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I repeat: It is the ordinary Muggle experience that the elapsed timefrom
A to B is independent of path. This is not true in general! Get used toimplicit
it. All the "proofs" that Moe's clock runs slow are based on the
assumption that time "should" be path-independent. However, there isanother
interpretation that is far simpler and far more consistent with thestructure
of the Lorentz group, namely that the clocks are not broken ordistorted, and
that the notion of elapsed time is path-dependent, just as path-length is
path-dependent.
Fayngold, Moses wrote:_______________________________________________
I find the kind of logics used by John here, rather unusual.
It's not unusual ... although it may be unfamilar to some people.
It boils=
down to this:
A moving car does not move. Why? Because the true properties of an =
object can be observed only in the object's rest frame, and any drive=
r knows that in the rest frame of a car the car does not move. Welcom=
e back to Zeno!
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.
The rotational analog of the twins "paradox" makes the "paradoxical" claim that
an ordinary one-foot ruler is less than an inch long, allegedly because I can
arrange two rulers end-to-end in a narrow V shape, with the starting point of
the first ruler only one inch from the ending point of the second ruler. This
V-shape is closely analogous to the travelling twin's world line, as diagrammed
at e.g.
http://www.av8n.com/physics/twins.htm#fig-twins-joe
I put "paradox" in scare quotes, because it's not really a paradox; it's just
silly. The rulers are being used improperly.
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.
=====================
Several people have asked me about the fact that elementary particles (e.g.
muons) are observed to decay more slowly when they are in motion relative
to the lab frame. Doesn't that mean that the muon's on-board clock is
running slow? I say no.
Consider another spatial analogy. Our heros, the twins Moe and Joe, drive
from point A to point B in separate cars. Each car has an odometer and a
clock.
-- Joe's odometer shows that he travelled a distance of 100 miles.
-- Moe's odometer shows that he travelled a distance of 200 miles.
Do we conclude from this that the odometer on Moe's care is somehow broken,
and measures some sort of weird "contracted" miles? Of course not. A far
simpler explanation is that Moe took the scenic route, while Joe took the
direct route.
In the non-relativistic case, we find that even though the odometers give a
path-dependent notion of elapsed distance, the clocks give a path-independent
notion of elapsed time.
The only halfway-tricky thing about relativity is that if the twins travel
at high speeds, their clock readings become just as path-dependent as their
odometer readings.
Do we conclude from this that Moe's clock is somehow broken, and measures
some sort of weird "slow" seconds? Of course not. A far simpler explanation
is that Moe took the scenic route, while Joe took the direct route.
Because of the hyperbolic geometry of Minkowski space, a scenic route will
always rack up _more_ distance and _less_ time than a direct route.
This is a property of the routes!!! It is not a property of the clocks or
odometers.
I repeat: It is the ordinary Muggle experience that the elapsed time from
A to B is independent of path. This is not true in general! Get used to
it. All the "proofs" that Moe's clock runs slow are based on the implicit
assumption that time "should" be path-independent. However, there is another
interpretation that is far simpler and far more consistent with the structure
of the Lorentz group, namely that the clocks are not broken or distorted, and
that the notion of elapsed time is path-dependent, just as path-length is
path-dependent.
=====================
Ludwik Kowalski wrote:
cutThe two approaches to SR (special relativity) described by JohnD