the frame at rest with
respect to an event. In this
case, the sneezing is the
event, so the proper time
is measured by a clock at
rest with respect to the
person sneezing. ∆T 2 , a
time measured in a different
inertial frame, can be
referred to as the dilated
time.Because the denominator of this fraction is never greater than 1, the value of γ is
never less than 1. So unless my spaceship was standing still relative to you, the
time that you’d measure between my sneezes would always be longer than the time
I’d measure. Seems strange? Of course, because, once again, we have no
understanding of moving at speeds close to the speed of light, so we don’t have any
experience for time dilation. In fact, even at v = c (which is about 67 million
miles per hour!) the relativistic factor γ is still only 1.005. But highly sensitive
atomic clocks have been flown on commercial jet airplanes, and it’s been found,
upon landing, that they are a little slow relative to their synchronized counterparts
that didn’t fly. Furthermore, the time difference has been shown to be just what the
relativistic formula above predicts.
For all ordinary speeds, where v is very, very small compared to c, the value of γ
is negligibly greater than 1, so ∆T 2 ≈ ∆T 1 , and we don’t notice a difference for
ordinary time intervals. But, as v gets closer and closer to c, γ gets bigger and
bigger. For example, for passengers on a spaceship moving at, say, v = 0.99c
relative to the earth, the value of γ is about 50. So, if someone on the ship says that
they’ve been on the ship for 2 years (as measured by them), we here on Earth
would say that the elapsed time is 50 × 2 = 100 years.
3) The Relativity of Length
Let’s once again go back to me on my spaceship, passing by the earth at a speed of
c. Let’s say I measure the length of my ship to be 100 meters. If you, on Earth,
were watching my ship fly by, you would not measure its length to be 100 m.
Length is relative. The length as measured by you would not agree with the length
as measured by me.
Proper Length and
Contracted Length