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2.2 Coincidence limits

It is useful to determine the limiting behaviour of the bitensors σ... as x approaches x ′. We introduce the notation
[Ω...] = lim Ω...(x,x′) = a tensor at x′ x→x′

to designate the limit of any bitensor ′ Ω...(x,x ) as x approaches ′ x; this is called the coincidence limit of the bitensor. We assume that the coincidence limit is a unique tensorial function of the base point x′, independent of the direction in which the limit is taken. In other words, if the limit is computed by letting λ → λ 0 after evaluating Ω (z,x′) ... as a function of λ on a specified geodesic β, it is assumed that the answer does not depend on the choice of geodesic.

2.2.1 Computation of coincidence limits

From Equations (53View Equation, 55View Equation, 56View Equation) we already have

[σ] = 0, [σ ] = [σ ′] = 0. (62 ) α α
Additional results are obtained by repeated differentiation of the relations (57View Equation) and (58View Equation). For example, Equation (57View Equation) implies σγ = gαβσ ασβγ = σβσ βγ, or (gβγ − σβγ)tβ = 0 after using Equation (55View Equation). From the assumption stated in the preceding paragraph, σβγ becomes independent of tβ in the limit ′ x → x, and we arrive at [σ αβ] = g α′β′. By very similar calculations we obtain all other coincidence limits for the second derivatives of the world function. The results are
[σαβ ] = [σα′β′] = gα′β′, [σαβ′] = [σ α′β] = − gα′β′. (63 )
From these relations we infer that [σ αα] = 4, so that [θ∗] = 3, where θ∗ was defined in Equation (61View Equation).

To generate coincidence limits of bitensors involving primed indices, it is efficient to invoke Synge’s rule,

[σ...α′] = [σ...];α′ − [σ...α], (64 )
in which “...” designates any combination of primed and unprimed indices; this rule will be established below. For example, according to Synge’s rule we have [σαβ′] = [σα];β′ − [σ αβ], and since the coincidence limit of σα is zero, this gives us [σ αβ′] = − [σ αβ] = − gα′β′, as was stated in Equation (63View Equation). Similarly, [σα′β′] = [σα′];β′ − [σα′β] = − [σ βα′] = gα′β′. The results of Equation (63View Equation) can thus all be generated from the known result for [σ ] αβ.

The coincidence limits of Equation (63View Equation) were derived from the relation δ σα = σ ασδ. We now differentiate this twice more and obtain σαβγ = σδαβγσδ + σδαβσδγ + σδαγσδβ + σδασδβγ. At coincidence we have

[ ] [ ] ′ [σαβγ] = σδαβ gδ′γ′ + σδαγ gδ′β′ + δδα′ [σ δβγ],

or [σγαβ] + [σβαγ] = 0 if we recognize that the operations of raising or lowering indices and taking the limit x → x ′ commute. Noting the symmetries of σα β, this gives us [σαγβ] + [σαβγ] = 0, or 2 [σ αβγ] − [Rδαβγσδ] = 0, or 2[σαβγ] = Rδ′α′β′γ′[σ δ′]. Since the last factor is zero, we arrive at

[σαβγ] = [σαβγ′] = [σαβ′γ′] = [σ α′β′γ′] = 0. (65 )
The last three results were derived from [σαβγ] = 0 by employing Synge’s rule.

We now differentiate the relation σ = σδ σ α α δ three times and obtain

σ = σ ε σ + σε σ + σε σ + σ ε σ + σε σ + σ ε σ + σε σ + σε σ . α βγδ αβγδ ε αβγ εδ αβδ εγ αγδ εβ αβ εγδ αγ εβδ αδ εβγ α εβγδ

At coincidence this reduces to [σ ] + [σ ] + [σ ] = 0 αβγδ αδβγ αγβδ. To simplify the third term we differentiate Ricci’s identity ε σαγβ = σαβγ − R αβγσε with respect to δ x and then take the coincidence limit. This gives us [σ αγβδ] = [σαβγδ] + R α′δ′β′γ′. The same manipulations on the second term give [σαδβγ] = [σαβδγ] + R α′γ′β′δ′. Using the identity σαβδγ = σ αβγδ − R εαγδσ εβ − R εβγδσ αε and the symmetries of the Riemann tensor, it is then easy to show that [σαβδγ] = [σαβγδ]. Gathering the results, we obtain 3[σ ] + R ′ ′′′ + R ′ ′′ ′ = 0 αβγδ α γβ δ α δβ γ, and Synge’s rule allows us to generalize this to any combination of primed and unprimed indices. Our final results are

1 [σαβγδ] = − -(R α′γ′β′δ′ + R α′δ′β′γ′), 3 1- [σ αβγδ′] = 3 (R α′γ′β′δ′ + Rα′δ′β′γ′) , [σαβγ′δ′] = − 1(R α′γ′β′δ′ + R α′δ′β′γ′), (66 ) 3 1 [σ αβ′γ′δ′] = − -(R α′β′γ′δ′ + R α′γ′β′δ′), 3 ′ ′′′ 1- ′′ ′′ ′′ ′′ [σα βγ δ] = − 3 (R αγ βδ + R α δβ γ).

2.2.2 Derivation of Synge’s rule

We begin with any bitensor ′ ΩAB ′(x,x ) in which A = α ...β is a multi-index that represents any number of unprimed indices, and ′ ′ ′ B = γ ...δ a multi-index that represents any number of primed indices. (It does not matter whether the primed and unprimed indices are segregated or mixed.) On the geodesic β that links x to x′ we introduce an ordinary tensor P M(z) where M is a multi-index that contains the same number of indices as A. This tensor is arbitrary, but we assume that it is parallel transported on β; this means that it satisfies A α P ;αt = 0 at x. Similarly, we introduce an ordinary tensor N Q (z) in which N contains the same number of indices as B ′. This tensor is arbitrary, but we assume that it is parallel transported on β; at x ′ it satisfies QB ′′tα′ = 0 ;α. With Ω, P, and Q we form a biscalar H (x,x′) defined by

′ ′ A B ′ ′ H (x,x ) = ΩAB ′(x, x )P (x )Q (x ).

Having specified the geodesic that links x to x ′, we can consider H to be a function of λ0 and λ1. If λ1 is not much larger than λ0 (so that x is not far from x ′), we can express H (λ1,λ0) as

|| H (λ1, λ0) = H (λ0,λ0) + (λ1 − λ0) ∂H-| + .... ∂ λ1|λ1= λ0

Alternatively,

| ∂H | H (λ1, λ0) = H (λ1,λ1) − (λ1 − λ0)∂-λ-|| + ..., 0 λ0= λ1

and these two expressions give

| | -d-H (λ ,λ ) = ∂H--|| + ∂H--|| , dλ0 0 0 ∂λ0 |λ0=λ1 ∂λ1 |λ1=λ0

because the left-hand side is the limit of [H (λ1,λ1) − H (λ0,λ0)]∕(λ1 − λ0) when λ1 → λ0. The partial derivative of H with respect to λ0 is equal to α′ A B′ ΩAB ′;α′t P Q, and in the limit this becomes ′ ′ ′ [ΩAB ′;α′]tα PA QB. Similarly, the partial derivative of H with respect to λ1 is ′ ΩAB ′;αtαP AQB, and in the limit λ1 → λ0 this becomes [ΩAB ′;α]tα′P A′QB ′. Finally, H (λ0,λ0 ) = [ΩAB ′]PA ′QB ′, and its derivative with respect to λ 0 is [Ω ′] ′tα′P A′QB ′ AB ;α. Gathering the results we find that

{ } α ′ A′ B ′ [ΩAB ′];α′ − [ΩAB ′;α′] − [ΩAB ′;α] t P Q = 0,

and the final statement of Synge’s rule,

[ΩAB ′];α′ = [ΩAB ′;α ′] + [ΩAB ′;α], (67 )
follows from the fact that the tensors M P and N Q, and the direction of the selected geodesic β, are all arbitrary. Equation (67View Equation) reduces to Equation (64View Equation) when σ... is substituted in place of ΩAB ′.
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