Only if they arrive at the same time "in which reference"?
> Regards, Laszlo Lemhenyi
[[Mod. note -- I have several comments.
First, note that Luigi's animation shows the entire spring responding
*instantaneously* to the release of the endpoints. That's not possible
(in special relativity). Rather, when each endpoint is released, a
wave of rarefaction will propagate along the spring away from the
endpoint. That propagation can't happen any faster than the speed of
sound in the material making up the spring, and for a realistic spring
would be considerably slower. I won't try to analyze this in detail
For the rest of what I'm going to write, let's set aside the
speed-of-sound issue, and consider instead some other interesting
physics questions posed by Luigi's animation.
First, it's useful to introduce a bit of terminology:
Let's call the release of the left end of the spring "event L".
And, let's call the release of the right end of the spring "event R".
Luigi's animation poses the following two questions:
(a) If event's L and R each send a photon (i.e., a signal which travels
at the speed of light) back to the wagon's central point, which
photon (left-end or right-end) arrives first?
(b) Which end of the spring is released first, i.e., which of events
L and R happens first?
Let's first consider question (a):
In relativity, the relative temporal ordering of different events
*along a single observer's worldline* is universal: all observers
agree on this ordering. (Since these events are all located along a
single observer's worldline, they are necessarily *timelike*-separated.)
That means that question (a) has a universal answer, i.e., the answer
to question (a) does *not* change from one reference frame to another.
This in turn means that we can compute the answer by using whatever
reference frame (RF) is most convenient. In this case, the wagon RF
is very convenient: the problem is fully symmetric, and it's clear
that in this frame both the left-end and right-end photons arrive
back at the wagon center at the *same* time. By the argument given
in the previous paragraph, that statement ("the left-end and right-end
photons arrive back at the wagon center at the *same* time") is
necessarily true in *any* RF.
Exercise for the reader: explicitly work out the photon propagation in
the ground RF and show that the two photons also arrive simultaneously
in this RF.
Now let's consider question (b):
In relativity, the relative temporal ordering of different
*spacelike-separated* events isn't universal: different observers (RFs)
will in general disagree on this ordering.
Events L and R are spacelike-separated, so they have *no* universal
temporal ordering. So, question (b) as I've written it is inherently
observer-dependent. As we've seen, in the wagon RF events L and R
are simultaneous. But in the ground RF, events L and R are *not*
simultaneous. That is:
(1) In the wagon RF, the time coordinate of event L is equal to the
time coordinate of event R. In other words, the two ends of the
spring are released at the same time, so the spring contracts
(2) In the ground inertial reference frame, the time coordinate of
event L is *not* equal to the time coordinate of event R. In
other words, the two ends of the spring are released at different
times, so the spring contracts *asymmetrically).
Both statements (1) and (2) are correct.
So, the statement
> Since the spring cannot contract in two different ways, one of the two
> contractions must be wrong: which of the two is correct and which is
is ill-posed. The correct statement is that *both* pictures are correct;
the notion of "symmetrical contraction" is observer-dependent, i.e.,
the answer to the question "is the spring contracting symmetrically"
varies from one RF to another.