Page 1269 - College Physics For AP Courses
P. 1269
Chapter 28 | Special Relativity 1257
(28.1)
This time has a separate name to distinguish it from the time measured by the Earth-bound observer.
Making Connections: GPS Navigation
For GPS navigation to work properly, satellites have to take into account the effects of both special relativity and general relativity. GPS satellites move at speeds of a few miles per second, and although these speeds are just tiny fractions of the speed of light, the accuracy of timing that is needed to pinpoint a position requires that we account for the effects of special relativity (that is, the slower motion of satellite time relative to an observer on Earth). Additionally, GPS satellites are in orbit roughly ten thousand miles above the Earth, where the gravitational force is weaker. From the theory of general relativity, the weaker gravitational force means that time on the satellite is ticking faster. If these two relativistic effects were not accounted for, GPS units would lose their accuracy in a matter of minutes.
Proper Time
Proper time is the time measured by an observer at rest relative to the event being observed.
In the case of the astronaut observe the reflecting light, the astronaut measures proper time. The time measured by the Earth- bound observer is
(28.2)
To find the relationship between and , consider the triangles formed by and . (See Figure 28.6(c).) The third side of these similar triangles is , the distance the astronaut moves as the light goes across her ship. In the frame of the Earth-
bound observer,
Using the Pythagorean Theorem, the distance is found to be
We square this equation, which yields
(28.3)
(28.4)
(28.5)
(28.6)
(28.7)
(28.8)
(28.9)
Substituting into the expression for the time interval gives
Note that if we square the first expression we had for , we get . This term appears in the preceding
equation, giving us a means to relate the two time intervals. Thus,
Gathering terms, we solve for : Thus,
Taking the square root yields an important relationship between elapsed times:
