On 1/5/2025 6:00 AM, 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??
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
*Brent; I have been studying your plots again. CMIIAW, but ISTM that
you've recapitulated the paradox, namely that the car fits in garage
in garage frame, but doesn't fit in garage in car frame. So fitting or
not is frame dependent. So, IYO, does the paradox simply rest on the
unfounded assumption that fitting or not is an absolute reality, and
cannot be frame dependent? TY, AG*
Not exactly. The paradox in the car/garage version is whether the doors
can be closed with the car inside (assuming instantaneously operating
doors). In the garage frame it seems clear that they can be for a short
time without being crashed into; so what happens in the car frame. It
happens that the simultaneous closing of the doors in the garage frame
is not simultaneous in the car frame. It's simpler to think about if we
start with the exit door closed. Then in the garage frame there is a
short time in which we can close the entrance door with the car inside
before we have to open the exit door to avoid the car crashing into it.
Lorentz transforming these same events to the car frame shows that the
exit door opens before the front of the car hits it and the back of the
car has not yet entered the garage. Then the entrance door closes after
the back of the car has passed, but at this time in the car's frame the
front of the car is well beyond the exit door. So whether the car was
ever completely inside the garage is frame dependent. There is no
objective fact, "The car fitted in the garage."
Brent
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