13 Distributions in curved spacetime
The distributions introduced in Section 12.5 can also be defined in a four-dimensional spacetime
with metric
. Here we produce the relevant generalizations of the results derived in that
section.
13.1 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 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) implies
and 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.
13.2 Light-cone distributions
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.