We first introduce , an invariant Dirac functional in a four-dimensional curved spacetime. This is defined by the relations

where is a smooth test function, any four-dimensional region that contains , and any four-dimensional region that contains . These relations imply that is symmetric in its arguments, and it is easy to see that where is the ordinary (coordinate) four-dimensional Dirac functional. The relations of Eq. (13.2) are all equivalent because is a distributional identity; the last form is manifestly symmetric in and .The invariant Dirac distribution satisfies the identities

where is any bitensor and , are parallel propagators. The first identity follows immediately from the definition of the -function. The second and third identities are established by showing that integration against a test function gives the same result from both sides. For example, the first of the Eqs. (13.1) impliesand on the other hand,

which establishes the second identity of Eq. (13.3). Notice that in these manipulations, the integrations involve scalar functions of the coordinates ; the fact that these functions are also vectors with respect to does not invalidate the procedure. The third identity of Eq. (13.3) is proved in a similar way.

For the remainder of Section 13 we assume that , so that a unique geodesic links these two points. We then let be the curved spacetime world function, and we define light-cone step functions by

where is one when is in the future of the spacelike hypersurface and zero otherwise, and . These are immediate generalizations to curved spacetime of the objects defined in flat spacetime by Eq. (12.14). We have that is one when is within , the chronological future of , and zero otherwise, and is one when is within , the chronological past of , and zero otherwise. We also have .We define the curved-spacetime version of the light-cone Dirac functionals by

an immediate generalization of Eq. (12.15). We have that , when viewed as a function of , is supported on the future light cone of , while is supported on its past light cone. We also have , and we recall that is negative when and are timelike related, and positive when they are spacelike related.For the same reasons as those mentioned in Section 12.5, it is sometimes convenient to shift the argument of the step and -functions from to , where is a small positive quantity. With this shift, the light-cone distributions can be straightforwardly differentiated with respect to . For example, , with a prime indicating differentiation with respect to .

We now prove that the identities of Eq. (12.16) – (12.18) generalize to

in a four-dimensional curved spacetime; the only differences lie with the definition of the world function and the fact that it is the invariant Dirac functional that appears in Eq. (13.8). To establish these identities in curved spacetime we use the fact that they hold in flat spacetime – as was shown in Section 12.5 – and that they are scalar relations that must be valid in any coordinate system if they are found to hold in one. Let us then examine Eqs. (13.6) – (13.7) in the Riemann normal coordinates of Section 8; these are denoted and are based at . We have that and , where is the van Vleck determinant, whose coincidence limit is unity. In Riemann normal coordinates, therefore, Eqs. (13.6) – (13.8) take exactly the same form as Eqs. (12.16) – (12.18). Because the identities are true in flat spacetime, they must be true also in curved spacetime (in Riemann normal coordinates based at ); and because these are scalar relations, they must be valid in any coordinate system.
Living Rev. Relativity 14, (2011), 7
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