vacuum. The matter fields can have also angular momentum and it can be transferred to the black hole, however the inequality (\[eq:JAt\]) remains true even for that case. It is important to note that the angular momentum that appears in (\[eq:JAt\]) is the gravitational one (i.e. the Komar integral). In fact this inequality is non-trivial even for the Kerr-Newman black hole, see the discussion in [@dain12].

Theorem \[t:2\] has the following history. The quasi-local inequality (\[eq:JAt\]) was first conjectured to hold in stationary spacetimes surrounded by matter in [@Ansorg:2007fh]. In that article the extreme limit of this inequality was analyzed and also numerical evidences for the validity in the stationary case was presented (using the numerical method and code developed in [@Ansorg05]). In a series of articles [@hennig08] [@Hennig:2008zy] the inequality (\[eq:JAt\]) (including also the electromagnetic charge) was proved for that class of stationary black holes. See also the review article [@Ansorg:2010ru].

In the dynamical regime, the inequality (\[eq:JAt\]) was conjectured to hold in [@dain10d] based on the heuristic argument mentioned in section \[sec:physical-picture\]. In that article also the main relevant techniques for its proof were introduced, namely the mass functional on the surface and its connections with the area. 