7.3.3 Multipolar Expansion of the Perturbation

The next step in our elaboration of the general solution (7.3.3) for the metric perturbation consists of its systematic multipolar expansion. This will enable us to single out the contributions to the wave from the various angular momentum components of the source deformation. Since these contributions have a faster and faster distance decay at increasing angular momentum , it follows that for weak fields only the first non-vanishing multipole needs to be considered. Due to the spin of the graviton which is s = 2 the first contributing multipole turns out to be the quadrupole as opposed to the case of electromagnetic radiation where the spin s = 1 of the photon selects the dipole as first non-vanishing and dominant contribution.

To implement the above outlined programme we begin by performing a partial Fourier (see Fig. 7.9) transform of the metric perturbation with respect to the time coordinate:

and since γμν is symmetric we can conclude:

Hence it suffices to determine the spatial components γij . The others are easily retrieved from these.

7.3.3.1 Multipolar Expansion

Let us consider the spatial part of the Green function, which apart from a factor 4π is just the Green function of the Laplace operator discussed in Sect. 7.2.1:

Fig. 7.9 Jean Baptiste Joseph Fourier (March 21, 1768–May 16, 1830) is one among the incredibly large number of scientific geniuses giving fundamental contributions to the development of Modern Physics, Mathematics and Chemistry contributed by the Revolutionary and Napoleonic France. Other members of that group are Laplace, Lagrange, Monge, Carnot (father and son), to mention some. Born at Auxerre, Fourier was educated in a monastery and studied mathematics. He took prominent part in the Revolution and was appointed professor first at the Ecole Normale Superieure, then at the Ecole Polytechnique where he inherited the chair of Laplace. He served as officer in the Napoleonic Army and was even appointed governor of Egypt. Fourier was elected member of the French Academy of Sciences in 1817. Besides the introduction of the Fourier series and of the Fourier transform, Fourier discovered the equation of heat propagation and was the first to point out the green-house effect of the atmosphere. He died in Paris at the age of sixty-two