
447Theory of Special Relativity
This quadratic expression is commonly known as an interval, and we denote it by s
2
; it is an invari-
ant, independent of the frame used to measure these coordinates. To see it, let us calculate the value
of x
2
− c
2
t
2
(for simplicity) in S frame in terms of x′ and t′ in S′ frame:
xctxut ctux c
cu
2222 22222
222
−=
′
+
′
−
′
+
=− −
′
γγ
γ
() ()
()
/
ttuxt ux txuc
xct
22
222
22+
′′
−
′′
+
′
−
=
′
−
′
(
.
/1)
As an interval is an invariant quantity, we may use it to classify the possible events in space–time.
In particular, we may divide events into three types: those for which the invariant label is positive,
zero, or negative. Events for which the ...