A variation 6Qi in the direction cosines produces a variation in the point x,
which is via Eq. (2-131) given by
J;aa/,xI -_n(i),a'6i. (2-136)
A pair of independent variations 61Qj and 652i in the direction cosines define an
element dQ of solid angle by the relation
QidQ = Cijk61Sj62Qk, (2-137)
where eijk is the three-dimensional Levi-Civita. This solid angle defines an element
of two-dimensional area on the surface of the sphere, enclosed by the parallelogram
formed from the corresponding displacements Jix" and 62x'. However, one is rather
interested in a three-dimensional surface element of the world-tube generated by
the sphere as proper time r varies. Then one shall construct a general displacement
of the point x on the world-tube, which is produced by independent variations of
- and Qi, with a linear combination of 61x", 62xa and the third displacement 63 x
orthogonal to the first and second, forming a parallelepiped:
gab1Xa63xb = 0, gab2X as3b = 0. (2-138)
Later, integrals over the world-tube will be evaluated to compute the energy-
momentum flow, and for this purpose one defines the directed surface element dEa,
which is a vector density, formed from independent displacements 61x", 62Xa and
63Xa
da = EabcdlX6b62X63Xd. (2-139)
In terms of the radius of tube c, variation of solid angle dQ and variation of proper
time dr, the surface element at x is expressed as
dEy(x) = 22 g-1/2 ()gaa'a' ( + C Rb'c' b' c dQdr + 0(e5), (2-140)
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