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An apparent horion ( S. W. Hawking & G. F. R. Ellis (1975). The large scale structure of space-time.) in General Relativity is a surface where all null vectors are pointing "inwards", i.e. it is the location of a marginally outer trapped surface (for a review see Thornburg, Jonathan; Living Rev.Rel. 10 (2007) 3). It is well known (see Hawking and Ellis) that if the Null Convergence Condition

\begin{equation} R_{ab}k^ak^b\geq0 \qquad \forall k^a \qquad s.t. \qquad k^ak_a=0 \end{equation}

holds everywhere in a spacetime, then black hole (i.e. event) horizons can only grow in size, and not shrink. Does a similar statement hold for apparent horizons?

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