4.2.1 Viewing it with a Space-Time Diagram (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.
4.2.1 Viewing it with a Space-Time Diagram (Car and Barn Paradox)
As we did with the twin paradox, here we will look at a space-time
diagram of the car and barn experiment in order to explain the paradox. We
will draw our diagrams such the relative velocity of Carol and Bob is 0.6 c.
In Diagram 4-4 I have drawn the situation keeping Bob's frame of reference
in mind. To keep the diagram from getting too cluttered, a second diagram
(Diagram 4-5) of the same situation will be used to mark points according to
Carol's frame of reference.
Diagram 4-4 Diagram 4-5
t ! t
e | < x > ! e | < x > .
e |< x > ! e |< x 4
e < x > ! e < x . > .
e <| x > ! e <| .4 > . .
e < | x > ! e < | . x 3 .
e < | x> ! e < |. .3>. .
3...3...|.......D ! e < . | . .D .
e < | >x ! e 4 |. . >x .
-------B-2-----o-----2-C----->x ! ---------B-<---.-o.---->-C------>x
e< | > x ! . e< . . | 1 x
A.......|...1...1 ! . A . | . > x
<e | > x ! . . <2 .| > x
< e | > x ! . .2 e . | > x
< e |> x ! . . < e. |> x
< e > x ! . < . 1 > x
< e >| x ! 1 e >| x
< e > | x ! . < e > | x
In the diagrams we have the following: The ">" marks indicate the
positions of the front of the car at different points in time, while the "<"
marks show the back of the car. The e's mark the entrance to the barn, and
the x's mark the exit of the barn. Hopefully it is apparent to the reader
that the car travels from left to right (with respect to the barn) and
passes through the barn. Also note that at the point where the entrance and
exit of the barn cross the x axis (i.e. when the front and back of the barn
are both at t = 0 in Bob's frame), both the front and back doors quickly
close and open again. Those points are labeled B and C.
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