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 Eq. (12.3) we appeal to 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 Eq. (12.3) 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 Eq. (12.3), we obtain , where is a surface element on . Assuming that the integral of over goes to zero in the limit , we have
To examine Eq. (12.5) 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 Eq. (12.5) reduces to
We have seen that Eq. (12.7) 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 Eq. (12.4) must therefore be such that Eq. (12.7) is satisfied. It follows immediately that must be a singular function, because for a smooth function the integral of Eq. (12.7) would be of order and the left-hand side of Eq. (12.7) 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
Differentiation of Eq. (12.8) and substitution into Eq. (12.7) 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 Eq. (12.3). 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 Eq. (12.3). We have , , , so that Green’s equation reduces to the ordinary differential equation
where we have used the identities of Eq. (12.9). The left-hand side will vanish as a distribution if we set
To solve Eq. (12.12) 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 Eq. (12.12) we see that . So we have found that the only acceptable solution to Eq. (12.12) is
To summarize, the retarded and advanced solutions to Eq. (12.3) are given by Eq. (12.4) with given by Eq. (12.10) and given by Eq. (12.13).
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 subsection, and in the next we shall use them to recover our previous results.
Let be a generalized step function, defined to be one when is in the future of the spacelike hypersurface and zero otherwise. Similarly, define to be one when 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 Eqs. (12.14) and (12.15) can be straightforwardly differentiated with respect to .
In the next paragraphs we shall establish the distributional identities
The derivation of Eqs. (12.16) – (12.18) relies on a “master” distributional identity, formulated in three-dimensional flat space:
To prove Eq. (12.16) 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 Eq. (12.19). It follows that
which establishes Eq. (12.16).
The validity of Eq. (12.17) 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 Eq. (12.18) 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 when . For these we make the ansatz
The functions that appear in Eq. (12.21) can be straightforwardly differentiated. The manipulations are similar to what was done in Section 12.4, and dropping all labels, we obtain , with a prime indicating differentiation with respect to . From Eq. (12.21) we obtain and . The identities of Eq. (12.9) can be expressed as and , and combining this with our previous results gives
Living Rev. Relativity 14, (2011), 7
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