In the context of special relativity,
-- I drew an analogy between space and time.
-- I drew an analogy between rulers and clocks.
-- I drew an analogy between projection onto Joe's X axis
and a projection onto Joe's time axis.
In the entire history of analogies, I can hardly think of an analogy
that is more appropriate, more useful, and/or more profound.
You have to take into account the (-+++) signature of the metric, but
other than that, what's the problem?
I'll put it to you as a question: If this analogy is false, is there
any analogy that is not false? I'm beginning to wonder whether "six"
is analogous to "half a dozen". "
=20
Clearly, the analogies above are not the one I wrote about.=20
By definition and construction of the space-time diagrams, if you hav=
e a 2-D interval (restricted, for the sake of argument, only to 2 dim=
ensions x and ct), then its x-projection (contravariant x-component) =
is the spatial distance between its end points, and its ct-projection=
(contravariant ct-component) is the time between its end points in t=
he given basis (x, ct), that is, in the given inertial RF. It follows=
immediately that in another RF, Lorentz-rotated relative to the firs=
t one, the corresponding components, and thereby corresponding distan=
ce and time interval between the same events, will be different from =
those in the first RF. As a special case, it follows that the instant=
distance between the edges of a longitudinally moving rod (DEFINED a=
s its length in the new RF) is different from (less than) its proper =
length (the moving rod contracts!), and the time between the two succ=
essive "ticks" of a moving clock (DEFINED as its period in the new RF=
) is different from (greater than) its proper period (the moving cloc=
k runs slow!).
John's statemet (moving clock does NOT run slow) flatly contradict=
s the latter conclusion. =20
Now, John wondered why I do not use the same logics to prove that,=
say, an x-projection of a rod in a 2-D space (x, y) is equal to the =
actual length of the rod. This is equivalent to demanding that I use =
the logics leading to the result "six" =3D "half a dozen" to prove th=
e result "six" =3D "quarter of a dozen". The only possible way to jus=
tify such a demand can be that, in John's view, the purely spatial sy=
stem (x, y) is TOTALLY analogous to the spacetime system (x, ct). Suc=
h analogy, if taken to this extreme, is false.