  Continuous sequences of asymptotically flat solutions to the Einstein-Maxwell
equations describing regular equilibrium configurations of ordinary matter can
reach a black hole limit. For a distant observer, the spacetime becomes more
and more indistinguishable from the metric of an extreme Kerr-Newman black hole
outside the horizon when approaching the limit. From an internal perspective, a
still regular but non-asymptotically flat spacetime with the extreme
Kerr-Newman near-horizon geometry at spatial infinity forms at the limit.
Interesting special cases are sequences of Papapetrou-Majumdar distributions of
electrically counterpoised dust leading to extreme Reissner-Nordstrom black
holes and sequences of rotating uncharged fluid bodies leading to extreme Kerr
black holes.
