

4.2 Distributions in curved
spacetime
The distributions introduced in Section 4.1.5 can also be defined in a
four-dimensional spacetime with metric
. Here we produce
the relevant generalizations of the results derived in that
section.
4.2.1
Invariant Dirac
distribution
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
Equation (259) 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 Equations (258) implies
and on the other hand,
which establishes the second identity of
Equation (260). 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 Equation (260) is proved in a
similar way.
4.2.2
Light-cone
distributions
For the remainder of Section 4.2 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 if
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
Equation (246). We have that
is one if
is an element of
, the chronological future of
, and zero otherwise, and
is one if
is an element of
, 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 Equation (247). 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 if
and
are timelike related, and positive if
they are spacelike related.
For the same reasons as those mentioned in
Section 4.1.5, 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
Equation (248, 249, 250) 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 Equation (265). To establish these
identities in curved spacetime we use the fact that they hold in
flat spacetime - as was shown in Section 4.1.5 - 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 Equations (263, 264) in the Riemann normal
coordinates of Section 3.1; 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,
Equations (263, 264, 265) take exactly the same
form as Equations (248, 264, 250). 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.

