  Wave turbulence is by nature a multiple time scale problem for which there is
a natural asymptotic closure. The main result of this analytical theory is the
kinetic equation that describes the long-time statistical behaviour of such
turbulence composed of a set of weakly nonlinear interacting waves. In the case
of gravitational waves, it involves four-wave interactions and two invariants,
energy and wave action. Although the kinetic equation of gravitational wave
turbulence has been published with the Hadad-Zakharov metric, along with their
physical properties, the detailed derivation has not been shown. Following the
seminal work of Newell (1968) for gravity/surface waves, we present the
multiple time scale method, rarely used to derive the kinetic equations, and
clarify the underlying assumptions and methodology. This formalism is applied
to a wave amplitude equation obtained using an Eulerian approach. It leads to a
kinetic equation slightly different from the one originally published, with a
wave equation obtained using a Hamiltonian approach; we verify, however, that
the two formulations are fully compatible when the number of symmetries used is
the same. We also show that the exact solutions (Kolmogorov-Zakharov spectra)
exhibit the same power laws and cascade directions. Furthermore, the use of the
multiple time scale method reveals that the system retains the memory of the
initially condition up to a certain level (second order) of development in
time.
