Finab;b Foutab;b ab;b Fradab;b = 0. (2-127)
Substituting Eqs. (2-83), (2-84), (2-85) and (2-97) into Eqs. (2-117) and
(2-118), one obtains
Aadv/ret = 4 G adv/reta/ 'd
-oO
S6 [Uaa',(7) Vaa',(-Cr)] a'(dT
Jr2
r / \ ~-11 *0
Te Uaa/ / v b' ,a'ZdT, (2-128)
ST= Tadv/ret Tadv/ret
where in the second line T- is the value of the proper time at the intersection of the
world-line of the particle with an arbitrary spacelike hypersurface E(x) containing
x, and in the third line Tadv/ret denotes the advanced or retarded proper time of the
particle relative to the point x. These potentials are the covariant Lienard- Wiechert
potentials. Corresponding to these, the field strength tensors is expressed as
F adv/ret ) "a \ [" b' c' 'bl' ( d')-3
Fvb = Te UbaI,;a -Uaa J;b) ~ ;b'c + cr;b 'Z (;d' Z
[(Ubai';a Uaa'/;b).b. za'%b' + (Uba/J;a Uaa'/;b) a I (u;cd^') 2
+ (Uba'u;a Uaa';b + I' I ,; Vaa';b) Z' (,0bl') 1TTjadv ret
T =adv/ret
e / ( ,, UVa;b) a 'd, (2-129)
Tadv/ret
where the last term is generally named I i!" term, which involves integration over
the entire past or future history of particle.
2.2.4 Derivation of Equations of Motion for an Electric Charge via
World-tube Method
Construction of world-tube.
In order to determine the effect of radiation reaction on the particle one must
keep a record of the energy-momentum balance between the particle and the field.