2
$\begingroup$

Suppose you are a massive observer falling into a Schwarzschild black hole (geodesically or not, it shouldn't matter). At any point on your personal worldline you can receive light signals from the past, which include light from your universe that you've just left behind (as described in this answer).

But if the black hole is actually part of a complete Kruskal manifold, then the black hole ("Region II" in the traditional Kruskal diagram) can also receive signals through another, second horizon that connects regions II and III. (Some people call this anti-horizon.)

How and when does the infalling, backwards looking observer see the second horizon? What puzzles me is that spacetime is regular everywhere away from the singularity, and so the observations along the infalling trajectory should be continuous. The sudden appearance of a second horizon seems like a discontinuity to me. Where is that second horizon, and what does it look like?

$\endgroup$
2
$\begingroup$

One thing to note is that this horizon would only be present in an idealized eternal black hole, for a realistic non-rotating black hole formed by a collapsing star the Kruskal-Szekeres diagram would look more like the right-hand diagram below (from Gravitation by Misner, Thorne and Wheeler), where the gray area represents the interior of the star and the diagonal line labeled r=2M, t=infinity represents the sole event horizon:

enter image description here

In the case of an eternal black hole, then with the understanding that light rays always travel at 45-degree angles on a Kruskal-Szekeres diagram, you can see that while you can always see light rays from events on the antihorizon, you can never see light rays from events on the horizon until you cross it, and afterwards. You may find the animated gif from this page on Andrew Hamilton's site helpful:

enter image description here

Here red wavy lines represent worldlines of light rays (with the waviness just being symbolic, they should really be straight lines) from various events on the antihorizon, and blue wavy lines represent worldlines of light rays from events on the horizon.

Andrew hamilton also has some animated images showing falls into a Schwarzschild black hole, with a red grid representing the visible position of the antihorizon, and a white grid representing the visible portion of the horizon. See the "Through the horizon" section on this page for an animation with fancier graphics, and this page for a simpler version. It's explained a little more clearly on the second page--the red grid representing the antihorizon always appears off at some distance ahead of them even after they cross the horizon, but at the moment they cross they get their first light from events on the horizon, and that light appears as a white dot on the graphic but "The white dot is actually a line which extends from us to the Schwarzschild surface still ahead, though we only ever see it as a dot, not as a line". Then after they cross, this line expands into an white ovoid "bubble" which appears to intersects the red antihorizon, with the observer appearing to be enclosed within it:

The Schwarzschild surface appears to split into two in a region around us. We seem to lie inside a bubble formed by the two surfaces. Looking outward, we see the Schwarzschild surface (white grid lines) we just passed through. If other observers fell through the surface after us, we could see them fall through here, and they would appear neither infinitely redshifted nor slowed.

Looking inward, we see the same Schwarzschild surface (red grid lines) apparently still ahead. Persons who fell through before us would continue to appear here. Already redshifted and slowed, such persons would continue to grow ever more redshifted and slow.

Persons who appear to us to be inside the Schwarzschild bubble have passed inside the horizon of the black hole. If they are sufficiently close to us, then we can communicate with them, but they must be close, for there's not much time left before we hit the central singularity, not much time left for light signals to travel between us.

Then the white bubble of the horizon continued to grow as the observer approaches the singularity.

So there is a sort of discontinuity in what you see as you cross the horizon, but in a way it's no different than the discontinuity when you cross any other light-like boundary like the surface of the future light cone of some event E (in fact the horizon and antihorizon are two sides of the future light cone of the event at the point where they cross). Prior to crossing an event's future light cone, you can't see light from events on the surface of that cone, but at the moment you cross you can see light from a single line of events going from E to your crossing-point, and after you've entered the cone you can see light from some spheroid of events in all directions away from you (the intersection of E's future light cone with your own past light cone).

$\endgroup$
  • $\begingroup$ Thanks a lot -- the final paragraph really made it clear: The geometry is continuous, but the causal relationships between events is discrete, and hence discontinuous, as intervals change from spacelike to lightlike to timelike. $\endgroup$ – Kerrek SB Nov 23 '14 at 22:57

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.