The van Vleck biscalar is defined by
Eqs. 4.2) and (5.13) imply that at coincidence, and . Equation (6.8), on the other hand, implies that near coincidence,
We shall prove below that the van Vleck determinant satisfies the differential equationincreases along the geodesics. If, on the other hand, , then the congruence expands more rapidly than it would in flat spacetime, and decreases along the geodesics. Thus, indicates that the geodesics are undergoing focusing, while indicates that the geodesics are undergoing defocusing. The connection between the van Vleck determinant and the strong energy condition is well illustrated by Eq. (7.4): the sign of near is determined by the sign of .
To show that Eq. (7.2) follows from Eq. (7.1) we rewrite the completeness relations at , , in the matrix form , where denotes the matrix whose entries correspond to . (In this translation we put tensor and frame indices on an equal footing.) With denoting the determinant of this matrix, we have , or . Similarly, we rewrite the completeness relations at , , in the matrix form , where is the matrix corresponding to . With denoting its determinant, we have , or . Now, the parallel propagator is defined by , and the matrix form of this equation is . The determinant of the parallel propagator is therefore . So we have
To establish Eq. (7.5) we differentiate the relation twice and obtain . If we replace the last factor by and multiply both sides by we find
In this expression we make the substitution , which follows directly from Eq. (7.1). This gives us
and taking the trace of this equation yields
We now recall the identity , which relates the variation of a determinant to the variation of the matrix elements. It implies, in particular, that , and we finally obtain
Living Rev. Relativity 14, (2011), 7
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