bei48482_FM

Relativity 7

Time dilation t (1.3)

Here is a reminder of what the symbols in Eq. (1.4) represent:

t 0 time interval on clock at rest relative to an observer proper time

ttime interval on clock in motion relative to an observer

speed of relative motion

cspeed of light

Because the quantity  1 ^2 c^2 is always smaller than 1 for a moving object, tis

always greater than t 0. The moving clock in the spacecraft appears to tick at a slower

rate than the stationary one on the ground, as seen by an observer on the ground.

Exactly the same analysis holds for measurements of the clock on the ground by

the pilot of the spacecraft. To him, the light pulse of the ground clock follows a zigzag

path that requires a total time tper round trip. His own clock, at rest in the spacecraft,

ticks at intervals of t 0. He too finds that

t

so the effect is reciprocal: everyobserver finds that clocks in motion relative to him

tick more slowly than clocks at rest relative to him.

Our discussion has been based on a somewhat unusual clock. Do the same conclusions

apply to ordinary clocks that use machinery—spring-controlled escapements, tuning

forks, vibrating quartz crystals, or whatever—to produce ticks at constant time intervals?

The answer must be yes, since if a mirror clock and a conventional clock in the space-

craft agree with each other on the ground but not when in flight, the disagreement

between then could be used to find the speed of the spacecraft independently of any

outside frame of reference—which contradicts the principle that all motion is relative.

t 0



 1 ^2 c^2

t 0



 1 ^2 c^2

0

t

t

2

t

2

v–

v

t

2

c– L 0

v

Figure 1.5A light-pulse clock in a spacecraft as seen by an observer on the ground. The mirrors are

parallel to the direction of motion of the spacecraft. The dial represents a conventional clock on the

ground.

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