6 Expansion of bitensors near coincidence

6.1 General method

We would like to express a bitensor Ω ′′(x,x′) αβ near coincidence as an expansion in powers of α′ ′ − σ (x,x ), the closest analogue in curved spacetime to the flat-spacetime quantity ′α (x − x). For concreteness we shall consider the case of rank-2 bitensor, and for the moment we will assume that the tensorial indices all refer to the base point x ′.

The expansion we seek is of the form

′ γ′ 1- γ′ δ′ 3 Ωα′β′(x,x ) = A α′β′ + Aα′β′γ′σ + 2A α′β′γ′δ′σ σ + O (šœ– ), (6.1 )
in which the “expansion coefficients” A α′β′, A α′β′γ′, and A α′β′γ′δ′ are all ordinary tensors at ′ x; this last tensor is symmetric in the pair of indices γ ′ and δ′, and šœ– measures the size of a typical component of σ α′.

To find the expansion coefficients we differentiate Eq. (6.1View Equation) repeatedly and take coincidence limits. Equation (6.1View Equation) immediately implies [Ωα′β′] = A α′β′. After one differentiation we obtain ′ ′ ′ ′ ′ ′ Ω α′β′;γ′ = A α′β′;γ′ + A α′β′šœ–′;γ′σ šœ–+ Aα′β′šœ–′σšœ–γ′ + 12A α′β′šœ–′ι′;γ′σ šœ–σι + A α′β′&, and at coincidence this reduces to [Ωα ′β′;γ′] = A α′β′;γ′ + Aα′β′γ′. Taking the coincidence limit after two differentiations yields [Ωα ′β′;γ′δ′] = A α′β′;γ′δ′ + A α′β′γ′;δ′ + A α′β′δ′;γ′ + Aα′β′γ′δ′. The expansion coefficients are therefore

[ ] Aα′β′ = [Ω α′β′,] A α′β′γ′ = Ω α′β′;γ′ − A α′β′;γ′, [ ] A α′β′γ′δ′ = Ω α′β′;γ′δ′ − A α′β′;γ′δ′ − A α′β′γ′;δ′ − Aα′β′δ′;γ′. (6.2 )
These results are to be substituted into Eq. (6.1View Equation), and this gives us Ω ′′(x,x′) αβ to second order in šœ–.

Suppose now that the bitensor is Ω α′β, with one index referring to ′ x and the other to x. The previous procedure can be applied directly if we introduce an auxiliary bitensor &tidle;Ωα′β′ := gββ′Ω α′β whose indices all refer to the point ′ x. Then &tidle; ′′ Ω αβ can be expanded as in Eq. (6.1View Equation), and the original bitensor is reconstructed as β′ Ω α′β = g β&tidle;Ω α′β′, or

( ) ′ β′ γ′ 1- γ′ δ′ 3 Ωα ′β(x,x ) = g β Bα′β′ + B α′β′γ′σ + 2 Bα′β′γ′δ′σ σ + O (šœ– ). (6.3 )
The expansion coefficients can be obtained from the coincidence limits of Ω&tidle;α′β′ and its derivatives. It is convenient, however, to express them directly in terms of the original bitensor Ω α′β by substituting the relation Ω&tidle;α′β′ = gβ ′Ωα′β β and its derivatives. After using the results of Eq. (5.13View Equation) – (5.15View Equation) we find
B ′′ = [Ω ′], αβ [ α β ] Bα′β′γ′ = Ωα ′β;γ′ − Bα′β′;γ′, [ ] 1 ′ B α′β′γ′δ′ = Ωα ′β;γ′δ′ + --Bα′šœ–′R šœ–β′γ′δ′ − Bα′β′;γ′δ′ − B α′β′γ′;δ′ − B α′β′&#x0 2
The only difference with respect to Eq. (6.3View Equation) is the presence of a Riemann-tensor term in Bα′β′γ′δ′.

Suppose finally that the bitensor to be expanded is Ω αβ, whose indices all refer to x. Much as we did before, we introduce an auxiliary bitensor &tidle;Ω ′ ′ = g α gβ Ω α β α′ β′ αβ whose indices all refer to x′, we expand &tidle;Ω α′β′ as in Eq. (6.1View Equation), and we then reconstruct the original bitensor. This gives us

( ) ′ α′ β′ ′ ′ ′′ ′ γ′ 1- ′ ′′′ γ′ δ′ 3 Ωαβ(x,x ) = g αg β Cα β + C αβ γσ + 2 Cα βγ δσ σ + O (šœ– ), (6.5 )
and the expansion coefficients are now
[ ] Cα ′β′ = Ωαβ , C α′β′γ′ = [Ωαβ;γ′] − C α′β′;γ′, C ′′ ′′ = [Ω ′′] + 1-C ′ ′R šœ–′′′′ + 1C ′′Ršœ–′′ ′′ − C ′′ ′ ′ − C ′′ ′′ − C ′ ′′ ′. (6.6 ) αβ γδ αβ;γδ 2 α šœ– βγ δ 2 šœ–β α γδ αβ ;γ δ αβ γ;δ α βδ ;γ
This differs from Eq. (6.4View Equation) by the presence of an additional Riemann-tensor term in C α′β′γ′δ′.

6.2 Special cases

We now apply the general expansion method developed in the preceding subsection to the bitensors σ α′β′, σ α′β, and σ αβ. In the first instance we have A α′β′ = gα′β′, A α′β′γ′ = 0, and A ′′ ′′ = − 1(R ′ ′′ ′ + R ′′ ′′) αβ γδ 3 α γβ δ αδ βγ. In the second instance we have B ′′ = − g ′′ αβ αβ, B ′ ′′ = 0 α βγ, and 1 1 1 1 B α′β′γ′δ′ = − 3(Rβ′α′γ′δ′ + R β′γ′α′δ′) − 2Rα′β′γ′δ′ = − 3R α′δ′β′γ′ − 6R α′β′γ&#x20. In the third instance we have C α′β′ = gα′β′, Cα′β′γ′ = 0, and 1 C α′β′γ′δ′ = − 3(R α′γ′β′δ′ + R α′δ′β′γ′). This gives us the expansions

1- γ′ δ′ 3 σα′β′ = gα′β′ − 3R α′γ′β′δ′σ σ + O (šœ– ), (6.7 ) β′( 1 ′ ′) σα′β = − g β gα′β′ + -R α′γ′β′δ′σγ σδ + O (šœ–3), (6.8 ) ( 6 ) σαβ = gα′gβ′′ gα′β′ − 1R α′γ′β′δ′σ γ′σ δ′ + O (šœ–3). (6.9 ) α β 3
Taking the trace of the last equation returns σα = 4 − 1R γ′δ′σγ′σ δ′ + O (šœ–3) α 3, or
1 ′ ′ šœƒ∗ = 3 − -R α′β′σ ασ β + O (šœ–3), (6.10 ) 3
where šœƒ∗ := σ αα − 1 was shown in Section 3.4 to describe the expansion of the congruence of geodesics that emanate from x ′. Equation (6.10View Equation) reveals that timelike geodesics are focused if the Ricci tensor is nonzero and the strong energy condition holds: when Rα′β′σα′σβ′ > 0 we see that šœƒ∗ is smaller than 3, the value it would take in flat spacetime.

The expansion method can easily be extended to bitensors of other tensorial ranks. In particular, it can be adapted to give expansions of the first derivatives of the parallel propagator. The expansions

gα ′ ′ = 1gα ′Rα′′′ ′σ δ′ + O (šœ–2), gα′ = 1gα ′g γ′R α′′′′σδ′ + O(šœ–2) (6.11 ) β ;γ 2 α βγ δ β ;γ 2 α γ βγ δ
and thus easy to establish, and they will be needed in part III of this review.

6.3 Expansion of tensors

The expansion method can also be applied to ordinary tensor fields. For concreteness, suppose that we wish to express a rank-2 tensor A αβ at a point x in terms of its values (and that of its covariant derivatives) at a neighbouring point ′ x. The tensor can be written as an expansion in powers of α′ ′ − σ (x,x ) and in this case we have

( ) α′β′ γ′ 1- γ′ δ′ 3 Aαβ(x ) = g αg β A α′β′ − Aα′β′;γ′σ + 2A α′β′;γ′δ′σ σ + O(šœ– ). (6.12 )
If the tensor field is parallel transported on the geodesic β that links x to x′, then Eq. (6.12View Equation) reduces to Eq. (5.10View Equation). The extension of this formula to tensors of other ranks is obvious.

To derive this result we express A μν(z ), the restriction of the tensor field on β, in terms of its tetrad components μ ν Aab(λ) = A μνeaeb. Recall from Section 5.1 that μ ea is an orthonormal basis that is parallel transported on β; recall also that the affine parameter λ ranges from λ0 (its value at x′) to λ1 (its value at x). We have A ′ ′(x′) = A (λ )ea eb α β ab 0 α′ β′, A (x) = A (λ )eaeb αβ ab 1 α β, and A (λ ) ab 1 can be expressed in terms of quantities at λ = λ0 by straightforward Taylor expansion. Since, for example,

|| ( ) | | (λ1 − λ0 )dAab| = (λ1 − λ0) A μνeμeν tλ|| = (λ1 − λ0)Aμν;λeμeνtλ|| = − A α′β′;γ′eα′eβ′σ γ′, dλ |λ0 a b ;λ λ0 a b λ0 a b

where we have used Eq. (3.4View Equation), we arrive at Eq. (6.12View Equation) after involving Eq. (5.6View Equation).


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