Hawking radiation from 3+1-dimensional black holes is studied using effective
field theory methods. The field equations that determine the time-dependent
renormalised stress tensor are numerically solved for a black hole formed by
an ingoing null shock wave. Hawking radiation is generated dynamically near
the event horizon and propagates to future null infinity. The outgoing energy
flux at future null infinity builds up from zero at early retarded times before the
black hole forms and approaches a constant late-time value, consistent with
analytic results for an Unruh state on a static Schwarzschild background of
the same mass. Preliminary numerical solutions of the full semiclassical field
equations, with back reaction included, show black hole formation and the
onset of Hawking evaporation but the solutions break down before the black
hole endpoint is reached.