Elements of Modern Physics
10
Also, since the relative velocity of the frames is v, t 1 – t 0 = l/v which leads to
1/ 2
2
1
0
2
1
l
v
t t
v
c
− =
−
′ ′
1
0
1/ 2
2
2
1
1
v
t t
v
c
− δ
− =
′
′
−
(1.26)
Therefore, the lag δ is given by
2
lv
c
δ =
(1.27)
The time dilation of moving clocks can be made more physical by considering
a clock which consists of a beam of light bouncing back and forth between two
mirrors kept at a distance l apart along the y′-direction (Fig. 1.3). In frame F′,
each round trip takes a time
2l
t
c
′
∆ =
(1.28)
Viewed from frame F, the beam travels a longer distance along a line
making an angle θ with the y-axis, given by sin θ =
v
c
. Therefore the
corresponding time observed for each trip is
1/ 2
2
2
2
1
l
t
v
c
c
∆ =
−
(1.29)
Combining Eqs. (1.28) and (1.29), one has
1/ 2
2
2
1
t
t
v
c
′
∆
∆ =
−
(1.30)
10
Also, since the relative velocity of the frames is v, t 1 – t 0 = l/v which leads to
1/ 2
2
1
0
2
1
l
v
t t
v
c
− =
−
′ ′
1
0
1/ 2
2
2
1
1
v
t t
v
c
− δ
− =
′
′
−
(1.26)
Therefore, the lag δ is given by
2
lv
c
δ =
(1.27)
The time dilation of moving clocks can be made more physical by considering
a clock which consists of a beam of light bouncing back and forth between two
mirrors kept at a distance l apart along the y′-direction (Fig. 1.3). In frame F′,
each round trip takes a time
2l
t
c
′
∆ =
(1.28)
Viewed from frame F, the beam travels a longer distance along a line
making an angle θ with the y-axis, given by sin θ =
v
c
. Therefore the
corresponding time observed for each trip is
1/ 2
2
2
2
1
l
t
v
c
c
∆ =
−
(1.29)
Combining Eqs. (1.28) and (1.29), one has
1/ 2
2
2
1
t
t
v
c
′
∆
∆ =
−
(1.30)