Analytic continuation of Spacetime Metrics

Beyond the Event Horizon
(M′, g′ ab
) is said to be an extension of (M, g ab
) if (M, g ab
) can be
isometrically embedded as a proper open subset of (M′, g′ ab
)
-Christodoulou proved CC for the spherically symmetric
Einstein-scalar field system->trapped regions. A point in
a trapped region corresponds to a trapped surface in
the 4d space-time manifold
-conditions for predictability for the Einstein equations
are related to the behavior of the unique solution of
the initial value problem on the boundary of this
region.
conformal representation of the manifold
into 2d Minkowski space
spacetime=future inextendible as a
manifold with continuous Lorentzian metric
-there always exists a maximal region of spacetime,
the maximal domain of development, for which the
initial value problem uniquely determines the solution.
-consequences of curvature singularity at inner horizon: the metric tensor is continuous,
the Riemann tensor diverges at inner horizon
Weak curvature singularity: although curvature diverges, the metric tensor has a welldefined,
continuous, non-singular limit at the singularity (Tipler, 1977)