Coincidence limits

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 (53 , 55 , 56 ) we already have

Additional results are obtained by repeated differentiation of the relations (57

) and (58

). For example, Equation (57

) implies $σγ = gαβσ ασβγ = σβσ βγ$ , or $(gβγ − σβγ)tβ = 0$ after using Equation (55

). 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

From these relations we infer that $[σ αα] = 4$ , so that $[θ∗] = 3$ , where $θ∗$ was defined in Equation (61

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

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 (63

). Similarly, $[σα′β′] = [σα′];β′ − [σα′β] = − [σ βα′] = gα′β′$ . The results of Equation (63

) can thus all be generated from the known result for $[σ ] αβ$ .

The coincidence limits of Equation (63 ) 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

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