CLASSICAL CHARGED PARTICLES IN ARBITRARY MOTION - Theoretical and Computational Research in the 21st Century - page 136

ab TTTT
−

=++

(31a)

ab ab ab

() () ()

2 − 3 − 4

where

( )

TFFqwawWk k =

c

=

24 2 −

−

− 22

(31b)

rn rn rc n

() ()()

− 2 − 1 − 1

( )

TFFFFqwk ak aw k k =

c

+

c

=

24 −

⎡ + + 2

− 2

− wWk v

− 1

+ k v ⎤

(31c)

⎣ ⎦

rn nr rn rn nr

()

rn

()() ()()

rc n nc r

− 3 − 1 − 2 − 1 − 2

()() (

) ( )

TFFFgqwgwv k =

c

− /4 =

24 −

⎡ +

− 1

+ v k − wk k

− 2

⎤

1

(31d)

⎣ ⎦

1 rn 2 rn r n n r r n

rn rc n

()

− 4 − 2 − 2

with the following properties:

( )

n
Tk
−

n
Tk
−

= 0

n 2 − 4

= 0

Tk =− q /2 wk

− − (31e)

r

rn rn

rn

()

()

()

3

− 4

2

From Eqs. (31a) and (31e), it is clear that

k is a null proper vector of the

Maxwell tensor:

( )

n 2 − 4

Tk = q /2 w k

, (32)

rn r

which was to be expected due to Eqs. (21d), (23a), and (30e):
(

)

- identical to Eq. (32)

Tk

n

=− qw Fk

− 2 n

+ q

2

/2 w k

− 4

=− qw k

2 − 4

1
2 rn rn r r

The notation

Ti
−

, = , , 4

evokes that Eqs. (31b), (31c), and (31d) vary

rn

()

i

like

w − , in consequence:

i

T
−

dominates when w →∞ (away from the charge) ab

()

2

(33)

T
−

T
−

(close from q)
Therefore, the Larmor formula comes from

& dominates when

w → 0

()

ab

ab

()

4

3

Ti
−

, = , 4

T
−

. This and

ab

()

ab

()

i

2

are responsible for the singularities in the point charge's position, there-
fore Teitelboim wrote Eq. (31a) in the following form:

TTT =+

, (34a) rn

rn rn

RB