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2.2 Coincidence limits

It is useful to determine the limiting behaviour of the bitensors s... as x approaches x'. We introduce the notation
[_O_...] = lim _O_...(x,x') = a tensor at x' x-->x'

to designate the limit of any bitensor ' _O_...(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 c --> c 0 after evaluating _O_ (z,x') ... as a function of c on a specified geodesic b, 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

[s] = 0, [s ] = [s '] = 0. (62) a a
Additional results are obtained by repeated differentiation of the relations (57View Equation) and (58View Equation). For example, Equation (57View Equation) implies sg = gabsasbg = sbsbg, or (gbg- sbg)tb = 0 after using Equation (55View Equation). From the assumption stated in the preceding paragraph, sbg becomes independent of tb in the limit ' x --> x, and we arrive at [sab] = ga'b'. By very similar calculations we obtain all other coincidence limits for the second derivatives of the world function. The results are
[sab] = [sa'b'] = ga'b', [sab'] = [sa'b] = - ga'b'. (63)
From these relations we infer that [saa] = 4, so that [h*] = 3, where h* was defined in Equation (61View Equation).

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

[s...a'] = [s...];a'- [s...a], (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 [sab'] = [sa];b' - [sab], and since the coincidence limit of sa is zero, this gives us [sab'] = - [sab] = -ga'b', as was stated in Equation (63View Equation). Similarly, [sa'b'] = [sa'];b'- [sa'b] = - [sba'] = ga'b'. The results of Equation (63View Equation) can thus all be generated from the known result for [s ] ab.

The coincidence limits of Equation (63View Equation) were derived from the relation d sa = s asd. We now differentiate this twice more and obtain sabg = sdabgsd + sdabsdg + sdagsdb + sdasdbg. At coincidence we have

[ ] [ ] ' [sabg] = sdab gd'g'+ sdag gd'b'+ dda'[sdbg],

or [sgab] + [sbag] = 0 if we recognize that the operations of raising or lowering indices and taking the limit x --> x' commute. Noting the symmetries of sab, this gives us [sagb] + [sabg] = 0, or 2 [sabg] - [Rdabgsd] = 0, or 2[sabg] = Rd'a'b'g'[sd']. Since the last factor is zero, we arrive at

[sabg] = [sabg'] = [sab'g'] = [sa'b'g'] = 0. (65)
The last three results were derived from [sabg] = 0 by employing Synge’s rule.

We now differentiate the relation s = sd s a a d three times and obtain

s = se s + se s + se s + se s + se s + se s + se s + se s . abgd abgd e abg ed abd eg agd eb ab egd ag ebd ad ebg a ebgd

At coincidence this reduces to [s ] + [s ] + [s ] = 0 abgd adbg agbd. To simplify the third term we differentiate Ricci’s identity e sagb = sabg- R abgse with respect to d x and then take the coincidence limit. This gives us [sagbd] = [sabgd] + Ra'd'b'g'. The same manipulations on the second term give [sadbg] = [sabdg] + Ra'g'b'd'. Using the identity sabdg = sabgd- Reagdseb- Rebgdsae and the symmetries of the Riemann tensor, it is then easy to show that [sabdg] = [sabgd]. Gathering the results, we obtain 3[s ] + R ' '''+ R ' '' '= 0 abgd a gb d a db g, and Synge’s rule allows us to generalize this to any combination of primed and unprimed indices. Our final results are

1 [sabgd] = - -(Ra'g'b'd'+ Ra'd'b'g'), 3 1- [sabgd'] = 3 (Ra'g'b'd'+ Ra'd'b'g') , [sabg'd'] = - 1(Ra'g'b'd'+ Ra'd'b'g'), (66) 3 1 [sab'g'd'] = - -(Ra'b'g'd'+ Ra'g'b'd'), 3 ' ''' 1- '' '' '' '' [sa bg d] = - 3 (Ra g bd + Ra db g).

2.2.2 Derivation of Synge’s rule

We begin with any bitensor ' _O_AB'(x, x ) in which A = a ...b is a multi-index that represents any number of unprimed indices, and ' ' ' B = g ...d 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 b that links x to x' we introduce an ordinary tensor PM (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 b; this means that it satisfies A a P ;at = 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 b; at x' it satisfies QB' 'ta'= 0 ;a. With _O_, P, and Q we form a biscalar H(x, x') defined by

' ' A B' ' H(x, x ) = _O_AB'(x, x )P (x)Q (x ).

Having specified the geodesic that links x to x', we can consider H to be a function of c0 and c1. If c1 is not much larger than c0 (so that x is not far from x'), we can express H(c1, c0) as

@H | H(c1, c0) = H(c0, c0) + (c1- c0)----|| + .... @c1 c1=c0

Alternatively,

| @H--| H(c1, c0) = H(c1, c1)- (c1- c0)@c0 |c0=c1+ ...,

and these two expressions give

d @H || @H || ---H(c0, c0) = ----| + ----| , dc0 @c0 c0=c1 @c1 c1=c0

because the left-hand side is the limit of [H(c1, c1) - H(c0, c0)]/(c1- c0) when c1 --> c0. The partial derivative of H with respect to c0 is equal to ' ' _O_AB';a'taP AQB, and in the limit this becomes [_O_AB';a']ta'PA'QB'. Similarly, the partial derivative of H with respect to c1 is _O_AB';ataP AQB', and in the limit c -- > c 1 0 this becomes [_O_ ' ]ta'P A'QB' AB ;a. Finally, H(c ,c ) = [_O_ ']P A'QB' 0 0 AB, and its derivative with respect to c0 is a' A' B' [_O_AB'];a't P Q. Gathering the results we find that

{ } ' ' ' [_O_AB'];a'- [_O_AB';a']- [_O_AB';a] ta P A QB = 0,

and the final statement of Synge’s rule,

[_O_AB'];a'= [_O_AB';a'] + [_O_AB';a], (67)
follows from the fact that the tensors M P and N Q, and the direction of the selected geodesic b, are all arbitrary. Equation (67View Equation) reduces to Equation (64View Equation) when s... is substituted in place of _O_AB'.

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