### 2.5 Van Vleck determinant

#### 2.5.1 Definition and properties

The van Vleck biscalar is defined by

As we shall show below, it can also be expressed as
where is the metric determinant at and the metric determinant at .

Equations (63) and (80) imply that at coincidence, and . Equation (89), on the other hand, implies that near coincidence

so that
This last result follows from the fact that for a “small” matrix , .

We shall prove below that the van Vleck determinant satisfies the differential equation

which can also be written as , or
in the notation introduced in Section 2.1.4. Equation (99) reveals that the behaviour of the van Vleck determinant is governed by the expansion of the congruence of geodesics that emanate from . If , then the congruence expands less rapidly than it would in flat spacetime, and increases 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 Equation (97): The sign of near is determined by the sign of .

#### 2.5.2 Derivations

To show that Equation (95) follows from Equation (94) 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 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

and Equation (95) follows from the fact that the matrix form of Equation (94) is , where is the matrix corresponding to .

To establish Equation (98) 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 Equation (94). This gives us

where we have used Equation (78). At this stage we introduce an inverse to the van Vleck bitensor, defined by . After multiplying both sides of Equation (101) by we find

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

which is equivalent to Equation (98) or Equation (99).