To prepare the way for our discussion of
Green’s functions in curved spacetime, we consider first the
slightly nontrivial case of a massive scalar field in flat spacetime. This field satisfies the wave
equation
To solve Equation (235) we use Lorentz
invariance and the fact that the spacetime is homogeneous to argue
that the retarded and advanced Green’s functions must be given by
expressions of the form
The Dirac functional on the right-hand side of
Equation (235) is a highly singular
quantity, and we can avoid dealing with it by integrating the
equation over a small four-volume
that contains
. This volume is bounded by a closed hypersurface
. After using Gauss’ theorem on the first term of
Equation (235
), we obtain
, where
is a surface
element on
. Assuming that the integral of
over
goes to zero in the limit
, we have
To examine Equation (237) we introduce
coordinates
defined by
and we let be a surface of constant
. The metric of flat spacetime is given by
in the new coordinates, where . Notice that
is a timelike coordinate when
, and that
is then a spacelike
coordinate; the roles are reversed when
.
Straightforward computations reveal that in these coordinates
,
,
,
,
, and
the only nonvanishing component of the surface element is
, where
. To
calculate the gradient of the Green’s function we express it as
, with the upper (lower) sign belonging
to the retarded (advanced) Green’s function. Calculation gives
, with a prime indicating
differentiation with respect to
; it should be noted that
derivatives of the step function do not appear in this
expression.
Integration of with respect to
is immediate, and we find that Equation (237
) reduces to
We have seen that Equation (239) properly encodes the
influence of the singular source term on both the retarded and
advanced Green’s function. The function
that enters into
the expressions of Equation (236
) must therefore be
such that Equation (239
) is satisfied. It
follows immediately that
must be a singular function,
because for a smooth function the integral of Equation (239
) would be of order
, and the left-hand side of Equation (239
) could never be made
equal to
. The singularity, however, must be
integrable, and this leads us to assume that
must be made out of Dirac
-functions and derivatives.
We make the ansatz
whereDifferentiation of Equation (240) and substitution into
Equation (239
) yields
where overdots (or a number within brackets)
indicate repeated differentiation with respect to . The limit
exists if
and only if
. In the limit we must then
have
, which implies
. We conclude that
must have the
form of
To determine we must go back
to the differential equation of Equation (235
). Because the singular
structure of the Green’s function is now under control, we can
safely set
in the forthcoming
operations. This means that the equation to solve is in fact
, the homogeneous version of
Equation (235
). We have
,
, and
, so
that Green’s equation reduces to the ordinary differential
equation
where we have used the identities of
Equation (241). The left-hand side
will vanish as a distribution if we set
To solve Equation (244) we let
, with
.
This gives rise to Bessel’s equation for the new function
:
The solution that is well behaved near is
, where
is a constant to be determined. We have that
for small values of
, and it follows that
. From Equation (244
) we see that
. So we have found that the only acceptable solution
to Equation (244
) is
To summarize, the retarded and advanced solutions
to Equation (235) are given by
Equation (236
) with
given by Equation (242
) and
given by Equation (245
).
The techniques developed previously to find Green’s functions for the scalar wave equation are limited to flat spacetime, and they would not be very useful for curved spacetimes. To pursue this generalization we must introduce more powerful distributional methods. We do so in this Section, and in the next we shall use them to recover our previous results.
Let be a generalized step
function, defined to be one if
is in the future of the
spacelike hypersurface
, and defined to be zero
otherwise. Similarly, define
to be one if
is in the past of the
spacelike hypersurface
, and zero otherwise. Then
define the light-cone step functions
The distributions and
are not defined at
and they
cannot be differentiated there. This pathology can be avoided if we
shift
by a small positive quantity
. We can therefore use the distributions
and
in some sensitive
computations, and then take the limit
. Notice
that the equation
describes a two-branch
hyperboloid that is located just within the light cone of the reference point
. The hyperboloid does not include
, and
is one everywhere on its
future branch, while
is one everywhere on its
past branch. These factors, therefore, become invisible to
differential operators. For example,
. This manipulation shows
that after the shift from
to
, the distributions of Equations (246
) and (247
) can be
straightforwardly differentiated with respect to
.
In the next paragraphs we shall establish the distributional identities
in four-dimensional flat spacetime. These will be used in the next Section 4.1.6 to recover the Green’s functions for the scalar wave equation, and they will be generalized to curved spacetime in Section 4.2.The derivation of Equations (248, 249
, 250
) relies on a “master”
distributional identity, formulated in three-dimensional flat
space:
To prove Equation (248) we must show that
vanishes as a distribution in the limit
. For this we must prove that a
functional of the form
where is a
smooth test function, vanishes for all such functions
. Our first task will be to find a more convenient
expression for
. Once more we set
(without loss of generality) and we note that
, where we have used Equation (251
). It follows that
which establishes Equation (248).
The validity of Equation (249) is established by a
similar computation. Here we must show that a functional of the
form
vanishes for all test functions . We have
To establish Equation (250) we consider the
functional
and show that it evaluates to . We have
The retarded and advanced Green’s functions for
the scalar wave equation are now defined as the limit of the
functions as
. For these
we make the ansatz
The functions that appear in Equation (253) can be
straightforwardly differentiated. The manipulations are similar to
what was done in Section 4.1.4, and dropping all labels, we
obtain
, with a prime indicating differentiation with
respect to
. From Equation (253
) we obtain
and
. The identities of Equation (241
) can be expressed as
and
, and combining this with
our previous results gives