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 equation
which can also be written as
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
http://www.livingreviews.org/lrr-2011-7 |
Living Rev. Relativity 14, (2011), 7
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