4.2.2 The explanation (Car and Barn Paradox)
Description
This article is from the Relativity and FTL Travel FAQ, by Jason W. Hinson jason@physicsguy.com with numerous contributions by others.
4.2.2 The explanation (Car and Barn Paradox)
We are interested in six different occurrences (though only 4 are shown
in the diagrams). The ones not shown in the diagrams are, first, the front
of the car enters the barn, and second, the back of the car exits the barn.
These would appear much lower and much higher (respectively) in the diagram
than is being shown here. The four events that we do note in the diagrams
are (A) the back of the car enters the barn, (B) the entrance door of the
barn closes and opens again, (C) the exit door of the barn closes and opens
again, and (D) the front of the car exits the barn. In the diagrams, I have
marked each of these events with the letters given and drawn lines of
simultaneity (marked with periods) for the observers.
In Diagram 4-4, we see that for Bob (whose lines of simultaneity are
drawn in that diagram), (A) is the first event which happens, and everything
that occurs simultaneous to (A) in Bob's frame of reference is marked with a
1. The next two events in Bob's frame are (B) and (C), which occur
simultaneously. Everything which occurs simultaneous to these events is
marked with a 2. Finally for Bob, (D) occurs, and everything which occurs
simultaneous to it is marked with a 3. Note that for Bob, as the back of the
car enters the barn--event (A)--the front of the car has yet to exit the
barn. Also, when the doors close and open--events (B) and (C)--simultaneous
in Bob's frame, the front and back of the car are inside the barn (they are
marked with 2's). Thus, in Bob's frame, the car is smaller than the barn,
and it is inside the barn when the doors close and open. Finally, after both
doors close and open, the front of the car exits the barn--event (D)--in
Bob's frame.
However, in Diagram 4-5 we see simultaneous events marked from Carol's
frame of reference. Again, the lines of simultaneity at each event are
marked with periods (but here they are drawn from Carol's frame). Now, we
see that the "lowest" line of simultaneity on the diagram from Carol's frame
of reference passes through the event (C), the exit door of the barn closes
and opens. Thus, this event occurs first in Carol's frame. Everything
occurring simultaneous with it in Carol's frame is marked with a 1. Next in
Carol's frame, event (D) occurs, followed by event (A), while event (B)
occurs last. The events occurring simultaneous with these events are marked
2, 3, and 4, respectively. Thus, according to Carol's frame, things happen
as follows: First, while the front of the car is in the barn, but before the
back of the car enters the barn, the exit door of the barn closes and opens.
Next, the front of the car exits the barn. (Note that while the front of the
car is then outside the exit of the barn at this point, the back of the car
has yet to enter the barn in Carol's frame--look along the x' axis, for
example. So for Carol, the barn is smaller than the car.) Next, the back of
the car enters the barn in Carol's frame. Finally, after the front of the
car has exited the barn and the back of the car has entered the barn, the
entrance door of the barn closes and opens.
And there you have it. In the end, each observer must agree that the
car gets through the barn without smashing into the doors. However, each
frame of reference offers a different explanation for how this comes to be,
because in each frame, different events are simultaneous with one another.
In Bob's frame, the car is in the barn all at once while the doors close and
open simultaneously. However, in Carol's frame, the doors do not close
simultaneously, and the car is never completely in the barn.
So, I hope you have seen the power of space-time diagrams when it comes
to explaining things in special relativity. When we simply say that moving
clocks run slower and moving rulers length contract, we miss a real
understanding of special relativity. That understanding comes from realizing
that the actual coordinates in space and time for events are different for
different observers who are moving with respect to one another. This
relationship can be viewed with space-time diagrams, and the answers to many
nagging questions in special relativity can be explained if one understands
these diagrams.
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