Trapped region in Kerr-Vaidya space-time

We review the basic definitions and properties of trapped surfaces and discuss them in the context of Kerr-Vaidya line element. Our study shows that the apparent horizon does not exist in general for axisymmetric space-times. The reason being the surface at which the null tangent vectors are geodesics and the surface at which the expansion of such vectors vanishes do not coincide. The calculation of an approximate apparent horizon for space-times that ensure its existence seems to be the only way to get away with this problem. The approximate apparent horizon, however, turned out to be non-unique. The choice of the shear-free null geodesics, at least in the leading order, seems to remove this non-uniqueness. We also propose a new definition of the black hole boundary.