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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 (U')?

       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.)

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Extra points for the awesome ascii art!

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)?

I was with you up to this logical step, which I think is wrong. The typical ray of light infalling from region III (the parallel universe) does not originate at $i^-$ (the bottom left corner), it originates at some point on $\mathscr{I}^-$ (the bottom left edge). These rays are not concentrated or focused at the horizon.

Looking at your diagram, I wonder if you're mixing up $i^-$ and $\mathscr{I}^-$. You don't have them labeled as separate things.

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  • $\begingroup$ You're right, I must have confused timelike and null "content" of the other universe: the timelike one starts at $i^-$ but spreads out due to its lower-than light speed, and the null content doesn't start at a single point. $\endgroup$ – Kerrek SB Jul 29 '18 at 22:20

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