4 Coincidence limits

It is useful to determine the limiting behaviour of the bitensors σ ⋅⋅⋅ as x approaches ′ x. We introduce the notation
[ ] ′ ′ Ω ⋅⋅⋅ = lxi→mx′Ω ⋅⋅⋅(x,x ) = a tensor at 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 → λ0 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.

4.1 Computation of coincidence limits

From Eqs. (3.1View Equation), (3.3View Equation), and (3.4View Equation) we already have

[ ] [ ] [ ] σ = 0, σα = σ α′ = 0. (4.1 )
Additional results are obtained by repeated differentiation of the relations (3.5View Equation) and (3.6View Equation). For example, Eq. (3.5View Equation) implies σγ = g αβσασβγ = σ βσβγ, or (gβγ − σβγ)tβ = 0 after using Eq. (3.3View 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 = σ α′β′ = gα′β′, σαβ′ = σ α′β = − gα′β′. (4.2 )
From these relations we infer that [σ α] = 4 α, so that [𝜃∗] = 3, where 𝜃∗ was defined in Eq. (3.9View Equation).

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

[ ′] [ ] [ ] σ ⋅⋅⋅α = σ ⋅⋅⋅;α′ − σ⋅⋅⋅α , (4.3 )
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 [σαβ′] = − [σ αβ] = − gα′β′, as was stated in Eq. (4.2View Equation). Similarly, [σα′β′] = [σα′];β′ − [σα′β] = − [σ βα′] = gα′β′. The results of Eq. (4.2View Equation) can thus all be generated from the known result for [σαβ].

The coincidence limits of Eq. (4.2View Equation) were derived from the relation σ = σδ σ α α δ. We now differentiate this twice more and obtain δ δ δ δ σ αβγ = σ αβγσδ + σ αβσδγ + σ αγσδβ + σ ασδβγ. At coincidence we have

[σ ] = [σδ ]g′ ′ + [σ δ ]g ′ ′ + δδ′[σ ], αβγ αβ δγ αγ δ β α′ δβγ

or [σ γαβ] + [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[σαβγ] − [R δαβγσδ] = 0, or 2[σαβγ] = R δ′α′β′γ′[σ δ′]. Since the last factor is zero, we arrive at

[ ] [ ] [ ] [ ] σαβγ = σ αβγ′ = σαβ′γ′ = σ α′β′γ′ = 0. (4.4 )
The last three results were derived from [σ ] = 0 αβγ by employing Synge’s rule.

We now differentiate the relation δ σα = σ ασδ three times and obtain

𝜖 𝜖 𝜖 𝜖 𝜖 𝜖 𝜖 𝜖 σα βγδ = σ αβγδσ𝜖 + σ αβγσ𝜖δ + σ αβδσ 𝜖γ + σ αγδσ𝜖β + σ αβσ𝜖γδ + σ αγσ 𝜖βδ + σ αδ&#x03

At coincidence this reduces to [σαβγδ] + [σαδβγ] + [σαγβδ] = 0. 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 αγ βδ α δβγ, and Synge’s rule allows us to generalize this to any combination of primed and unprimed indices. Our final results are

[ ] 1 ( ) [ ] 1( ) σ αβγδ = − -- Rα′γ′β′δ′ + R α′δ′β′γ′ , σαβγδ′ = --R α′γ′β′δ′ + Rα′δ′β′γ′, [ ] 3 ( ) [ ] 3 ( ) σ αβγ′δ′ = − 1- Rα′γ′β′δ′ + R α′δ′β′γ′ , σαβ′γ′δ′ = − 1-R α′β′γ′δ′ + R α′γ′β′δ′ , 3 3 [ ] 1-( ) σα ′β′γ′δ′ = − 3 Rα′γ′β′δ′ + R α′δ′β′γ′ . (4.5 )

4.2 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 PA;αtα = 0 at x. Similarly, we introduce an ordinary tensor QN (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 B ′ α′ Q ;α′t = 0. 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 | H (λ1,λ0) = H (λ0,λ0 ) + (λ1 − λ0)---|| + ⋅⋅⋅. ∂ λ1 λ1=λ0

Alternatively,

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

and these two expressions give

| | d ∂H | ∂H | ---H (λ0,λ0 ) = ---|| + ----|| , dλ0 ∂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 ′ ′ ΩAB ′;α′tα P AQB, 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 λ → λ 1 0 this becomes [Ω ′ ]tα′P A′QB ′ AB ;α. Finally, H (λ ,λ ) = [Ω ′]PA ′QB ′ 0 0 AB, and its derivative with respect to λ0 is α′ A′ B′ [ΩAB ′];α′t P Q. Gathering the results we find that

{ [ ] [ ] [ ]} ′ ′ ′ ΩAB ′ ;α′ − ΩAB ′;α′ − ΩAB ′;α tα P A QB = 0,

and the final statement of Synge’s rule,

[ ′] [ ′ ′] [ ′ ] ΩAB ;α′ = ΩAB ;α + ΩAB ;α , (4.6 )
follows from the fact that the tensors P M and QN, and the direction of the selected geodesic β, are all arbitrary. Equation (4.6View Equation) reduces to Eq. (4.3View Equation) when σ ⋅⋅⋅ is substituted in place of ΩAB ′.
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