in this region and Eq. (2-6) is satisfied throughout it. The integral
fIm 8 (8Tac/8xc) dxodx dx2dx3 (2-15)
over the region of spacetime between the two world-tubes of a certain length can
be expressed as a surface integral over the three-dimensional surface of this region.
Then the difference in the flows of energy (or momentum) across the surfaces of
the two tubes should depend only on conditions at the two ends of the length
considered. Thus the information provided by the conservation laws is well defined.
For easier calculations, the simplest configuration of the world-tube is chosen,
with a spherical surface and of a constant radius c for each instant of the proper
time in that Lorentz frame of reference in which the electron is at rest. Also, we
note the following elementary equations for later use
VaV' = 1, (2-16)
Vai = 0, (2-17)
v +vv =a 0, (2-18)
where Va dza/ds and dots denote differentiations with respect to s. After rather
lengthy calculations with the integral of the stress tensor Tac over the world-tube,
one can show that the flow of energy and momentum out from the surface of any
finite length of tube is given as
I 2 2 -1" a Vbfab ds, (2-19)
where terms that vanish with c are neglected. Since this integral must depend only
on conditions at the two ends of the length of tube, the integrand must be a perfect
differential, i.e.,
1 = B. (2-20)
te2 C_- ia CVb fab Ba. (2-20)
2