Asymptotic Symmetries of the Null Infinity and the Isolated Horizon

The common intrinsic geometry shared by all the null hypersurfaces gives rise to the asymptotic symmetries found on the null infinities ${\mathcal{I}}^{\pm }$ and the isolated horizons $\mathrm{\Delta }$. In this work, the properties of a null hypersurface are reviewed and the invariance of its intrinsic geometry ($n^a{n}^b{h}_{ab}$) is revealed under the spacetime conformal transformation. The generators, i.e., infinitesimal symmetries, of the conformal transformation tangent to the null hypersurface are defined and classified by their effects on the induced metric and the normal vector field. Two particular examples and their symmetries are discussed, that is, the null infinities ${\mathcal{I}}^{\pm }$ of an asymptotic flat spacetime, and the isolated horizon $\mathrm{\Delta }$ of a black hole.