I've seen similar discussions for the inner horizons of Kerr and Reissner-Nordstöm metrics, but I've never seen mention for the simple maximally extended Schwarzschild geometry: As an infalling observer coming from one of the exterior regions (
U) crosses the future horizon of the black hole, does the observer start seeing the entire history of the other exterior region (
I+ ~~~~~~ I+ /\ /\ / \ / \ I0 / \/ \ I0 \ U' /\ U / \ / \ / \/ \/ I- ~~~~~~ I-
In the Penrose-Carter diagram for the spacetime, null geodesics starting at past timelike infinity in the other exterior region would just "ride along" the white hole horizon (bottom-left
I- to top-right
I+), which is the black hole horizon on the future that our observer crosses; any light emitted later than at past infinity would meet the infalling observer a little further on her way into the interior region.
If this is the case, would that mean that the energy density at the future horizon is infinite and that the horizon should be unstable (notwithstanding the fact that this metric models empty space)?
(Similar energy divergence has been described and analyzed at the inner (Cauchy) horizon of the other, more complicated black-hole geometries that feature time-like singularities.)