  A Kerr black hole with mass $M$ and angular momentum $J$ satisfies the
extremality inequality $|J| \le M^2$. In the presence of matter and/or
gravitational radiation, this bound needs to be reformulated in terms of local
measurements of the mass and the angular momentum directly associated with the
black hole. The isolated and dynamical horizon framework provides such
quasi-local characterization of black hole mass and angular momentum. With this
framework, it is possible in axisymmetry to reformulate the extremality limit
as $|J| \le 2\,M_H^2$, with $M_H$ the irreducible mass of the black hole
computed from its apparent horizon area and $J$ obtained using approximate
rotational Killing vectors on the apparent horizon. The $|J| \le 2\,M_H^2$
condition is also equivalent to requiring a non-negative black hole surface
gravity. We present numerical experiments of an accreting black hole that
temporarily violates this extremality inequality. The initial configuration
consists of a single, rotating black hole surrounded by a thick, shell cloud of
negative energy density. For these numerical experiments, we introduce a new
matter-without-matter evolution method.
