All you have to do is solve for the speed at which the Lorentz
contraction is 10/12 so that the car is ten feet long in the garage frame.
Brent
On 12/23/2024 1:10 PM, Alan Grayson wrote:
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
--
You received this message because you are subscribed to the Google
Groups "Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send
an email to [email protected].
To view this discussion visit
https://groups.google.com/d/msgid/everything-list/1211e981-797c-40fa-8926-12d6d45b6697n%40googlegroups.com
<https://groups.google.com/d/msgid/everything-list/1211e981-797c-40fa-8926-12d6d45b6697n%40googlegroups.com?utm_medium=email&utm_source=footer>.
--
You received this message because you are subscribed to the Google Groups
"Everything List" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To view this discussion visit
https://groups.google.com/d/msgid/everything-list/35e7fec6-cad4-428d-a27c-e7d27a06d7b8%40gmail.com.
