# What is the real meaning of length contraction?

Suppose one inertial observer measures a rod at rest w.r.t. him and another observer is moving w.r.t. rod. We then say that length will be shorter for moving observer but at the instants the first observer is measuring the length, the second observer doesn't even get the length of the rod, he just gets distance between two points in space after Lorentz Transformations because simultaneity is a relative concept. So how is it a length contraction in literal sense? Isn't it a misnomer ?

• The question as to whether length contraction really exists or not is misleading. It doesn't "really" exist, in so far as it doesn't exist for a comoving observer; though it "really" exists, i.e. in such a way that it could be demonstrated in principle by physical means by a non-comoving observer. —Albert Einstein, 1911 – gregsan Jan 28 '14 at 19:32

No it really is a length contraction - but this is easier to see with the classic example of the measured cosmic ray muon flux.

High energy muons shower down on Earth from the upper atmosphere:

• Muons have a mean lifetime of approximately 2.2 microseconds in the laboratory.
• The distance muons needs to travel from the upper atmosphere to the Earth is approximately 15km.

Let's consider two situations:

# 1. Before taking length contraction into account:

A muon travelling close to the speed of light would be expected to travel approximately 660m before decaying. Hence we wouldn't expect the measure any muons at ground level. However the measured flux of muons at ground level is actually 1 cm$^{-2}$ min$^{-1}$... so where are these muons coming from?

# 2. After taking length contraction into account:

If you consider a muon with energy 20 GeV, it has a length contraction factor of 189 - so the distance that a 20 GeV muon observes is from the atmosphere to the earth not 15km, it is 79m! The length has contracted. Hence you would expect the majority of muons at this energy to survive - which is what is observed.

• This answer is not good enough given current understanding of relativity. In the number 1 above, radio active decay has been used to define time: So length contraction with muon can be seen as time dilation. When an object moves, its relative time (defined by its radioactive decay) is lengthened which easily explains why there are still so many muons on the ground. Less radio active time has passed to them so they decay much less. – user124111 Jul 25 '16 at 13:27
• In special relativity length contraction and time dilation occur simultaneously in order preserve volume in 4D spacetime, so you can happily convert between length contraction and time dilation arguments as you like – kd88 Aug 14 '16 at 17:01

Second observer is not supposed to care explaining the result of measurement of the first observer. He has to do his own measurement. If both measure in the same way, the resulting length is shorter for the second observer.

[...] is it a length contraction in literal sense? Isn't it a misnomer ?

The short answer is that people who speak of "length contraction" (or likewise of "time dilation") are thereby not strictly and exclusively referring to proper quantities, but they are making (therefore) "improper statements" (in a specific technical sense. It is therefore not really considered odious to make such "improper statements" in the context of RT; but it is certainly possible, less confusing, and thus preferrable to strictly stick to "proper statements" instead).

To explain in more detail:

Suppose one inertial observer measures a rod at rest w.r.t. him

Of course, the two ends of this rod (let's call them $A$ and $B$) can and should be considered observers in their own right; and they themselves, first of all, should have been able (at least in principle) to determine that they were at rest to each other. Consequently it can be said that $A$ and $B$ are characterized by a distance from each other: "distance $AB$". (Using terminology which permits making "improper statements", and which thus allows to distinguish "improper" from "proper" in the first place, the distance $AB$ would also be called "proper length of rod $AB$".)

and another observer is moving w.r.t. rod.

Let's give this other observer an explicit name, too: say $J$.
We require of course that $J$ moved uniformly (straight, without acceleration) w.r.t. $A$ and $B$ (and others, too, who were at rest w.r.t. $A$ and $B$).
If so, there are many additional observers identifiable (say $K$, $P$, $Q$ ...) who were at rest w.r.t. $J$ (and who consequently were moving w.r.t. $A$ and $B$, just as $J$ was).

Now, of particular interest here is the case that $J$ moved "along the rod"; say first passing $A$ and subsequently passing $B$.

Then let $P$ be the observer (at rest w.r.t. $J$) who also moved "along the rod", first passing $A$ and subsequently passing $B$, such that

• $J$'s indication of passing $B$ was simultaneous to $P$'s indication of passing $A$.

An let $Q$ be the observer (at rest w.r.t. $J$) who also moved "along the rod", first passing $A$ and subsequently passing $B$, such that

• $B$'s indication of passing $J$ was simultaneous to $A$'s indication of passing $Q$.

From those setup conditions, together with Einstein's definition of how to measure "simultaneity" of indications between suitable pairs of participants, follows the value for the distance ratio

$\frac{JP}{JQ} = (1 - \beta^2)$,

where the number $\beta$ quantifies the speed at which the rod (participants $A$ and $B$) and $J$, $P$, $Q$ etc. were moving against each other; in comparison to the speed of light. (I may add an explicit derivation later as an appendix.)

As far as the motion of $J$, $P$, $Q$ etc. w.r.t. $A$ and $B$ can be considered equivalent to the motion of $A$ and $B$ w.r.t. $J$, $P$, $Q$ etc. (and in particular, if the value of the refractive index in the region containing these participants was found as $n = 1$) then the corresponding distance relations may be considered as mutually equivalent as well, viz.

$\frac{JP}{AB} = \frac{AB}{JQ}$,

and therefore

$\frac{JP}{AB} = \sqrt{ \frac{JP}{AB} \frac{AB}{JQ} } = \sqrt{ \frac{JP}{JQ} } = \sqrt{ 1 - \beta^2 }$.

$JP$ denotes first of all plainly the distance of $J$ and $P$ to each other. Of course there is some particlar relation to $A$ and $B$ due to the setup prescription above (especially due to the requirement that $J$'s indication of passing $B$ was simultaneous to $P$'s indication of passing $A$).
As shown above: the distance $JP$ is not equal to the distance $AB$ (if $\beta \ne 0$).

We then say that length will be shorter for moving observer [...]

That's an "improper statement" since participants $J$ and $P$ are plainly other participants than $A$ and $B$ (if $\beta \ne 0$).
Nevertheless, this "improper statement" is referring precisely to the setup prescription, calculations, and results shown above.

Relativistic optical distortions are from the observer's point of view. Additional POV effects occur,

http://bkocay.cs.umanitoba.ca/Students/Theory.html James Terrell, "The Terrell Effect" Am. J. Phys, 57(1) 9–10 (1989) http://www.youtube.com/watch?v=JQnHTKZBTI4

For a relativistc broomstick fitting within a shorter width barn, it depends upon the observer. Perception is maleable.