Conformal boundary and geodesics for $AdS_5\times S^5$ and the plane wave: Their approach in the Penrose limit

Projecting on a suitable subset of coordinates, a picture is constructed in which the conformal boundary of $AdS_5\times S^5$ and that of the plane wave resulting in the Penrose limit are located at the same line. In a second line of arguments all $AdS_5\times S^5$ and plane wave geodesics are constructed in their integrated form. Performing the Penrose limit, the approach of null geodesics reaching the conformal boundary of $AdS_5\times S^5$ to that of the plane wave is studied in detail. At each point these null geodesics of $AdS_5\times S^5$ form a cone which degenerates in the limit.