Katherine watches all this and measures the time between the throw and the catch
using her stopwatch. This is shown in Concept 2.
If they compare their observations afterwards and had incredibly precise equipment,
Katherine will discover that she measured more time passing between the two events
than the professor does with his stopwatch. If the professor moves at everyday speeds
as we show above, the effect will be very small, far too small to be measured on any
typical stopwatch. But if the professor were moving past at 87% of the speed of the
light, Katherine would measure over twice as much time between the two events as the
professor does.
The professor’s time is said to be “dilated”, which means it is passing more slowly. This
is not due to some malfunctioning nor change in functioning of his clock. Any clock in
his frame, including the professor’s “biological clock”, will record less time passing.
Time dilation can be quantified, as the equation at the right shows. To understand the
equation you must understand another term: proper time. The proper time between two
events is the time measured by a clock in the reference frame where the two events
occur in the same place. “Proper” comes from the German for the events’ “own time”.
The proper time is measured with a single clock, which is at rest in the reference frame
in which the events occur at the same xyz coordinates. The professor’s stopwatch
measures the proper time.
Another observer, the student Katherine in this case, watches the professor (and his
stopwatch) pass by at a velocity v. On the right side of the equation is the proper time
interval,t 0. Katherine measures an interval of time t between the events.
We also present the equation in another way. The term that multiplies the proper time,
that is, the reciprocal of the square root term, is frequently used in relativity. It is
represented by the Greek letter Ȗ (gamma) and is called the Lorentz factor.
Physicists have confirmed the principle of time dilation by many methods, such as
studying muons. Muons are subatomic particles that have an average half life of about
2.2 microseconds. That is, in a sample of muons that are stationary relative to an
observer, half of them will decay (change) into other particles during this interval of time.
Fast-moving muons are produced when cosmic rays enter the Earth’s atmosphere and
collide at high speeds with atoms there. The muons travel toward the ground at about
0.999c. Scientists observe that these moving muons decay more slowly than stationary
muons.
Why? The fraction of muons that decay is a function of time, and less time elapses in
the reference frame of the moving muons. In fact, time passes about 22 times more
slowly in the moving muons’ reference frame. If 2.2 μs have passed according to the
scientists’ clock, then only 0.1 μs have passed in the moving muon reference frame,
which is a time interval much shorter than the half life of the muons. The scientists see
that far less than one-half of the muons have decayed in 2.2 μs, and conclude that the
moving muons decay more slowly. In the reference frame of the muons, the decay rate
is unchanged. Time dilation explains the discrepancy.
You can confirm the “22 times” ratio using the time dilation equation. The proper time is
being measured by the muons’ “clock”, which is observed via their decay rate.
Experiments with highly precise clocks have also confirmed Einstein’s conclusions
concerning the effect of motion on time. For instance, scientists have measured a
difference in the time interval measured by a clock in a plane to that measured by a
clock on the ground. Einstein’s general theory of relativity and its predictions figure
prominently into the results, but his two theories account for the discrepancies between
the time intervals measured by the clocks.
Do you experience time dilation? Yes, but very small amounts of it, given how slowly
you move compared to the speed of light. Using the equation to the right, you could
determine that if you travel for an hour in an airplane flying at 1000 km/h, you would
have aged about 0.0000000015 (that is, 1.5×10í^9 ) seconds less than a person who
remained stationary on the ground. To provide another sense of the magnitudes, if you
moved at a speed of 90 km/hr away from a twin for a 100-year lifespan, you would have
lived 11 microseconds less. On the other hand, if you could move at one-tenth the
speed of light during a century of his life, you would be six months younger, and if you
moved at 90 percent of the speed of light, 270,000,000 meters every second, you would
be 44 years old when he was 100.“Moving” observer/clock measure less
timet = “stationary” observer time
t 0 = proper time in “moving” frame
v = speed of reference frame
c = speed of light
t = Ȗt 0
20 seconds have passed
between two events on the
rocket. How much time has
passed on Earth?
t = 25 s
(^644) Copyright 2007 Kinetic Books Co. Chapter 35