7 van Vleck determinant

7.1 Definition and properties

The van Vleck biscalar Δ (x,x ′) is defined by

[ ′ ] ′ ′ Δ (x,x′) := det Δ αβ′(x, x′), Δ αβ′(x,x′) := − gαα(x′,x)σαβ′(x,x′). (7.1 )
As we shall show below, it can also be expressed as
[ ] ′ det − σ αβ′(x,x ′) Δ (x,x ) = −----√---√----′---, (7.2 ) − g − g
where g is the metric determinant at x and g′ the metric determinant at x′.

Eqs. 4.2View Equation) and (5.13View Equation) imply that at coincidence, [Δ α′′] = δα′′ β β and [Δ ] = 1. Equation (6.8View Equation), on the other hand, implies that near coincidence,

′ ′ 1 ′ ′ ′ Δ αβ′ = δαβ′ +--Rαγ′β′δ′σ γσ δ + O (𝜖3), (7.3 ) 6
so that
1- α′ β′ 3 Δ = 1 + 6R α′β′σ σ + O (𝜖). (7.4 )
This last result follows from the fact that for a “small” matrix a, 2 det(1 + a ) = 1 + tr(a) + O (a ).

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

1-( α) Δ Δ σ ;α = 4, (7.5 )
which can also be written as α α (ln Δ ),ασ = 4 − σ α, or
d --∗-(ln Δ ) = 3 − 𝜃∗ (7.6 ) dλ
in the notation introduced in Section 3.4. Equation (7.6View Equation) reveals that the behaviour of the van Vleck determinant is governed by the expansion of the congruence of geodesics that emanate from x′. If 𝜃∗ < 3, then the congruence expands less rapidly than it would in flat spacetime, and Δ increases along the geodesics. If, on the other hand, ∗ 𝜃 > 3, then the congruence expands more rapidly than it would in flat spacetime, and Δ decreases along the geodesics. Thus, Δ > 1 indicates that the geodesics are undergoing focusing, while Δ < 1 indicates that the geodesics are undergoing defocusing. The connection between the van Vleck determinant and the strong energy condition is well illustrated by Eq. (7.4View Equation): the sign of Δ − 1 near x′ is determined by the sign of α′ β′ R α′β′σ σ.

7.2 Derivations

To show that Eq. (7.2View Equation) follows from Eq. (7.1View Equation) we rewrite the completeness relations at x, αβ abα β g = η ea eb, in the matrix form −1 T g = E ηE, where E denotes the 4 × 4 matrix whose entries correspond to eαa. (In this translation we put tensor and frame indices on an equal footing.) With e denoting the determinant of this matrix, we have 1∕g = − e2, or e = 1∕ √−-g. Similarly, we rewrite the completeness relations at x′, gα′β′ = ηabeα′eβ′ a b, in the matrix form ′− 1 ′ ′T g = E ηE, where ′ E is the matrix corresponding to α′ ea. With ′ e denoting its determinant, we have 1∕g ′ = − e′2, or √ ---- e′ = 1 ∕ − g ′. Now, the parallel propagator is defined by ′ gαα′ = ηabgα ′β′eαaeβb, and the matrix form of this equation is ˆg = E ηE ′Tg′T. The determinant of the parallel propagator is therefore ˆg = − ee′g′ = √ −-g′∕√ −-g. So we have

√ ---- √ --- [ α ] − g′ [ α′] − g det g α′ = -√----, det gα = √----′, (7.7 ) − g − g
and Eq. (7.2View Equation) follows from the fact that the matrix form of Eq. (7.1View Equation) is Δ = − ˆg−1g −1σ, where σ is the matrix corresponding to σ αβ′.

To establish Eq. (7.5View Equation) we differentiate the relation 1 γ σ = 2σ σ γ twice and obtain γ γ σ αβ′ = σ ασ γβ′ + σ σ γαβ′. If we replace the last factor by σαβ′γ and multiply both sides by α′α − g we find

′ ′( ) Δ αβ′ = − g αα σ γασ γβ′ + σγσ αβ′γ .

In this expression we make the substitution σ ′ = − g ′Δα′′ αβ αα β, which follows directly from Eq. (7.1View Equation). This gives us

Δ α′′ = gα′gγ ′σ αΔ γ′′ + Δ α′′ σγ, (7.8 ) β α γ γ β β;γ
where we have used Eq. (5.11View Equation). At this stage we introduce an inverse ′ (Δ − 1)αβ′ to the van Vleck bitensor, defined by α′ −1 β′ α′ Δ β′(Δ ) γ′ = δ γ′. After multiplying both sides of Eq. (7.8View Equation) by − 1β′ (Δ )γ′ we find
α′ α′ β α − 1γ′ α′ γ δ β′ = g αg β′σ β + (Δ )β′Δ γ′;γσ ,

and taking the trace of this equation yields

4 = σ α + (Δ −1)β′Δ α′ σγ. α α′ β′;γ

We now recall the identity −1 δ ln detM = Tr(M δM ), which relates the variation of a determinant to the variation of the matrix elements. It implies, in particular, that ′ ′ (Δ −1)βα′Δ αβ′;γ = (ln Δ ),γ, and we finally obtain

4 = σα + (ln Δ ),ασα, (7.9 ) α
which is equivalent to Eq. (7.5View Equation) or Eq. (7.6View Equation).


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