On Sunday, January 5, 2025 at 6:02:30 PM UTC-7 Brent Meeker wrote:
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
*You claim there is no objective fact. The car fitted in the garage. But
that's only from the garage frame. It doesn't fit from the car frame,
regardless of the doors, which IMO can be dispensed with. So, as I see it,
the paradox follows from the belief that there can't be disagreement about
what the frames conclude. Isn't this the claim that must be disproven to
resolve the paradox, and a constructive proof that the frames disagree
using the LT is insufficient? AG*
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