In theorem 1, for the fist time, the sameradius definition is used for both bodies and black holes

The hypothesis of asymptotic flatness is necessary:

A re-gion between two concentric balls is said to beuntrappedifθ+θ−>0 on thatregion. The region it is said to betrappedifθ+θ−<0. The outer boundaryof a trapped region on an asymptotically flat data is called ahorizonand itsatisfiesθ+θ−= 0.

we restrict ourselves to spherically symmetric initial data where the3-dimensional Riemannian manifold is taken to beR3. We call themreg-ular spherically symmetricinitial data. We also assume that the data areasymptotically flat.

∂Bbe a sphere centered at the origin with area radiusR. That is,the area of∂Bis given by 4πR2.