We can obtain the concomitant effect of LORENTZ CONTRACTION without too much trouble^{23.8} using the following Gedankenexperiment, which is so simple we don't even need a Figure:
Suppose a spaceship gets a nice running start and whips by the Earth at a velocity u on the way to Planet X, a distance x away as measured in the Earth's reference frame, which we call O. [We assume that Planet X is at rest with respect to the Earth, so that there are no complications due to their relative motion.] If the spaceship just "coasts" the rest of the way at velocity u [this is what is meant by an INERTIAL REFERENCE FRAME], then by definition the time required for the voyage is t = x/u. But this is the time as measured in the Earth's reference frame, and we already know about TIME DILATION, which says that the duration t' of the trip as measured aboard the ship (frame O') is shorter than t by a factor of : .
Let's look at the whole trip from the point of view of the observer
O' aboard the ship: since our choice of who is at rest and who
is moving is perfectly arbitrary, we can choose to consider the
ship at rest and the Earth (and Planet X) to be hurtling
past/toward the ship at velocity u. As measured in the ship's
reference frame, the distance from the Earth to Planet X is x'
and we must have u = x'/t' by definition. But we also must
have u = x/t in the other frame; and by symmetry they are
both talking about the same u, so
Of course, the effect works both ways. The length of the
spaceship, for instance, will be shorter as viewed
from the Earth than it is aboard the spaceship itself,
because in this case the length in question is in
the frame that moved with respect to the Earth.
The sense of the contraction effect can be remembered by this
mnemonic:
(23.1) |