where
Qa a' ni' (2-141)
K -j;a'b'Za'' a;.) 1/2 (2-142)
The equations of motion .
The conservation law of energy and momentum, whose differential form was
given by Eq. (2-114), can be expressed in integral form using the bi-vector of the
geodetic parallel displacement, in which the contributions to the integral at the
variable point x is referred back to some fixed point z. This integral is a local
covariant vector at z, and Gauss's theorem can be employ, -1 Then, one may write
0 gaa' ab;b 4
ga+ + ) g dbb 9a ;bg ab 4x, (2-143)
where Ei and E2 are the hypersurfaces or "caps" at the proper times r1 and T2,
respectively, and E represents the surface of the world-tube between E1 and E2,
and V is the volume of the tube, enclosed by Ei, E2 and E. Now by taking the
limit c -i 0, the integrals over Ei, E2 and V will retain contributions only from the
particle stress-energy tensor. Furthermore, taking the fixed point z to lie on the
particle's world-line at a proper time T, which is r- < r < T2, will give
0 limr r'
0 -lim 9 gaa" abd + in0 [gb"Ia ((z"),_ -)) 9" (,T)] T//
-0 J4 T r"=T1
-mo gb1a';," (z(r"), z(r)) b" (-") c" (r")dr", (2-144)
where the replacement has been made,
I j j (2-145)
//S J 47r