CLASSICAL CHARGED PARTICLES IN ARBITRARY MOTION - Theoretical and Computational Research in the 21st Century - page 132

( )

−
=++

2 1 c

,

U B Bwksk

k =− w τ

r , r r , r c r

which substituted in Eq. (21b) implies that

( )

(22)

FqBx =− τ = q τ B B − τ

rc , r , c , r , c , c , r

with the form Eq. (21b), meaning that τ and B correspond to functions β

and . Expression (22) was first obtained by Plebañski [6].

Now we shall consider the eigenvalue problem of F  . For this purpose,

ψ

we stem from Eq. (21b), and due to this

r
kk

Uk

r

=− 1 = 0

,

Then, we immediately have one of the two null proper vectors of a nonnull
field (different type from C ) [23]:

proper value = qw −

2

F k

m

= qw

− 2

k

, (23a)

rm r

This suggests that U r may be an eigenvector, but it is not:

( )

b − 2 − 2 2 2

FU =− qw U − qw a − B k

, (23b)

rb r r

( )

Nevertheless, if we multiply Eq. (23a) by and add the resulting

equation to Eq. (23b), we obtain the other null proper vector [24, 25]:

2 2

1
2

aB −

m − 2

qw −

2

F η =− qw η

,

proper value =−

with (23c)

rm r

( )

η η

r

= 0

η

r

=+ Ua

r 2

− Bk

2 r

1
2

, ;

r

It is not usual to find explicitly in the literature. It is clear that these two

proper vectors are independent because:

η

r

k

r

η = 1

(23d)

r

γ

r

Remember that two null vectors

r

and are proportional if and only

ξ

r

if

ξγ = 0

, therefore Eq. (23d) implies the nonparallelism of such proper r

vectors.
With Eqs. (10) and (21b), it is possible to prove that