" 1) First of all, in spacetime, a car at rest *does* move. It move=
s
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 starti=
ng point
of many a calculation.
2) While we are discussing analogies, I find it odd that people wh=
o 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 th=
e other.
I'm not talking about Lorentz contraction here; I'm talking about pl=
ain 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 _d=
oes_.
The projection of the clock or ruler onto your field of view will dep=
end
on your viewpoint, but that is a property of the projection, not a pr=
operty
of the clock or ruler."
***********************
John=92s reference to 4-velocity =93u=94 was irrelevant to my point=
. I also tell the students in my Relativity class that, in terms of 4=
-velocity, we all =93move through spacetime=94 at the speed of light =
(or at the universal speed |u| =3D 1, depending on choice of units), =
and the only difference between a photon and a boulder in this respec=
t is in the tilt of their respective world lines. This, however, has =
NOTHING TO DO with my previous example. My example was about 3-veloci=
ty v, which is indisputably different in different RF, but must be id=
entically zero in all of them according to John's logics.=20
The same is true for the temporal and spatial components of a 4-int=
erval. It is true that the PROPER period to of a clock (determined, s=
ay, by its two consecutive =93ticks=94) is relativistic invariant =
=96 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 =96 by the similar definition; and the same ca=
n be said about the rest energy of an object. But this does not mean =
that the time interval between the same two =93ticks=94 remains to in=
another RF. There is clearly defined operational procedure of its di=
rect measurement in case when the clock in question is moving: it is =
the time between the corresponding instant readings of the two identi=
cal synchronized clocks in the second RF at the moments of passing of=
the test clock by them. Similarly, there is an operational definitio=
n 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 d=
efinitions have not been specially designed to suit the needs of the =
relativity theory, - they are natural procedures that anyone with com=
mon sense would use for a direct measurement of the period of a movin=
g clock in a Lab or the length of a moving car in the non-relativisti=
c domain as well. These two attributes, uniquely defined by the opera=
tional procedure of their measurement, turned out to be as relative a=
s is 3-velocity, and this discovery was one of the deepest insights o=
f Einstein, and forms an essential part of what Relativity is about. =
Even though the Lorentz transformations, forming the back-bone of SR,=
had been already known before Einstein=92s work, it is Einstein, who=
is credited as the author of SR, because he was the first to realize=
that the time interval between two events (say, two ticks of a clock=
, or creation and subsequent decay of a mu-meson) is really different=
in different RF, and accordingly, the time variable introduced by Lo=
rentz in his transformations is indeed real time in another RF, rathe=
r than a convenient auxiliary variable, as Lorentz himself had initia=
lly 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 obv=
iously false statement that the projection of a stick in John=92s exa=
mple is also the length of the stick. This claim results from the fal=
se analogy. The projection of a stick onto another direction is not i=
ts length because applying a ruler to this projection would not const=
itute a direct length measurement of the stick. You do not directly m=
easure the length of the stick by applying the ruler at a finite angl=
e (let alone perpendicular) to it at its edge.=20
Note the word =93direct=94 here. We could still measure it indirect=
ly by the additional measuring of the corresponding angle, say, betwe=
en the tilted rod and its shadow cast on the floor by a vertical beam=
of light, and adding the additional computational operation =96 the =
division by the cosine of this angle. Measuring the shadow alone does=
not constitute the length measurement, and so the length of the shad=
ow is not the length of the rod. In contrast, the properly taken proj=
ection of the 4-interval (necessarily including time!) between the tw=
o events in spacetime does give physical time interval and the spatia=
l distance between these events in the corresponding RF, simply becau=
se this projection represents graphically the operational definition =
of the direct measurement of the corresponding attributes.=20
This comparison directly illustrates my point in one of my previous=
messages on this subject, - that the pseudo-Euclidean geometry of sp=
acetime is not identical to the geometry of space alone, and the anal=
ogy between them has important limitations.=20
As I said in the previous message, the whole discussion might be me=
rely the matter of different interpretations of the same thing. Now I=
see that the situation is more serious =96 namely, the disagreement =
between us actually reflects fundamentally different attitudes toward=
s basic operational definitions of what constitutes distance and time=
between two events as considered from different RF. These definition=
s are so clear and unambiguous, that they leave no room for different=
interpretations. If you accept them, and reason correctly, you immed=
iately arrive at the conventional relativistic statements. If not, ju=
st say so, and we move apart peacefully without any farther dispute.