Causal Nature and Dynamics of Trapping Horizons in Black Hole Collapse
  In calculations of gravitational collapse to form black holes, trapping
horizons (foliated by marginally trapped surfaces) make their first appearance
either within the collapsing matter or where it joins on to a vacuum exterior.
Those which then move outwards with respect to the matter have been proposed
for use in defining black holes, replacing the global concept of an "event
horizon" which has some serious drawbacks for practical applications. We here
present results from a study of the properties of both outgoing and ingoing
trapping horizons, assuming strict spherical symmetry throughout. We have
investigated their causal nature (i.e. whether they are spacelike, timelike or
null), making contact with the Misner-Sharp- Hernandez formalism, which has
often been used for numerical calculations of spherical collapse. We follow two
different approaches, one using a geometrical quantity related to expansions of
null geodesic congruences, and the other using the horizon velocity measured
with respect to the collapsing matter. After an introduction to these concepts,
we then implement them within numerical simulations of stellar collapse,
revisiting pioneering calculations from the 1960s where some features of the
emergence and subsequent behaviour of trapping horizons could already be seen.
Our presentation here is aimed firmly at "real world" applications of interest
to astrophysicists and includes the effects of pressure, which may be important
for the asymptotic behaviour of the ingoing horizon.
