As one sees, no singularity was met in this integration and the Green function of the Laplacian has the form of the Newtonian central potential generated by a pointlike mass. As a consequence the gravitational or electric potential generated by an arbitrary distribution of masses or charges can be written as:

Let us now return to the relativistic case and let us observe the differences.

7.2.2 The Relativistic Propagators

From (7.2.8), by separating the integration on the various momentum components, we obtain:

(7.2.16) and the singularities on the integration path of k0 become evident. They occur at k0 = ±|k|. In order to give a meaning to the integral it is necessary to give a prescription to deform the integration path in such a way as to avoid the singularities. There exist three possibilities depicted in Fig. 7.8.

The upper path CR yields the retarded Green function GR (x − x ), the lower path yields the advanced Green function GA(x − x ), while the middle path CF yields the Feynman propagator GF (x − x ). As discussed in all courses in Quantum Field Theory and Quantum Electrodynamics, the Feynman choice is that appropriate for perturbative quantum calculations. This prescription is such that for positive energy the propagator captures the contribution of particles advancing in time while for negative energies it captures that of anti-particles receding in time. The retarded and

Fig. 7.8 The integration path in the k0-plane corresponding to the three choices of relativistic propagator, advanced, retarded and Feynman

advanced prescriptions pertain instead to classical physics. They are both meaningful and simply correspond to the solutions of two different problems: that of emission and that of propagation of classical waves. In the retarded case we pick up the contribution of all events that are in the past light-cone of the event we consider. In this way we predict the value of a field at a certain space-time point knowing the behavior of the source in the past. Alternatively in the advanced case we pick up the contribution of all events that are in the future light-cone of the considered event. In this way knowing the value of a field at a certain space-time point we determine its influence on future events, for instance on those pertaining to an antenna which is supposed to receive signals. Relevant to us, while discussing gravitational waves emitted from astrophysical sources, is the retarded potential.

7.2.2.1 The Retarded Potential

We choose the retarded integration path CR . The integral can be calculated with the residue theorem, using Jordan’s lemma and closing the contour in the upper or in the lower half-plane. If x0 − x0 < 0 Jordan’s lemma allows us to close it in the upper half plane, while for x0 − x0 > 0 we have to close it in the lower half plane. In the first case the integration contour contains no poles and the integral is simply zero, while in the second case the two poles are both encircled and we have non-vanishing contributions from the residues. This amounts to saying that the retarded potential is proportional to a step function θ (x0 − x0). Indeed, using polar coordinates as in the previous case of the Laplacian Green function, we obtain:

Restoring the original variables in the final result of the above calculation we obtain the following expression for the retarded Green function: