# Is the calculation done in Hawking's original paper is done using perturbation?

In the paper Particle creation by black holes when calculating $$p_{\omega}$$, Hawking does the calculation in following manner let

• $$p_{\omega}^{(1)}$$ denotes the radiation which gets scattered by Black hole outside of event horizon and ends up on $$\mathcal{I^-}$$ with the same frequency
• $$p_{\omega}^{(2)}$$ are those radiations which enter the collapsing body and gets partly scattered and partly reflected through the centre before emerging on $$\mathcal{I^-}$$

Is the above calculation a part of perturbative series, $$p_{\omega}$$ is expanded in terms of $$p_{\omega}^{(1)}$$, $$p_{\omega}^{(2)}$$...? Can I continue above logic and say $$p_{\omega}^{(3)}$$ are those set of radiation which are re-reflected from the outer boundary of collapsing star (reflection of $$p_{\omega}^{(2)}$$ on outer boundary of sphere) and then gets scattered and reflected from the center, just like $$p_{\omega}^{(2)}$$ and then reaches $$\mathcal{I^-}$$?