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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 $\mathcal{I^-}$?

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