"Reality" of length contraction in SR I was in argument with someone who claims that length contraction is not "real" but only "apparent", that the measurement of a solid rod in its rest reference frame is the "real length" of the rod and all other measurements are somehow just "artificial" and "apparent". Seemed to me like a bad conversation about poorly defined words and not really relevant to physics, but then some quotes were offered:

...so that the back appears closer to the front. But of course nothing has happened to the rod itself.

(Rindler)

The effects are apparent (that is, caused by the motion) in the same sense that proper quantities have not changed.

(Resnick & Halliday)
At the same time, the IEP entry on SR insists that length contraction and time dilation are "real", with observable consequences:

Time and space dilation are often referred to as ‘perspective effects’ in discussions of STR. Objects and processes are said to ‘look’ shorter or longer when viewed in one inertial frame rather than in another. It is common to regard this effect as a purely ‘conventional’ feature, which merely reflects a conventional choice of reference frame. But this is rather misleading, because time and space dilation are very real physical effects, and they lead to completely different types of physical predictions than classical physics.
[...] However, this does not mean that time and space dilation are not real effects. They are displayed in other situations where there is no ambiguity. One example is the twins' paradox, where proper time slows down in an absolute way for a moving twin. And there are equally real physical effects resulting from space dilation. It is just that these effects cannot be used to determine an absolute frame of rest.

I went through a lot of Einstein, Minkowski and Lorentz original material, and didn't find anything about what is "real" and what is not. Finally, I know about muons, where the effects of SR seem to be very real (from Wikipedia, but I had seen it in a physics class before):

When a cosmic ray proton impacts atomic nuclei in the upper atmosphere, pions are created. These decay within a relatively short distance (meters) into muons (their preferred decay product), and muon neutrinos. The muons from these high energy cosmic rays generally continue in about the same direction as the original proton, at a velocity near the speed of light. Although their lifetime without relativistic effects would allow a half-survival distance of only about 456 m (2,197 µs×ln(2) × 0,9997×c) at most (as seen from Earth) the time dilation effect of special relativity (from the viewpoint of the Earth) allows cosmic ray secondary muons to survive the flight to the Earth's surface, since in the Earth frame, the muons have a longer half life due to their velocity. From the viewpoint (inertial frame) of the muon, on the other hand, it is the length contraction effect of special relativity which allows this penetration, since in the muon frame, its lifetime is unaffected, but the length contraction causes distances through the atmosphere and Earth to be far shorter than these distances in the Earth rest-frame. Both effects are equally valid ways of explaining the fast muon's unusual survival over distances.

So which is which? Why do Rindler, Resnick & Halliday use the word "apparent"?
 A: Lorentz contraction is easy to understand once you realise that it is not a contraction at all. Instead it is a rotation and the length of the object, or more precisely its proper length, doesn't change at all.
To see this take the usual example of a rod of length $2a$ aligned along the $x$ axis. We'll draw the rod at time $t=0$ in its rest frame $\mathbf S$:

So the ends of the rod are at the positions $(t=0, x=-a)$ and $(t=0, x=a)$.
Now consider a frame $\mathbf S'$ moving at a velocity $v$ with respect to $\mathbf S$ and as usual we'll take the origins of the frames to coincide at $t=0$. To find the positions of the ends of the rod in $\mathbf s'$ we use the Lorentz transformations:
$$\begin{align}
 t' &= \gamma \left( t - \frac{vx}{c^2} \right ) \\
 x' &= \gamma \left( x - vt \right)
\end{align}$$
and with some minor algebra this gives the positions of the ends in $\mathbf  S'$ as:
$$ (0,-a) \rightarrow \left(\gamma a \frac{v}{c^2}, -\gamma a \right) $$
$$ (0,a) \rightarrow \left(-\gamma a \frac{v}{c^2}, \gamma a \right) $$
So in $\mathbf S'$ at $t=0$ the rod looks like:

So in the $\mathbf S'$ frame the rod has been rotated. However it is a rotation in spacetime, not just in space, so as well as moving in $x'$ one end of the rod has rotated forward in the $t'$ coordinate while the other has rotated backwards in $t'$.
The proper length of the rod $\Delta s$ is given by:
$$ \Delta s^2 = \Delta x^2 - c^2 \Delta t^2 $$
So:
$$ \Delta x = \gamma a - -\gamma a = 2\gamma a $$
$$ \Delta t = -\gamma a \frac{v}{c^2} - \gamma a \frac{v}{c^2} = -2\gamma a \frac{v}{c^2} $$
And substituting for $\Delta x$ and $\Delta t$ in our expression for the proper length $\Delta s$ gives:
$$\begin{align}
 \Delta s^2 &= 4\gamma^2a^2 - c^2\,4\gamma^2a^2 \frac{v^2}{c^4} \\
            &= 4\gamma^2a^2 \left(1 - \frac{v^2}{c^2}\right) \\
            &= 4\gamma^2a^2 \frac{1}{\gamma^2} \\
            &= 4a^2
\end{align}$$
And we find that the proper length of the rod is $\Delta s = 2a$, so the proper length of the rod hasn't changed at all. In fact let me emphasise this:

In $\mathbf S'$ the proper length of the rod hasn't changed at all

So why then do we talk about a Lorentz contraction? It's because if you are the observer in the frame $\mathbf S'$ you are not seeing the two ends of the rod at $t'=-\gamma a v/c^2$ and $t'=\gamma a v/c^2$, you are seeing them both at $t'=0$.
Consider the far end of the rod at $(t'=\gamma a v/c^2, x'=-\gamma a)$. To get the position at $t'=0$ we have to subtract off the distance moved in the time $\gamma a v/c^2$, that is:
$$\begin{align}
 x'(t=0) &= -\gamma a + v\,\gamma a v/c^2 \\
         &= -\gamma a \left(1 - \frac{v^2}{c^2} \right) \\
         &= -\frac{a}{\gamma}
\end{align}$$
And likewise for the other end, though I won't go through the details we get $x'(0)=a/\gamma$, so if we view the ends of the rod at $t'=0$ we find the length is:
$$ \ell = \frac{2a}{\gamma} $$
And this is less than the proper length $2a$, and that's why we say the length of the rod has been decreased by Lorentz contraction. It hasn't really been contracted, it's just that due to the rotation in spacetime we are viewing the two ends at different times.
A: 
The laws of physics have the same form for all, but there are different measurements which are equally "real"?

Correct. Having said that, it is often sensible to differentiate between 'apparent' and 'proper' (or 'intrinsic') values, the latter normally measured in the rest frame of the object in question and giving an upper or lower bound for an observable that varies continuously from frame to frame.
However, this does not imply that 'apparent' values are less real: For example, arguably all massive objects have zero proper (3-)momentum, but if you get hit by a train, its apparent momentum will feel quite real to you ;)
Also, proper values need not always exist, in particular in case of light. Eg, there's no way to decide on physical grounds which wavelength should be considered the intrinsic one of a photon: The one at time of emission, or the Doppler-shifted one at time of absorption? The process is time-symmetric and as there is no rest frame for light-like particles, basically the whole continuum of wavelengths is equally (im)proper.
A: I think it's worth noting that the contraction is (at least arguably) not something that "happens to" the moving object. Special relativity is largely just a set of coordinate transformation rules, telling the right definition for length and duration in a coordinate system that is moving relative to one you already know about.
In Newtonian mechanics, position and velocity are only relative to a given coordinate system. If something flies past us along the x axis, we don't say "its x coordinate isn't really changing because it is fixed in the object's rest frame." Instead, we just specify the relevant coordinate system along with the behavior of x. 
In the same way, in special relativity, an object's length is also relative to the coordinate system. But it's really the coordinate system that's different, not the object itself. 
As for Resnick and Halliday using the term "apparent", I think it's probably an unfortunate choice of words, but I think you have to assume they are implicitly referring to the length at rest. In other words, it looks to the naive observer as if it has physically gotten shorter in a way that ought to require crushing, but in fact it's just shorter as a natural result of the coordinate system change.
A: This is a very simple geometric phenomenon, and there's no reason to be confused about it at all.
Forget spacetime for a moment, and consider a long cylinder (a dowel) in ordinary space. If you slice through it at a right angle, the cross section is a circle. If you slice through it at an oblique angle, the cross section is an ellipse.
If the dowel is more generally shaped—square instead of circular, for example—the oblique cross section will still be an elongated version of the right-angle cross section. It will be elongated only in the direction of the slant of your cut, so a square may become a rectangle, a diamond/rhombus, or a more general parallelogram, depending on how you cut it.

That's all that there is to length contraction. Because of the minus sign in the spacetime version of the Pythagorean theorem, the cross section is shortened instead of lengthened. But the basic geometric reason for it is exactly the same. Is it real, or just apparent? Answer that question in the Euclidean case, and you've answered it in the spacetime case.
Another example: suppose you have seven flexible dowels (licorice or thick twist ties, perhaps), bunched together with one in the middle and the other six around it. Now twist them as a unit so that the one in the center is still straight but the others spiral around the center. It may not be obvious, but the bunch will expand when you do this. If you take a right-angle cross section through the bunch, you'll see why: the center tube still has a circular cross section, but the other six have elongated cross sections, so fewer of them can fit around the circle, unless the radius also increases. This is the Ehrenfest paradox, aside from (again) a change of sign. Does it mean space(time) is curved? Answer that in the Euclidean case and you've answered it in the spacetime case. (The answer is no.)
You may think that I've left something out of this analogy: why do people who are moving relative to an object (i.e. human-shaped tubes that are tilted relative to the dowel) "observe" that particular diagonal slice through it to begin with? They don't. The Euclidean version is this: choose a Cartesian coordinate system with the z axis parallel to the human-shaped tube. Take a constant-z slice of the whole space. The dowel part of that slice is an ellipse, not a circle. But why did we pick that particular coordinate system? Why did we take a constant-z slice? My reason was that teachers of special relativity do it. The teachers' reason is, as far as I can tell, that they don't understand geometry. They think that there is a law of physics that makes you "see" that slice. This is exactly like believing that human-shaped tubes in Euclidean geometry "see" planar slices perpendicular to them (or, even more ridiculously, that they see something else but can "compensate" for that in order to derive what's on the One True Planar Slice).
In Euclidean geometry, you can use any coordinate system to solve any problem. Some may be easier than others, but they all work. Many problems can be solved in a coordinate-free way, as people did in the thousands of years between Euclid and Descartes. The same is true of special relativity. The whole point of the equivalence of reference frames is that you can pick any one. They all work. You don't need to pick one that's aligned with a human-shaped tube. Even after you've picked a coordinate system, there's no reason to take constant-x or constant-y or constant-z slices through it. The coordinate system just assigns tuples of numbers to things. It doesn't alter reality.
I don't think that this mess is Einstein's fault. In his original paper, he explicitly constructed Euclidean coordinate systems from clocks and metersticks. When you measured the length of the dowel, you did it with physical objects that were actually present at the endpoints of the interval you were measuring. This has nothing to do with the modern pedagogical idea that if you're walking down the street on Earth, you are somehow magically "seeing" a planar slice extending out to Andromeda.
In Einstein's original paper, he also carefully distinguished coordinate systems from observers. An observer, in the paper, is just a scientist seeing (actually seeing) things and writing them down. What they see is coincidences: an object passing by a clock as the clock reads noon, for example. Because the object and the clock are in the same place when this happens, the light from them reaches the observer (scientist) at the same time regardless of the scientist's motion. The observations are therefore independent of the scientist's motion. Almost no one after Einstein seems to have understood this.
Atmospheric muons
The original question asked about atmospheric muons, so I'll add a section on that.
This is almost the same as the elongation/contraction of the dowel. In spacetime it's a timelike interval instead of a spacelike interval, but the Euclidean analogy is the same.
Imagine a chasm whose boundaries are two parallel lines 1 meter apart. If you try to bridge it perpendicularly with a plank of length 1 meter + epsilon, it will just reach. If you try to bridge it diagonally, it won't reach. If you have a bunch of planks with a median length of 1 meter, half of them will bridge the chasm perpendicularly, but fewer than half will bridge it diagonally. (Again, the sign of the effect is opposite in spacetime.)
The analogue of the length-contraction explanation of muon lifetime is that "relative to the plank" (i.e., measured along a line parallel to the plank), the chasm is wider if the plank is diagonal. The analogue of the time-dilation explanation is that the length of the plank is "used up more quickly" relative to the chasm if it's diagonal. I think that these descriptions are somewhat silly, and I would suspect that someone who endorsed them didn't entirely grok geometry (especially if they didn't seem to know any other way to solve the problem). But neither explanation is wrong as such. You can get numerically correct answers from them, so if you find it easier to think that way then I suppose it's fine.
A: The existing answers do an excellent job of explaining how the so-called 'length contraction' actually arises because of using a relatively rotated (though equally valid) view-point. I would like to build upon them and add just a little note-point on the use of the words real and apparent that I feel are particularly confusing you.
By saying that lorentz contraction is not real, one means that it doesn't really 'do' anything to the rod itself. Four dimensionally speaking, you are looking at the rod from a different 'angle' when you use a different inertial frame. You are not at all affecting the structure of the rod itself. 
If you were to apply forces to the rod (say, put the rod in a non-uniform gravitational field) then you would really change the structure of the rod. The contraction/expansion resulting in such a case would be called 'real' in the sense that it really affects the rod itself. 
A: Discussions about SR are fraught with misunderstandings, in part as a result of ambiguous terms. In the case of the current question, one ambiguous term is the word 'real', which is the cause of much of the confusion.
Suppose you were to look at an ant through a magnifying glass. The enlarged image you see is a real one- it can be photographed or projected upon a screen. It is not an illusion. However, the ant itself has not enlarged. So if you were to ask 'is magnification real'? you would need to clarify the meaning of the question. 
Bear in mind that the effects of relativity are entirely reciprocated, so if a rocket ship appears foreshortened in your reference frame, you appear foreshortened in the frame of reference in the rocket ship. Clearly you are not any shorter in your own reference frame. 
More generally, you exist in an infinite number of reference frames in each of which you are moving with some arbitrary velocity, and in each of which you appear length-contracted to some arbitrary degree and in some arbitrary direction. Clearly you cannot 'really' be length contracted by different amounts simultaneously. Your length varies in each of the reference frames because the observers within them measure your end-points at different times.
A: In special relativity, it is crucial to distinguish between frame independent (proper) quantities and frame dependent (coordinate) quantities.
The proper length of a rod is frame independent, while the coordinate length of a rod is frame dependent.
In a frame in which the rod is at rest, the proper length and coordinate length are equal.
In a relatively moving frame, the coordinate length of the rod is smaller than the proper length.  This phenomenon is length contraction and it is real - the coordinate length of a rod is largest (and equal to the proper length) in the rest frame of the object.
Assuming uniform motion, length contraction is symmetric.  Two identical rods in relative motion have identical proper length but, in the rest frame of each rod, the coordinate length of the other rod is smaller than the rest length.
As with similar questions about (symmetric) time dilation, the question of "which rod is really contracted" implies a misunderstanding of the nature of length contraction.
Like many relativistic results, length contraction can be fully understood in the context of the relativity of simultaneity (which follows from the invariance of the two-way speed of light and the Einstein synchrony convention).
From the rest frame of the rod, a relatively moving frame is measuring the location of the two ends of the rod at different times while, according to a relatively moving frame, the measurements are made at the same time thus accounting for the difference between the measured coordinate length and proper length. 
Update:  In the comments, Frank questions the appropriateness of my describing proper length as frame independent which Frank takes to mean invariant.
While I have yet to grok precisely what Frank objects to, I do wish to quote from Leonard Susskind's lecture notes found here:

We shall show that only the proper length of an object, which can be
  defined to be the length as measured by an observer at rest relative
  to the object, is invariant, and the coordinate length, as measured by
  observers moving relative to the rod, is not the same for all.

A: I will focus on the question "is length contraction real or only apparent"? This amounts to asking, "is an object really shorter than its rest length when observed in a frame relative to which it is moving?" To answer this we need to say what would be required in order for it to be deemed to be "really shorter". I think a good definition of something being shorter than $L$ is if that something can fit inside a container of length less than $L$. But when the thing is moving we need to add a further comment to clarify the timing. I will take it that
if some object $A$ can be entirely inside a container of length $L$ all at the same moment, then $A$ is shorter than $L$.
Now let's illustrate Lorentz contraction by setting up a thought-experiment. We suppose some futuristic scenario where rockets can move very fast and doors can be opened and closed very fast. Suppose there is a spray-painting chamber of length 10 metres in a space-station. Suppose the spray-painting chamber only operates when both its doors are closed. An actuator at the centre of the chamber can be arranged to send light signals to both doors, ensuring that they open and close simultaneously in the space-station's rest frame. Suppose also that there is a space-ship of rest length 30 metres. The question is, can this space-ship fit inside the spray-painting chamber, with both chamber doors closed, and receive a spray-paint?
The unambiguous answer of special relativity to this question is "yes". The space-ship needs to move at about $0.97c$ relative to the space-station. It will then comfortably fit inside and there is time to operate the doors while it flies very fast through the chamber.
There is a short moment while both doors are closed (in the space-station's rest frame) with the visiting space-ship entirely contained inside the chamber.
This answer satisfies the definition I gave earlier as to what it means to be "really shorter". So the Lorentz-contraction is a real effect.
Another way to convince someone is to imagine accelerating some physical body from rest. Let $S$ be the frame in which it is initially at rest. e.g. Earth if you like. If the acceleration is carefully arranged so as to keep the proper length constant, then the body will get shorter and shorter in frame $S$. Yes it really gets shorter. There is no other way to say it. This experiment has been done, to all intents and purposes, in various particle accelerators when a bunch of particles is accelerated in such a way that the proper length of the bunch stays constant.
A: An observer cannot change the whole universe just by accelerating his spaceship. This is why "apparent" means "real for the observer". Spacetime is relative, and the relative spacetime diagram of an observer is changing.
In short: The only absolute, undilated time value is the proper time of an object. Proper time is dilated by time dilation for observers which are moving at a relative velocity v. The observed time is relative and differs from one observer to another. Length contraction is not more and not less than a secondary effect of time dilation which is due to the fact that the relative velocity between observed object and observer is the same.
An example is showing the mechanism between time dilation and length contraction: 
An astronaut (or a muon) is traveling near light speed at a velocity of 0,8 c (i.e. the relative velocity Earth - spaceship), to an exoplanet at a distance of 100 light years. According to the Earth frame, at this speed the travel will take him 125 years. Due to the effects of time dilation his proper time will be only
125 x 0,6 = 75 years. 
As the relative velocity of Earth with regard to the spaceship is the same as the relative velocity of the spaceship with regard to Earth (i.e. 0,8 c), the distance is contracted from the point of view of the spaceship to 
75 x 0,8 = 60 light years.
As a result, the astronaut did not travel 100 light years in 75 years of his life (that would lead to superluminal velocity) but only 
100 x 0,6 = 60 light years. 
By the way, 0,6 is called the reciprocal Lorentz factor, it depends on the relative velocity v.
A: After checking various sources, including Einstein, this link, or this link, it seems that the length an observer measures for a rod in a reference frame where the rod is at rest is "real" (that term though seems to be usually avoided by physicists), but equally "real" is the contracted length of the same rod measured in another reference frame in uniform translation w.r.t. to the first one. "Real" should probably be replaced by "having measurable effects". A related phenomenon, time dilation, has measurable effects as explained here and here. The results of the measurements of the muon in particular are evidence that both view points (the point of view of an observer moving with the muons, and an observer on Earth) are equally "real". 
Side note: while I was checking various references, I noticed that the terminology used in SR sometimes varies between authors. The following might be useful clarifications:


*

*frame of reference (for an object or observer): the set of all points which have the same state of motion from the point of view of the object or observer. 

*rest frame of reference (for an object or observer): the set of all points where the object or observer is stationary. 

*coordinate system: a system of coordinates (t,x,y,z) (in SR) in a given reference frame. Sometimes "reference frame" and "coordinate system" are not distinguished, or used fairly interchangeably by some physicists. 

*invariant (for SR): a measurement whose value is the same when measured directly by all observers. $c$, the speed of light, is an invariant. 

*proper length: the length of an object measured in its rest reference frame (often called $L_0$). For observers in uniform motion (constant speed) w.r.t. to the object, the measured length is $L = L_0/\gamma$ where $\gamma=1/\sqrt{1-v^2/c^2}$. 

*proper distance or proper interval, or invariant interval: $d\sigma = \sqrt{dx_\mu dx^\mu}$. $d\sigma$ is an invariant w.r.t. the Lorentz transformations, but $L_0$ is not. 

*again, various authors may define "proper" length/distance in different ways. I would suggest it is best to use a more precise, mathematical language ($L_0, d\sigma$), as there is complete agreement between the authors about those, and their behavior. In doubt, request a clear definition.

A: Due to the structure of reality, while within it, you cannot distinguish between absolute rest and uniform motion, you cannot distinguish between absolute length and apparent length. Due to these absolutes not being detectable, the absolute structure of reality itself also becomes undetectable. 
With this being the case, many folk conclude that the absolutes do not exist. However, if reality only exists relativistically, then this also means that reality has no absolute structural foundation. But promoting the idea that reality absolutely does not exist, is somewhat disturbing.
However, it is to be seriously noted that if you fully examine the concept of absolute motion, of an object of absolute length, that takes place within an absolute 4 dimensional Space-Time environment, the eventual outcome of this exploration is a full exposure and understanding of Special Relativity, along with a fully independent derivation of all of its equations.  This is what is accomplished if reality is explored from ground up. See http://goo.gl/fz4R0I if interested ( 1hr 39min ).
Meanwhile, Special Relativity is most often not fully understood by those who attempt to view it from up to ground instead of from ground up. 
Meaning, if you examine Special Relativity by starting with the examination of such things as the speed of light or the speed of electromagnetic waves, the understanding of Special Relativity is most often limited. This is due to having started with one or more of the bizarre counterintuitive outcomes of Special Relativity itself.  Thus in this case one is still confined to being within Special Relativity, rather than having stepped outside of it and thus in turn having proceeded to fully encompass it. 
Instead, you simply can expose Special Relativity at work, and that is all.
A: 
Seemed to me like a bad conversation about poorly defined words and not really relevant to physics

Indeed. Wouldn't it be nice to have some common source, a canon, of definitions which presents them unambiguously and consistently, in relation to each other, and (by applying Ockham's razor of course not more notions than are actually distinct)?       
Of course, there wouldn't be any need for using the word "proper" in those definitions, because they'd all be suitable by virtue of having been unambiguously and consistently defined;
and consequently there wouldn't be any need for using the word "apparent" in those definitions either (at least not as a mere verbal distinction to otherwise eponymous "proper" terms).

but then some quotes were offered: 

And (worse) some insisted on such appeals to (what they considered) authority or precedence, rather than taking responsibility themselves. 

that the measurement of a solid rod in its rest reference frame is the "real length" of the rod 

This requires of course first of all a definition how to determine whether or not a pairs of given "ends" had been "at rest" with respect to each other; without involving an appeal to any determinations or values of "length". 
So, any pair of distinct "ends" which are (separate but) at rest to each other shall by characterized by a value their "length", or (synonymously) "distance (from each other)".
Now the fun really starts when considering several (more than two) "ends" which are all (measured to be) pairwise at rest to each other. Then we may consider and determine the real-number values of their length ratios, comparing one pair to another.
Finally (to make a rather lengthy story very short) we can come up with measurement methods for comparing pairs (of ends which are at rest to each other) where each member of one pair is not at rest wrt. each member of the other pair. At least if the motion of these two pairs wrt. each other is characterized by a suitable, common, mutually agreed number $\beta$. The result of comparing these pairs in terms of their (individual) "length" values is (of course, also just) a real length ratio.
A: Length contraction, as a consequence of Einstein's 1905 false constant-speed-of-light postulate, is ABSURD - it implies that unlimitedly long objects can be permanently trapped inside unlimitedly short containers:
http://www.youtube.com/watch?v=uQHPAeiiQ3w 
 "How fast does a 7 m long buick need to go to fit in a 2 m deep closet?"
http://math.ucr.edu/home/baez/physics/Relativity/SR/barn_pole.html
 "These are the props. You own a barn, 40m long, with automatic doors at either end, that can be opened and closed simultaneously by a switch. You also have a pole, 80m long, which of course won't fit in the barn. (...) If it does not explode under the strain and it is sufficiently elastic it will come to rest and start to spring back to its natural shape but since it is too big for the barn the other end is now going to crash into the back door and the rod will be trapped IN A COMPRESSED STATE inside the barn."
A: At first glance it might appear that the question about “reality” of length contraction in Special Relativity, is meaningless.
To take the basic example, two 100 unit long spaceships A and B pass each other, at a respectable fraction of the speed of light. In A’s frame of reference B is much shorter than 100 units, say, only 70 units. And the situation is entirely symmetrical, so in B’s frame of reference it’s A that’s shorter; A is only 70 units as measured from B.
Both viewpoints are as real as it gets, 100% real.

Now we


*

*increase the relative speed, so that B is measured to be only 33 1/3 units long, in the A frame of reference (and vice versa),

*place A end to end with itself in a very small finite but unbounded universe of diameter 100 units, and

*triplicate B and place the three instances placed end to end, circling along right beside A (there’s just enough room for that!).
Well, effectively the size of the universe in the direction of A has now been tripled.
So we jump over on one of the B instances, and now measure this universe's diameter to be 300 units, in the spaceship nose direction (we can do this measurement by physically moving along the lengths of all three B spaceships, around the universe). Then we ask our spaceship producer to make more spaceships, for repeating this exercises in other directions. Each time inflating the universe’s size in some direction by 200% or the like, at will.
Then the “reality” of it all is, like, very questionable…
But what this shows is that Special Relativity needs an infinite universe to play out in. Apply it in a little finite universe and you get all sorts of contradictions. E.g., here's a very much simpler one: send one ray of light A in some direction, and one called B in the exact opposite direction. If you keep still (no acceleration) they would hit you simultaneously after circumnavigating that little universe. But for some reason you start moving in the same direction as A, which means that ray A will have some catching up to do, while ray B will need to go a shorter distance, so B will now hit you first. Which means that the effective speeds are not both c with respect to your new moving frame.
Which again means, that Special Relativity doesn't play well here: it needs more room, an infinite amount of room.
A: Regarding the case of muons created in the upper atmosphere you can treat them as "test-particles" moving in a Schwarzschild (spherically symmetric) gravitational field of the Earth. This is not within the framework of special relativity but switching to general relativity.
Under this assumption the muons created in the atmosphere actually has a longer lifetime than muons that are stationary on the face of the Earth and there is no need for "length contraction". You still need time dilation though. 
