Behavior of curvature and matter in the Penrose limit

In the Penrose limit construction a constant parameter $\Omega$ is introduced by which coordinates as well as the metric are rescaled in a suitable way. It results in a metric which apart from a possible coordinate dependence has an additional dependence on this constant parameter $\Omega$. Here the scaling with $\Omega$ of the components of the Weyl tensor and the energy-momentum tensor is determined in the Penrose limit, i.e. $\Omega \to 0$. In both cases, a type of peeling-off behavior in $\Omega$ is found. Examples of different types of matter are provided. The expansion and shear of the congruence of null geodesics along which the Penrose limit is taken are determined. Finally, the approach to the singularity in the Penrose limit of cosmological space-times is discussed.