use the gamma factor, my question again is how relative simultaneity (which is most certainly affected with the gamma factor) comes into play when defining length, since from what you're saying I still can't grasp why different simultaneity surfaces, or spacelike slices cannot have a constant distance between its endpoints, and I know this can't be true...

We are drawing different lines between different points on the world-tube of the object, so there's no reason to expect the distance along these lines to be the same. The gamma factor just allows us to calculate the difference in length between these two different lines when we have chosen the endpoints of our two different lines in a particular way.

You might want to go back to Einstein's train thought experiment on the relativity of simultaneity. Call the point where the lightning strikes the rear of train point A. Now ask yourself:
1) where is the front of the train at the same time that the lightning bolt hits the rear of the train according to the train observer's "at the same time"? Call this point B.
2) where is the front of the train at the same time that the lightning bolt hits the rear of the train according to the platform observer's "at the same time"? Call this point C.
These are different points.

Because B and C are different, clearly the distance between point A and point B (length of train in train frame) will be different than the distance between point A and point C (length of train in platform frame). The gamma factor just tells us how different they'll be, and that's because the gamma factor is part of the simultaneity calculation; it tells us how different the two two observers' "at the same time" will be, ad therefore how different points B and C will be, and therefore the difference in lengths.

 

[PeterDonis]

But why does it get contracted in the sense of different simultaneity use?

Because "different simultaneity use" is what happens when you move relative to the rod.

Imagine that we take a space-like slice while traveling with 0.5 c relative to the rod, we still take each worldline, just their different time stages from the initial rest frame

Yes.

and it this case it can be stated that the distance between the end and the front will remain the same

No, it can't, because you can't just arbitrarily "state" what the new distance is; the metric, i.e., the geometry of spacetime, *tells* you what it is.

it seems to me that simultaneity itself isn't sufficient to explain why a different spacelike slice has a new length that is shorter than the original one.

If "simultaneity itself" does not include the metric, i.e., the geometry of spacetime, then yes, you're right. "simultaneity itself" isn't sufficient. You also need to understand how the geometry of spacetime works. Posts #23 (by Nugatory) and #26 (by Fredrik) give good discussions of that.

 

[analyst5]

I am not sure what you mean. There are an infinite number of points between any two points. Points are not countable.

I'm confused about this, Dale, and I know this isn't a relativistic issue but a geometrical one but I'll ask it anyway. Sorry if it sounds too trivial.

So if a line contains infinitely many points, this implies that it isn't possible to measure the distance between some points on the line, since Planck length is defined as the smallest unit of distance. If the Planck length contains also infinitely many points that we cannot measure the distance between the points 'on the middle' of a such length, of course if that's possible. And let's suppose that we have a line that is 1 cm long. Can we measure the distance between any two points on the line, despite this may imply that we come do the realm of decimals in our measurement which may go lower than the value of Planck length? I hope you understand my question, if it needs more detail I'll try harder.

One of the reasons I'm asking this is because I'm trying to connect this to the case of the rotating disc, where each point has a different velocity and therefore a different time dilation value. Since this depends on the distance from centre, and there are infinitely many points, can a distance from the centre be calculated for each point (and therefore the velocity) despite the number of points not being countable?

Regards

 

[Nugatory]

Planck length is defined as the smallest unit of distance.

It is not. It might be the smallest distance that we can measure, but even that is by no means a settled question.

One of the reasons I'm asking this is because I'm trying to connect this to the case of the rotating disc, where each point has a different velocity and therefore a different time dilation value. Since this depends on the distance from centre, and there are infinitely many points, can a distance from the centre be calculated for each point (and therefore the velocity) despite the number of points not being countable?

We can certainly calculate a distance for any single point - write down its coordinates, do some calculation (trivial if we're using $r,\theta$ polar coordinates) and we have the distance.

The fact that we have infinite number of points moving at different velocities just means that we may have to do some integration to get an answer to some questions, such as the distance between two points with different $r$ coordinates. This is the sort of problem that integral calculus was invented for.

 

[Fredrik]

So if a line contains infinitely many points, this implies that it isn't possible to measure the distance between some points on the line, since Planck length is defined as the smallest unit of distance.

The Planck length has no special significance in SR. It may have significance in the real world, but if you're trying to learn SR, you should focus on trying to understand what the theory says, and not let the real world confuse you while you're doing it.