Re: SR cannot determine Contraction

On Feb 26, 8:06 pm, Tom Roberts <tjroberts...@xxxxxxxxxxxxx> wrote:


Uniform motion makes objects be measured to be shorter than when they
are at rest. And if the "measurement apparatus" includes doors that
close simultaneously in their frame, then for appropriate values of the
parameters a moving object can fit between closed doors that are closer
together (in their rest frame) than the proper length of the object --
at least in principle.

Remember how the length of a moving object is measured: its front and
rear locations are marked SIMULTANEOUSLY in the measuring frame, and
then the distance between the marks is measured with rulers at rest in
the measuring frame. But the order can be reversed: one can pre-position
doors an appropriate distance apart, and "mark" the ends of the object
by opening/closing them. See my description of the gedanken without
accelerations, and my solution to the problem you posed.


Tom Roberts


The measurement of the length of moving objects is nothing more than
an exercise in projective geometry. Depending how you do the
projections, you get a DIFFERENT result.
The final result depends on the "cut", the plane of simultaneity use.
Here is an illustration.

Let the barn frame be S' and the rod frame be S. The rod moves along
the positive x' axis at speed v.

The Lorentz tranforms are :

x'=g(x+vt) (1)
t'=g(t+vx/c^2)

Using (1) we can determine the realtionships in terms of spatial and
temporat separation via differentiation:

d_x'=g(d_x+v*d_t)
d_t'=g(d_t+v/c^2*d_x) (2)

Theory of measurement tells us that if we want to make a correct
determination of the length of the moving rod d_x' in S' we need to
mark both ends simultaneously in S':


d_t'=0 i.e. d_t=-v/c^2*d_x (3)

Substituting (3) into the first equation of (2) we obtain :

d_x'=g*d_x*(1-(v/c)^2)=g^-1*d_x (4)

(4) is the standard expression for length contraction.


Now, let's try something different: what happens if we mark both ends
of the rod simultaneously in S? So, we now take d_t=0
Substituting in the second equation of (2) we obtain:

d_t'=g*v/c^2*d_x (5)


The first equation of (2) becomes simply:

d_x'=g*d_x (6)

This is a very surprising result since it shows length dilation!
.

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