bei48482_FM

The Lorentz Transformation 43

The observer in S, however, measures the end of the same time interval to be

t 2 

so to her the duration of the interval tis

tt 2 t 1 

This is what we found earlier with the help of a light-pulse clock.

Velocity Addition

Special relativity postulates that the speed of light cin free space has the same value

for all observers, regardless of their relative motion.“Common sense” (which means

here the Galilean transformation) tells us that if we throw a ball forward at 10 m/s

from a car moving at 30 m/s, the ball’s speed relative to the road will be 40 m/s, the

sum of the two speeds. What if we switch on the car’s headlights when its speed is ?

The same reasoning suggests that their light, which is emitted from the reference frame

S (the car) in the direction of its motion relative to another frame S(the road), ought

to have a speed of cas measured in S. But this violates the above postulate, which

has had ample experimental verification. Common sense is no more reliable as a guide

in science than it is elsewhere, and we must turn to the Lorentz transformation equa-

tions for the correct scheme of velocity addition.

Suppose something is moving relative to both Sand S. An observer in Smeasures

its three velocity components to be

Vx Vy Vz

while to an observer in S they are

V (^) x V (^) y V (^) z
By differentiating the inverse Lorentz transformation equations for x,y,z, and t, we
obtain
dx dydy dzdz dt
and so Vx

d
d
x
t


1 
c

2 
d
d
x
t

dx dt

dt 

c
d
2
x

dx

dt
dt 

c
d
2
z


 1 ^2 c^2
dx dt

 1 ^2 c^2
dz

dt
dy

dt
dx

dt
dz

dt
dy

dt
dx

dt
t 0

 1 ^2 c^2
t 2 t (^1)

 1 ^2 c^2
t 2 
c
x
2
^

 1 ^2 c^2
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