" 1) First of all, in spacetime, a car at rest *does* move. It moves
toward the future at the rate of 60 minutes per hour.
That is, its four-vector velocity (u) is not zero. The components of (u) are
[1, 0, 0, 0].
This may be unfamiliar to some people, but it is useful as the starting point
of many a calculation.
2) While we are discussing analogies, I find it odd that people who defend the
notion that "moving clocks run slow" don't seem willing to defend the notion
that a ruler is shorter when viewed nearly end-on.
If you want to be logically consistent, you can't have one without the other.
I'm not talking about Lorentz contraction here; I'm talking about plain old
rotation. Note that the rotation group is a subgroup of the Lorentz group.
Most people consider it obvious that the proper length of a ruler is independent of the viewing angle.
...........
There is a profound analogy between rotations and boosts. Viewing a ruler
end-on doesn't change what the ruler _is_. Similarly viewing a clock from
some boosted reference frame doesn't change what the clock _is_ or _does_.
The projection of the clock or ruler onto your field of view will depend
on your viewpoint, but that is a property of the projection, not a property
of the clock or ruler."
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John’s reference to 4-velocity “u” was irrelevant to my point. I also tell the students in my Relativity class that, in terms of 4-velocity, we all “move through spacetime” at the speed of light (or at the universal speed |u| = 1, depending on choice of units), and the only difference between a photon and a boulder in this respect is in the tilt of their respective world lines. This, however, has NOTHING TO DO with my previous example. My example was about 3-velocity v, which is indisputably different in different RF, but must be identically zero in all of them according to John's logics.
The same is true for the temporal and spatial components of a 4-interval. It is true that the PROPER period to of a clock (determined, say, by its two consecutive “ticks”) is relativistic invariant – by its definition as the result of its direct measurement in the rest frame of the clock; and the PROPER length of a rod, too, is the relativistic invariant – by the similar definition; and the same can be said about the rest energy of an object. But this does not mean that the time interval between the same two “ticks” remains to in another RF. There is clearly defined operational procedure of its direct measurement in case when the clock in question is moving: it is the time between the corresponding instant readings of the two identical synchronized clocks in the second RF at the moments of passing of the test clock by them. Similarly, there is an operational definition of length of a moving rod as the distance between two simultaneous marks of its front and rear edge. It is
important to note that both def
ther RF,
rather than a convenient auxiliary variable, as Lorentz himself had initially thought. In particular, t is not equal to to, and l is not equal to lo. The opposite statement essentially takes us back to Newton.
John claims that if I were consistent, I should also defend the obviously false statement that the projection of a stick in John’s example is also the length of the stick. This claim results from the false analogy. The projection of a stick onto another direction is not its length because applying a ruler to this projection would not constitute a direct length measurement of the stick. You do not directly measure the length of the stick by applying the ruler at a finite angle (let alone perpendicular) to it at its edge.
Note the word “direct” here. We could still measure it indirectly by the additional measuring of the corresponding angle, say, between the tilted rod and its shadow cast on the floor by a vertical beam of light, and adding the additional computational operation – the division by the cosine of this angle. Measuring the shadow alone does not constitute the length measurement, and so the length of the shadow is not the length of the rod. In contrast, the properly taken projection of the 4-interval (necessarily including time!) between the two events in spacetime does give physical time interval and the spatial distance between these events in the corresponding RF, simply because this projection represents graphically the operational definition of the direct measurement of the corresponding attributes.
This comparison directly illustrates my point in one of my previous messages on this subject, - that the pseudo-Euclidean geometry of spacetime is not identical to the geometry of space alone, and the analogy between them has important limitations.
As I said in the previous message, the whole discussion might be merely the matter of different interpretations of the same thing. Now I see that the situation is more serious – namely, the disagreement between us actually reflects fundamentally different attitudes towards basic operational definitions of what constitutes distance and time between two events as considered from different RF. These definitions are so clear and unambiguous, that they leave no room for different interpretations. If you accept them, and reason correctly, you immediately arrive at the conventional relativistic statements. If not, just say so, and we move apart peacefully without any farther dispute.