On 12/11/2024 12:16 AM, Alan Grayson wrote:
On Wednesday, December 11, 2024 at 12:44:05 AM UTC-7 Alan Grayson wrote:
On Tuesday, December 10, 2024 at 11:40:10 PM UTC-7 Alan Grayson wrote:
On Tuesday, December 10, 2024 at 11:15:16 PM UTC-7 Brent
Meeker wrote:
Do I not only have provide a diagram I also have to
explain it in detail just to end this silly thread??
*Yes you do. Providing plots without the numerical values in
the LT, is useless. I can't tell if you're drawing plots to
satisfy your biases, or if the numbers support the case you're
making. Lesson learned; always do a real proof, which means
supplying the arguments, or STFU. AG *
*Brent; your numbers check out. The car fits with ease from the
pov of the garage frame, but not from the pov of the car frame.
But this bothers me since we know that all frames are equivalent
in SR. How then can two, so-called equivalent frames, gives
different results? Using the LT, measurements in different frames
generally differ, but here something more fundamental seems to be
happening; namely, that the car fits and doesn't fit, depending on
the frame being analyzed. AG *
*
*
*What I'm getting at is this; if one could do the experiment with a
real car, it would either fit, or wouldn't fit. Do you agree? *
*No, that's the whole point. If you define fit in terms of both ends of
the car simultaneously being within the garage the answer depends on
which reference frame you use to define "simultaneously", the car's
frame or the garages.*
*But in SR, the result is frame dependent. How would you reconcile
this apparent problem or contradiction? AG
*
*There's no contradiction because the first of your propositions "it
would either fit, or wouldn't fit" is false.
Brent
*
First note by comparing the two diagrams that the car is
longer than the garage, 12' vs 10'. So the car doesn't
fit at small relative speed. What does "fit" mean? It
means that the event of the front of the car coinciding
with the right-hand end of the garage is after or at the
same time as the rear of the car coinciding with the
left-had end of the garage. In both diagrams the car is
moving to the right at 0.8c so \gamma=sqrt{1-0.8^2}=0.6.
Consequently, in the car's reference frame, the garage is
contracted to 6' length and when the rear of the car is
just entering the garage, the front is /*simultaneously*/,
in the car's reference frame, already 6' beyond the
right-hand end of the garage.
Then in the garage's reference frame the car's length is
contracted to 0.6*12'=7.2' so at the moment the front of
the car coincides with the right end of the garage, the
rear of the car will simultaneously, in the garage
reference system, be 2.8' inside the garage as shown below.
Note that in the above diagram I have marked two
simultaneous events with small \delta's. The diagram
below is just the Lorentz transform of the one above. The
two simultaneous \delta's are also in the diagram below.
You can confirm they are the same events by referring to
the time blips along the world lines, which are also just
the Lorentz transforms of those above. But clearly the
events marking the simultaneous locations of the rear and
front of the car above are NOT simultaneous in the garage
frame below. Conversely, the front and rear simultaneous
locations of the car below are not simultaneous in the
above diagram, as the reader is invited to confirm by
plotting them. Simultaneity is frame dependent.
Incidentally, when I was in graduate school this was still
know as the "Tank Trap Paradox". The idea was that if one
dug a tank trap shorter than the enemy tank, then the tank
would just bridge the hole, UNLESS the tank were going
very fast in which its contracted length would allow it to
fall into the trap. This was being explained to me by
Jurgen Ehlers, whom you may correctly infer from his name
was a German professor recently hired at Univ Texas. I
said, "What is it with you Germans, illustrating things
with tank traps and cats in boxes with poison gas?"
Jurgen who was too young to have fought in the war didn't
realize I was pulling his leg and he was struck speechless.
Brent
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