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In Figure (6.1), we have not indicated this last vector since it might be
outside or inside the light cone.
From an event
X
r
outside
C
, we trace its null cone's past sheet which
intersects to
C
in the point
x
r
called “retarded event associated with
X
r
”,
therefore we say that
()
xxX
r
=
r
i
(2a)
Thus with
X
i
given, the retarded point over
C
is automatically determined.
This allows us to introduce the vector:
(2b)
r
r
r
kXx
=−
of magnitude zero, because it rests over the cone:
,
(2c)
kk
r
=
0
r
Therefore,
k
r
indicates the propagation direction of the photons emitted
by the particle. The null or light-type vector Eq. (2b) is truly important in
electrodynamics: we could say that studying the Maxwell field is almost
equivalent to an analysis of the null cone and its relation to the world line.
From
X
r
, we can build two distances widely used in the study of charg-
es in Minkowski space:
FIGURE 6.2(A)
Instantaneous distance from
X
r
to
C
.
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