to emphasize is that one can choose any light-like geodesic of $\mbox{AdS}_5 \times S^5$ for implementing the procedure. While the pp-wave background always corresponds to the maximally supersymmetric pp-wave background of type IIB supergravity [@Blau:2001ne], different choices of light-like geodesics can give this background in different coordinate systems [@Bertolini:2002nr]. Naively this should not matter, however, the different coordinate systems can correspond to different choices of lightcone time on the pp-wave background. And this corresponds moreover to different dictionaries between the physical quantities of the $\mbox{AdS}_5\times S^5$ background and of the maximally supersymmetric pp-wave background. Therefore, the different coordinate systems for the pp-wave background are connected to the fact that the different Penrose limits that we consider correspond to zooming in to different regimes of type IIB string theory on $\mbox{AdS}_5\times S^5$. This in turns corresponds to zooming in to different regimes of ${\mathcal{N}}=4$ SYM. Furthermore, as we discuss in section \[sec:decsectors\], the different Penrose limits correspond to different decoupling limits of ${\mathcal{N}}=4$ SYM on ${\mathbb{R}}\times S^3$.

In the literature the “canonical” coordinate system used for the maximally supersymmetric pp-wave background is that of [@Blau:2001ne; @Blau:2002dy; @Berenstein:2002jq] which we here dub the [*BMN pp-wave background*]{}. This coordinate system is such that 