here - Department of Physics, HKU

CHAPTER 3. LORENTZ TRANSFORMATIONS 19
√
The trajectory of another end is x 2 = x ′ 2 1 − v 2 /c 2 + vt. The distance in
the stationary system is
√
x 2 − x 1 = (x ′ 2 − x ′ 1) 1 − u 2 /c 2
< x ′ 2 − x′ 1 . (3.24)
This is length contraction.
In Newtonian mechanics, distance between two points does not change
with observers. In special relativity, this is not true anymore. Let two events
be labeled by subscripts 1 and 2, and ∆t ′ ≡ t ′ 2 − t′ 1 , etc. Then,
c 2 (∆t ′ ) 2 − (∆x ′ ) 2
= c 2 (t ′ 2 − t′ 1 )2 − (x ′ 2 − x′ 1
⎛
)2
⎞
= c 2 ⎝ t 2 − t 1 − u/c 2 (x 2 − x 1 )
√
⎠
1 − u 2 /c 2
2
−
⎛
⎞
⎝ x 2 − x 1 − u(t 2 − t 1 )
√
⎠
1 − u 2 /c 2
= c2 (∆t) 2 − 2u∆t∆x + u 2 /c 2 (∆x) 2 − ((∆x) 2 − 2u∆x∆t + u 2 (∆t) 2 )
1 − u 2 /c 2
= (c2 − u 2 )(∆t) 2 − (1 − u 2 /c 2 )(∆x) 2
1 − u 2 /c 2
= c 2 (∆t) 2 − (∆x) 2 . (3.25)
This is called the invariant interval between two events, because the value
does not change with observers.
• If ∆s 2 ≡ c 2 (∆t) 2 − (∆x) 2 > 0, the two events are time-like related;
• if ∆s 2 = 0, they are light-like related;
• if ∆s 2 < 0, they are space-like related.
(For three dimensional space, ∆s 2 ≡ c 2 (∆t) 2 − (∆x) 2 − (∆y) 2 − (∆z) 2 .)
Very often, we compare one event with the origin of the frame. Referring
to Fig. 3.1, all events in the upper or lower wedges in the diagram are timelike
related to the origin. All events in the left or right wedges are space-like
related to the origin. All events on the two diagonal lines are light-like related
to the origin. Their significance will be further discussed in the next section.
3.2 Simultaneity
How can we say that two events far apart from each other happen at the
same time? Since the speed of light is constant, we can send light pulses
from the middle point to the two events. If the two light pulses arrive at the
same time with the events, they happen simultaneously, Fig. 3.2.
2