Physics 325
Lecture 21
More on Minkowski Diagrams
The Minkoski diagrams make some of the strange things we’ve been investigating very
apparent.
Consider for example (Kogut sec. 3.2) the world lines of two clocks at rest in
frame S:
Clocks 1 (on the left) and 2 (on the right) are at rest in the frame S.
Their world lines
take them from points along the
x
-axis at
ct=0
to some later time
d=ct
away.
The picture
has been drawn intentionally such that the
x’
axis intersects clock 1 at
t=0
and clock 2 at
t=d
.
This means that both clocks lie on the
x’
axis in S’ at
t’=0
.
At this time, clock 1
reads
t=0
, but clock 2 reads
t=d/c
.
If the separation between the clocks in S is
0
A
(the
proper distance between the clocks), then
0
tan
d
ϕ
β
=
=
A
.
so that
0
d
β
=
A
and the time on clock 2 is
0
0
2
v
d
t
c
c
c
β
=
=
=
A
A
This is the same result that we got before:
The two clocks are at rest and synchronized in
S (they are at constant
t
in the Minkowski diagram), but separated by
0
A
.
When viewed
from S’, the clocks are moving to the left at velocity
v
.
When clock 1 reads
t=t’=0,
clock
2 reads
2
0
t
v c
=
A
.
x
′
x
ct
1
tan
ϕ
β
−
=
d
1
2

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