ABSTRACT
The (infinitesimal) geometry of this situation is shown in Figure 9.1.
Since dτ is the length of the hypotenuse, it follows immediately that
dτ = 1
coshα dt. (9.1)
Equivalently, since the position of the moving observer doesn’t change
in the moving observer’s own rest frame, we have, from the invariance of
the interval, that
dx2 − c2dt2 = 0− c2dτ2, (9.2) so that
dτ2 =
( 1− 1
c2
( dx
dt
)2) dt2, (9.3)
or, equivalently,
dτ =
√ 1− u
c2 dt =
γ dt =
coshα dt. (9.4)
This computation yields the same answer (for τ , not for t) in any ref-
erence frame; proper time is independent of reference frame.
