Analytic continuation of Spacetime Metrics

Closed trapped surfaces
Cauchy horizon (CH)
=surface of infinite blueshift with respect to the EH of the BH ->
leads to a dynamical instability, referred to as mass inflation,
which replaces CH by a null singularity that turns spacelike deep
inside the BH
=light-like boundary of the domain of validity of a Cauchy
problem (separates closed timelike geodesic and closed spacelike
geodesic regions)
-while approaching EH, when stress-energy tensor diverges at the
horizon, CH prevents spacetime from developing closed time-like
Schwarzschild singularity replaced by Cauchy horizon
curves that would otherwise be feasible. Under the averaged weak energy condition, CH are unstable.
geodesic completeness (g-completeness)
- every geodesic can be extended to arbitrary values of its affine parameter.
- 3 kinds: timelike, null and spacelike geodesics. If one cuts a regular point out of space-time, the
resulting manifold is incomplete in all three ways ->a spacetime which was complete in one of them would
be complete in the other two.
timelike and null g-completeness -minimum conditions for singularity-free space-time -> if a space-time
is timelike or null geodesically incomplete ->it has a singularity. The advantage of taking timelike or null
incompleteness as being indicative of the presence of a singularity is that on this basis one can establish
a number of theorems about their occurrence.
closed trapped surface T
=a closed (i.e. compact, without boundary) spacelike two-surface such that the two families of null
geodesies orthogonal to T are converging at T. One may think of T as being in such a strong gravitational
field that even the 'outgoing' light rays are dragged back and are, in fact, converging. Since nothing can
travel faster than light, the matter within T is trapped inside a succession of two-surfaces of smaller area -
> something must go wrong.
Trapped surface formation is intimately connected with the question of singularities.