CLASSICAL CHARGED PARTICLES IN ARBITRARY MOTION - Theoretical and Computational Research in the 21st Century - page 123

− 1

τ =− wk

where , (4)

r

r =∂ ∂ / X

, r r

therefore we have that , r is a null vector because it is antiparallel to k r .

Every event X r over the same cone has an associated unique value of

τ

τ , that

is, the light cone is the , r is the vector

normal to the cone even though our Euclidian eyes do not see it like that.
Due to Eq. (4) it makes no sense to look for a unitary normal to the cone.
Thanks to Eq. (4), it is easy to obtain these useful relationships:

τ = constant surface; therefore, τ

, , , ,

r − 1 r

r − 1 r

r −

=+

r 1 r

x =− wv k

r − 1 r

a =− ws k

k δ w v k

v =− wa k

, j j , j l

, b b

, C C C

( ) , , ,

WW awksk ==−+

− 1 r

,

Bw =

− 1

1 − W

Wka =−

r

wv =− + k , b b b r b

, C C c r

( )

, , (5)

C

U =+ va Uk

C

=− 1

Uv =− B

⎡ ⎤

BwU Bwks =

− 1

−

2

+

− 1 r

k

c C

⎣ ⎦

C C C , C C r c

, , , .

Uw

c

= 0

C

C 2 C 2 2

U = 0

Ua = a

UU =− a B

, c

, C

C C

In relativity, a spatial triad of vectors is also important at each point of
the curve because this triad is a local frame of reference for an observer
mounted on the particle (see Figure 6.3):

FIGURE 6.3 Orthonormal tetrad.

( )

( )

(6a)

ee

r

= D g 1, 1, 1, − 1

() ()

a b r