On Sunday, December 22, 2024 at 10:05:54 PM UTC-7 Jesse Mazer wrote:
BTW, since you seem to be interested in a scenario where the car and garage
are exactly matched in length in the garage frame, something which isn't
true in Brent's scenario, here's a different scenario you could look at,
where I'm again using units where c=1, let's say nanoseconds for time and
light-nanoseconds (i.e. distance light travels in one nanosecond) for
distance.
--Car's rest length is 25, garage's rest length is 20, car and garage have
a relative velocity of 0.6c, so gamma factor is 1/sqrt(1 - 0.6^2) = 1.25
*OK. *
--In garage rest frame, garage has length 20 and car has length 25/1.25 =
20. In the car rest frame, the garage has length 20/1.25 = 16 and the car
has length 25.
*OK, assuming car is moving, but I wouldn't call that "in the car rest
frame" since you have garage length as contracted. AG *
| - In both frames, set the origin of our coordinate system to be the
point where the back of the car passes the front of the garage--then that
point will have coordinates x = 0 and t = 0 in the garage frame, x' = 0 and
t' = 0 in the car frame.
*OK.*
--In the garage frame, at t = 0 the front of the car is at the same
position as the back of the garage, at position x = 20, so that's the
position and time of the event of the front of the car passing the back of
the garage in the garage frame.
*OK. *
--In the car frame, at t' = 0 the back of the garage must be at x' =
16 (since we know the front of the garage is at position x' = 0 at time
t'=0, and using Lorentz contraction in the car frame we know the garage
has length 16 in this frame), and the front of the car is at rest at x' =
25, so a distance of 25-16 = 9 from the back of the garage, which in this
frame has already passed the front of the car at that moment.
?
--If the back of the garage is moving at 0.6c in the -x' direction and at
t' = 0 is now a distance 9 away from the front of the car, we can conclude
that in this frame it must have passed the front of the car at 9/0.6 = 15
nanoseconds earlier. So at t' = -15 in the car frame, the back of the
garage was at the same position as the front of the car, which has a fixed
position of x' = 25 in the car frame.
?
--Since all the car clocks are synched to coordinate time t' in the car
frame, this tells us that when the front of the car was passing the back of
the garage, the clock at the front of the car showed a reading of -15
nanoseconds.
?
--And this prediction about the reading on the clock at the front of car
when it passes the back of the garage, which was calculated above just
using the garage's contracted length and velocity combined with the idea
that the front of the garage was at position x' = 0 at time t' = 0 in the
car frame, matches up with what you'd get if you instead used the LT to
calculate the answer, using the knowledge that in the garage frame, the
front of the car was at position x = 20 at time t = 0. If you apply the LT
equation t' = gamma*(t - vx/c^2) here, you get t' = 1.25*(0 - 0.6*20) =
-15. So, it all works out consistently.
?
Jesse
Perhaps you can rewrite the text on the sections I don't follow. About
ambiguities in your defintion of local events, I was referring to the
comparison of a spacetime event which is transformed to another frame using
the LT. Is the transformed event also local? AG
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