--- On Fri, 7/3/09, John Denker <jsd@av8n.com> wrote:
"There has been a long discussion of the mass × velocity issue,
based on the utterly false assumption that using the spacetime
approach makes this more complicated. It doesn't! I seems that
people who have never tried the spacetime approach are presuming
it guilty of a crime it did not commit, indeed a crime that never
took place"
No serious physics professional denies simplicity and elegance of the space-time description of the world. It is the denial by some professionals
of the intuitive appeal of the relativistic mass and its direct connection with experiment, that triggered the discussion. Another component triggering the discussion is the (ungrounded!) claim that the geometry is something primary whereas in fact it itself has a dynamical aspect, and in GR the space-time geometry explicitly originates from the dynamics (energy-momentum tensor, which includes forces, motion, stress, pressure etc.).
"While we're on the subject, it strikes me as beyond bizarre that
anyone would ask students to give up the idea that the length of
a ruler is /the/ length of the ruler ... and that the length is
invariant with respect to rotations. The spacetime approach says
the length is invariant under rotations in the XY plane and also
rotations in the XT plane, i.e. boosts, i.e. changes of velocity."
Nobody is asking the students to do it, as far as the XY rotations are concerned.
As to the XT rotations, the proper length is NOT invariant under these rotations!
At least not in the sense usually associated with the word "invariant" - as unchangeable either in direct measurement when at rest and in motion (experimental invariant) or as unchangeable under a specified computational procedure (computable invariant) (see below).
In this respect it is different from the proper time between two events. The characteristic analogous to proper time in this respect is the PROPER DISTANCE, not the POPER LENGTH of an object. They are totally different characteristics! Read this:
Suppose that events 1 and 2 are
separated by a space-like interval, that is
s(1,2)^2 = c^2*(t2-t1)^2 - r(1,2)^2
(1)
Then we can find a RF where
both events happen simultaneously, and the interval reduces to pure distance r(1,2) between them. In this RF we have
t2 = t1 and get
r(1,2) = s(1,2) (2)
Generally, the proper distance is the norm of the 4-vector (1). Note that the proper distance between two events is
generally not the same as the proper length of an object whose end
points happen to be respectively coincident with these events. Consider a solid
rod of constant proper length l(0). If you are in the rest frame K0 of the
rod, and you want to measure its length, you can do it by first marking its
end-points. And it is not necessary
that you mark them simultaneously in K0. You can mark one end now (at a moment t1) and the other end later (at a moment t2) in K0, and then quietly measure the distance
between the marks. We can even consider such measurement as a possible
operational
definition of proper length. From the viewpoint of the experimental physics,
the requirement that the marks be made simultaneously is redundant for a
stationary object with constant shape and size, and can in this case be dropped from such definition. Since the rod is stationary in K0, the
distance between the marks is the PROPER LENGTH of the rod regardless of the
time lapse between the two markings. On the other hand, it is not the PROPER DISTANCE between the
marking events if the marks are not made simultaneously in K0. Moreover,
we can make the time separation between the markings so big, that the
corresponding interval becomes the time-like interval! Consider a stationary
rigid rod of 1 m proper length. In its rest frame, we mark its left end now and
its right end 1 million years from now. The interval between the markings is
definitely time-like, in which case it cannot even be assigned a proper
distance. There is no such thing as a proper distance for a time-like interval!
And yet we can measure the distance between the marks, and low and behold, we
obtain exactly 1 m, which is, according to definition, the proper length of the
rod. So the two characteristics are more than just different: in this example one exists whereas the other does not!
Consider now a reciprocal situation: let the
rod move with velocity v relative to
a reference frame K, and let its length be along the direction of its motion.
We want to measure the proper length of the rod, while remaining in K. We can
do it, again, by marking the end-points of the rod and then measuring the
distance between the marks; but now, since the rod is moving, it is absolutely imperative that the instant positions of its leading and trailing edges be
marked simultaneously in K. In this case, the spatial separation r(1,2) between the marks is,
by definition, the proper distance
between the marking events; but it is not
the proper length of the rod! Indeed, the described procedure constitutes the
length measurement of the Lorentz-contracted
rod; the rod’s proper length l(0) (assuming its speed is
known) is obtained by multiplying the measured distance l by the corresponding Lorentz-factor:
l(0) = l*gamma (v) , (3)
which is not equal to r(1,2).
Thus, the proper distance and the proper
length are both relativistic invariants. But they, generally, describe
different characteristics of a process or an object. The proper distance
relates to a pair of events in space-time, which are connected by a space-like
interval; the proper length describes geometrical properties of a material
object observed in its rest frame.
"If the component you get by _projecting_ the ruler onto this or
that set of axes is shorter than /the/ length of the ruler, that's
fine. The foreshortening of the projection has got nothing to do
with how the ruler is constructed, and everything to do with the
geometry of the projection".
As follows from the preceding excerpt, the relativistically contracted length
of the rod measured from a moving RF has nothing to do with "projection" of its proper length since the latter is not the norm of any space-like interval. What John says, pertains to the proper distance, whereas the quantity discussed here is the proper length. The proper length turns out to be a more subtle characteristic than usually thought. It can be called the "conventional invariant", not computable invariant (as is a proper time or invariant mass), let alone directly measurable invariant (like electric charge).
"In contrast, the contraction/dilation approach says that rotation
changes "the" length of the ruler ... not just the length of its
shadow, but "the" length of the ruler itself. It says rulers
can't be trusted, and clocks can't be trusted either. You've got
to be kidding!"
Not at all. The relativistically-contracted length cannot not be denied by giving names (it is rod, not its shadow that is measured, albeit sometimes we can also measure it physical shadow as well - but physical shadow produced by physical illumination, and direct one at that, not oblique shadow, and this direct physical shadow shows the same effect.)
"Some 2300 years ago, Plato pointed out that the shadow of a thing
is not the same as the thing itself."
True, but again, irrelevant to the physics discussed. We measure the mass of a moving object, its length, and time between the start and the end of a process. These are not shadows, and the measurement results form the foundation of relativity. It is wonderful that the experimental results can be collected into and described by an astonishingly elegant theory, but, paraphrasing Plato, the theory of things is not the same as the things themselves.