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Re: [Phys-l] how to explain relativity

In addition to my previous email:

Another way to see that the rope will stretch is to note that the world lines
AA' and BB' in John Denker's diagram are the two temporal axes (shifted relative
to one
another) of system S'. As S' accelerates, the axes bend., "acquiring" more and
more of spatial dimension, in JD terminology, when considered in the Lab frame.
By the same token, the spatial x-axis, when drawn in the Lab frame, will also
bend,
"acquiring" more and moreof temporal dimension. Both effects together will form
the 4-rotation.
As a result, the rope as observed by the S'-observer, will lie along the
x'-axis
which is TILTED RELATIVE TO X in the Lab frame. So, in contrast to AB, A'B' does
not
represent the proper length of the rope, but merely its Lorentz-contracted
length.
The proper length will be represented by AM, where M is the intersection point
between x'
and BB'. In the spirit of Jeffrey Schnick's argument, the unchanged
Lorentz-contracted length
requires stretched proper length (see Ref. [3, 4] in my previous message and the
figures therein).
My impression from discussing this topic is that the formal use of space-time
diagrams alone
is of little help and even worse, can totally mislead the students, if it is not
combined with clear
presentation of underlying physics as the basis for geometric formulation.

Moses Fayngold,
NJIT

________________________________
From: Moses Fayngold <moshfarlan@yahoo.com>
To: Forum for Physics Educators <phys-l@carnot.physics.buffalo.edu>
Sent: Mon, August 2, 2010 8:49:51 PM
Subject: Re: [Phys-l] how to explain relativity

After a long vacation, I want to share with all participants some of my thoughts
on the topic of how to explain relativity, which was discussed here more than a
month ago. I think this topic warrants even late comments since it pops up
periodically on this List.

On 06/17/2010, 4:50 am John Denker wrote:

I don't think length contraction has any physical reality.

It has.

In my experience, all the arguments for its physical reality
are grossly flawed. It's just a question of how many
femtoseconds of thought are required to find the flaw.


It is not only a question of arguments or thought, it is
(to a far greater extent!) a question of existing scientific evidence.
The length contraction is an experimental fact (See. e.g., Ref. [1, 2]).

The fact is, if you rotate a ruler, "the" length of the
ruler does not change. The projection of the ruler onto
this-or-that coordinate system might change, but the actual
length -- the proper length -- does not change.

True.

Similarly if you boost a ruler, "the" length of the ruler
does not change. The projection onto this-or-that coordinate
system might change, but the actual length -- the proper
length -- does not change.


I see many subtle substitutions of terms here.
First, it is not similar, because "the" length of the ruler is NOT an
invariant under 4-rotations. There is a widely spread confusion between
the proper length and the proper distance, which IS such an invariant,
but here we discuss the proper length, which IS NOT (Ref. [3]).
Second, even if it were, by writing "the actual length - the proper length"
you identify two different concepts. The Lorentz-contracted length and the
proper length
are different, but neither of them is less actual (better to say, less real)
than the other.
Third, if you pilot an ultra-relativistic rocket passing by the Earth's
resident Bob, then the 50-m

proper length of the rocket may not change for you, and not even for Bob if you
had used
a specific acceleration program and then email to him your experimental
results;
but this does not undermine 3-m length of your rocket as measured by Bob.
So it may be true that the PROPER length of the rocket does not change after the

boost, but this is totally consistent with the Lorentz-contraction of this
length for another

observer. Different but equally objective values of a certain physical
characteristic
observed from different reference frames is the essence of Relativity.
Fourth, precisely in case discussed below (equal and synchronous
accelerations
of two spaceships in the Lab frame), even the PROPER length of the rope
connecting them
will change (Ref. [4]).

It has been known for more than 100 years that in spacetime,
physics is much more closely connected to the proper length and
proper time than it is to the projections onto whatever
coordinate frame (if any!) is being used at the moment.

True, but not always, and only if we use these connections carefully.

Minkowski (1908) said quite clearly that the right way to think
about relativity -- including position, time, and velocity --
was in terms of invariances.

True, but again, the proper length is not the invariant under 4-rotations - it
is NOT the norm of a 4-displacement (Ref. [3]).

One spaceship accelerates for one day
of its proper time. The other spaceship accelerates for one
day of /its/ proper time. The two motions are congruent,
differing only in a change of position. We can represent this
using a super-simple spacetime diagram:

A' B'
/ /
/ /
| |
| |
A B

The initial length of the rope is the proper distance between
A and B. The final length of the rope is the proper distance
between A' and B'. We can easily evaluate both of these lengths
in the lab frame. But proper length is a Lorentz scalar, so
it is the same in /any/ frame. So the rope does not stretch.

This conclusion does not follow from your argument because the premise of
proper length being a Lorentz scalar is not correct, and because in this
specific process
even the proper length (measured in the rest frame of the system S' (rope + both
spaceships))
will undergo dynamic change IN TOTAL AGREEMENT with geometry of 4-rotations.
Indeed,
if you examine carefully the upper part of your diagram, you will realize that
the events A' and B', while simultaneous

in the Lab frame, are not simultaneous in frame S'. Due of relativity of
simultaneity, accelerations of the edges that occurred
synchronously (simultaneously) in the Lab frame, are not synchronous in the
S'-frame (except for the very first moment,
when the system S' has not yet picked up velocity). After that, each
incremental increase of velocity at B' occurred earlier in S'
than the corresponding increase at A'. As a result, the rope will undergo
stretch and eventually snap.


At this point, I agree with the analysis of Jeffrey Schnick, who wrote:

By the way, the rope breaks. Consider a taut horizontal rope segment of
length L with the sun directly overhead. It is casting a shadow of
length L. A person rotates the rope about a horizontal axis that is
perpendicular to the rope while at the same time stretching the rope so
that its shadow remains at length L. The rope breaks.

In the given problem the rope is rotated in spacetime by boosting its
ends in such a manner that the projection of the length of the rope on
frame (S/O/earth) remains constant. To keep the projection constant one
must stretch the rope. The rope breaks.

This more formal analysis is based on John Denker's analogy of the
Lorentz-contracted length with a shadow. Such analogy is not accurate (Ref.
[3]), but

(ironically!) its logics,when reversing the argument, works well in this case.
If a rod with fixed size

can cast shadows of different length under different angles of illumination, the
shadow of fixed

length under changing angle of illumination will require a rod with
appropriately changing size.

The references below provide with more details illustrating the subtle
relations between the geometry
and dynamics in Relativity.

1. P. Tipler, Elementary Modern Physics, Worth Publishers, 1992
2. M. Fayngold, Special Relativity and How It Works, Wiley-VCH, 2008 (Sec. 7.1,
7.2,, 9.5)
3. M. Fayngold, Three +1 Faces of Invariance, arXiv:1001.0088 (Dec. 2009)
4. M. Fayngold, The Dynamics of Relativistic Length Contraction and the
Ehrenfest Paradox, arXiv:0712.3891 (Dec. 2007)


Moses Fayngold,
NJIT

___________________________________________
Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l




_______________________________________________
Forum for Physics Educators
Phys-l@carnot.physics.buffalo.edu
https://carnot.physics.buffalo.edu/mailman/listinfo/phys-l