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2.4 Expansion of bitensors near coincidence

2.4.1 General method

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

The expansion we seek is of the form

′ γ′ 1- γ′ δ′ 3 Ω α′β′(x,x ) = Aα′β′ + A α′β′γ′ σ + 2 A α′β′γ′δ′ σ σ + 𝒪 (ε ), (83 )
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 ε measures the size of a typical component of σ α′.

To find the expansion coefficients we differentiate Equation (83View Equation) repeatedly and take coincidence limits. Equation (83View Equation) immediately implies [Ω α′β′] = A α′β′. After one differentiation we obtain ′ ′ ′ ′ ′ ′ Ω α′β′;γ′ = A α′β′;γ′ + A α′β′ε′;γ′σ ε+ Aα′β′ε′σεγ′ + 12 Aα′β′ε′ι′;γ′σεσ ι+ A α′β′&#x03B, and at coincidence this reduces to [Ωα ′β′;γ′] = A α′β′;γ′ + Aα′β′γ′. Taking the coincidence limit after two differentiations yields [Ω α′β′;γ′δ′] = A α′β′;γ′δ′ + Aα′β′γ′;δ′ + Aα′β′δ′;γ′ + A α′β′γ′δ′. The expansion coefficients are therefore

A ′ ′ = [Ω ′ ′], α β α β A α′β′γ′ = [Ωα′β′;γ′] − A α′β′;γ′, (84 ) Aα′β′γ′δ′ = [Ωα′β′;γ′δ′] − A α′β′;γ′δ′ − A α′β′γ′;δ′ − Aα′β′δ′;γ′.
These results are to be substituted into Equation (83View Equation), and this gives us ′ ′ ′ Ωα β (x, x ) to second order in ε.

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′ ≡ g β′Ω α′β β whose indices all refer to the point ′ x. Then &tidle; Ω α′β′ can be expanded as in Equation (83View Equation), and the original bitensor is reconstructed as ′ Ωα′β = gββ&tidle;Ω α′β′, or

( ) ′ β′ γ′ 1- γ′ δ′ 3 Ω α′β(x,x ) = g β B α′β′ + Bα′β′γ′ σ + 2 B α′β′γ′δ′ σ σ + 𝒪 (ε ). (85 )
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; ′ ′ = gβ Ω ′ α β β′ α β and its derivatives. After using the results of Equation (80View Equation, 81View Equation, 82View Equation) we find
Bα′β′ = [Ω α′β], B α′β′γ′ = [Ω α′β;γ′] − Bα′β′;γ′, (86 ) 1 ′ B α′β′γ′δ′ = [Ω α′β;γ′δ′] +--B α′ε′R εβ′γ′δ′ − B α′β′;γ′δ′ − Bα′β′γ′;δ′ − Bα′β 2
The only difference with respect to Equation (85View Equation) is the presence of a Riemann-tensor term in B α′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 α β &tidle;Ωα′β′ = g α′g β′Ω αβ whose indices all refer to ′ x, we expand &tidle;Ω α′β′ as in Equation (83View Equation), and we then reconstruct the original bitensor. This gives us

( ) Ωαβ (x, x′) = gα ′gβ′ C α′β′ + Cα′β′γ′ σ γ′ + 1-C α′β′γ′δ′ σγ′σδ′ + 𝒪 (ε3), (87 ) α β 2
and the expansion coefficients are now
C α′β′ = [Ωαβ], Cα′β′γ′ = [Ωαβ;γ′] − Cα′β′;γ′, 1 ε′ 1 ε′ C α′β′γ′δ′ = [Ωαβ;γ′δ′] + 2-Cα′ε′R β′γ′δ′ + 2-Cε′β′R α ′γ′δ′ − Cα′β′;γ′δ′ − C α′β′
This differs from Equation (86View Equation) by the presence of an additional Riemann-tensor term in C α′β′γ′δ′.

2.4.2 Special cases

We now apply the general expansion method developed in the preceding Section 2.4.1 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 α′β′γ′δ′ = − 1(Rα′γ′β′δ′ + R α′δ′β′γ′) 3. In the second instance we have B α′β′ = − gα′β′, B α′β′γ′ = 0, and ′′ ′′ 1 ′ ′′ ′ ′′ ′′ 1 ′ ′′ ′ 1 ′′ ′′ 1 ′′ ′′ B α βγδ = − 3(Rβ α γδ + R βγ αδ ) − 2Rα βγ δ = − 3R αδ βγ − 6R αβ γδ. 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′ − 3 Rα′γ′β′δ′ σ σ + 𝒪 (ε), (88 ) ′( 1 ′ ′) σα′β = − g ββ gα′β′ + -R α′γ′β′δ′ σ γσδ + 𝒪 (ε3), (89 ) ( 6 ) α′ β′ 1 γ′ δ′ 3 σ αβ = gαg β′ gα′β′ −--R α′γ′β′δ′ σ σ + 𝒪 (ε ). (90 ) 3
Taking the trace of the last equation returns ′ ′ σαα = 4 − 13R γ′δ′ σγ σδ + 𝒪 (ε3), or
∗ 1 α′ β′ 3 θ = 3 − 3 R α′β′ σ σ + 𝒪 (ε ), (91 )
where θ∗ ≡ σα − 1 α was shown in Section 2.1.4 to describe the expansion of the congruence of geodesics that emanate from x ′. Equation (91View Equation) reveals that timelike geodesics are focused if the Ricci tensor is nonzero and the strong energy condition holds: When α′ β′ R α′β′ σ σ > 0 we see that ∗ θ 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

α 1 α α′ δ′ 2 α 1 α γ′ α′ δ′ 2 g β′;γ′ = --gα′R β′γ′δ′ σ + 𝒪 (ε ), gβ′;γ = -g α′g γR β′γ′δ′ σ + 𝒪 (ε ) (92 ) 2 2
and thus easy to establish, and they will be needed in Section 4 of this review.

2.4.3 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

( ) Aαβ (x ) = gα′gβ′ A α′β′ − A α′β′;γ′ σ γ′ + 1-Aα ′β′;γ′δ′ σ γ′σδ′ + 𝒪 (ε3). (93 ) α β 2
If the tensor field is parallel transported on the geodesic β that links x to ′ x, then Equation (93View Equation) reduces to Equation (77View Equation). The extension of this formula to tensors of other ranks is obvious.

To derive this result we express A (z ) μν, the restriction of the tensor field on β, in terms of its tetrad components μ ν Aab(λ) = A μνeaeb. Recall from Section 2.3.1 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 α′β′(x′) = Aab(λ0)eaα′ebβ′, A αβ(x) = Aab(λ1)eaαebβ, and Aab (λ1) can be expressed in terms of quantities at λ = λ0 by straightforward Taylor expansion. Since, for example,

| dAab | μ ν λ|| μ νλ|| α ′β′ γ′ (λ1 − λ0)-----|| = (λ1 − λ0 )(Aμνeaeb);λt | = (λ1 − λ0 )A μν;λe aebt| = − Aα′β′;γ′ea eb σ , dλ λ0 λ0 λ0

where we have used Equation (56View Equation), we arrive at Equation (93View Equation) after involving Equation (73View Equation).

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