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". "
Clearly, the analogies above are not the one I wrote about.
By definition and construction of the space-time diagrams, if you have a 2-D interval (restricted, for the sake of argument, only to 2 dimensions 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 the given basis (x, ct), that is, in the given inertial RF. It follows immediately that in another RF, Lorentz-rotated relative to the first one, the corresponding components, and thereby corresponding distance 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 as its length in the new RF) is different from (less than) its proper length (the moving rod contracts!), and the time between the two successive "ticks" of a moving clock (DEFINED as its period in the new RF) is different from (gre
ater than) its proper period (th
John's statemet (moving clock does NOT run slow) flatly contradicts the latter conclusion.
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" = "half a dozen" to prove the result "six" = "quarter of a dozen". The only possible way to justify such a demand can be that, in John's view, the purely spatial system (x, y) is TOTALLY analogous to the spacetime system (x, ct). Such analogy, if taken to this extreme, is false.