Can Black Holes Truly Exist for Earth Observers?

[PeterDonis]

http://arxiv.org/abs/astro-ph/0512211

Interesting paper. What strikes me about it is that it gives a way of getting around the question doubters typically ask: "How could we ever tell there was a horizon, since it takes an infinite amount of time for light from the horizon to get out to us?" This paper looks at the consequences of having a surface at some R > 2M on the *spectrum* of the observed radiation coming out, and shows that they are not consistent with the actual observed spectrum. Basically, if there is a surface at some R > 2M (but close enough to 2M that we can't see it directly), there is no possible mass flow rate of infalling matter onto the surface that will match the observed spectrum: a flow rate low enough to match the small observed luminosity in the near infrared will be far too low to match the larger observed luminosity in the radio spectrum at sub-millimeter wavelengths. This is nice because it links the hypothesis that there is a surface there, as opposed to an event horizon, to testable consequences.

 

[PAllen]

Edit: Reading PAllen's referenc...g.de/~rezzolla/lnotes/mondragone/collapse.pdf quickly, they do talk about when an event horizon forms. They also use language like "Note that the apparent horizon is formed after the event horizon but not when the stellar surface crosses R = 2M". Is some simultaneity convention being used here? In which case, couldn't one attach a "when" to the existence of an event horizon?

I think you can sensibly attach a when, but not a unique when. For a BH that forms, rather than being eternal, and for a given distant observer world line, there is an earliest event on the world line from which a light signal will cross an event horizon before being absorbed by matter or reaching a singularity. Then, since there is no sense in which ingoing light is trapped (only outgoing light is trapped), you can adopt some convention for how long after sending such a signal you consider that the event horizon has formed. You can also image the collapse and see when all evidence of surface disappears (as described in your references). The latter is more direct.

However, in the article I linked, these time comparisons are, if memory doesn't fail me, referred to the point of view of observers going with the collapsing body. Especially for a hypothetical observer near the center of the collapsing body, there is a precise when for the growing event horizon passing them; similarly, for an observer falling with the surface of collapsing body, there is a precise time of crossing both EH and AH.

 

[PeterDonis]

They also use language like "Note that the apparent horizon is formed after the event horizon but not when the stellar surface crosses R = 2M". Is some simultaneity convention being used here? In which case, couldn't one attach a "when" to the existence of an event horizon?

They are using "comoving" coordinates (which are basically what Oppenheimer and Snyder used: they are equivalent to Painleve coordinates in the vacuum region and to FRW coordinates inside the collapsing matter), so that's the simultaneity convention to use when interpreting their statements about "when". It's a nice convention to use in this problem because the coordinate time under this convention corresponds to the proper time of observers who are freely falling inward, so statements about "when" things happen have an obvious interpretation in terms of those observers.

But note that this tells you when the event horizon *forms*, but that's different from asking whether or not there *is* an event horizon somewhere in the spacetime. The "when" statement is still coordinate-dependent; there will be coordinates, like Schwarzschild coordinates, in which the EH never forms, because the coordinates don't cover that portion of the spacetime. But the statement about there being an EH somewhere in the spacetime is independent of coordinates.

 

[Dale]

It's hard to see how it could be true unless the manifold were really pathological

Yes, I was thinking of pathological manifolds, like flat ones with holes. If you had a flat manifold with two spacelike separated events and removed a large enough section in between then you could make it so that the shortest possible path is timelike everywhere. Or perhaps a path which is somewhere timelike and somewhere spacelike.

PAllen seemed to be thinking along similar lines.

 

[PAllen]

Even more (or less??) interesting is whether you can have a (pseudo-reimannian) manifold such that there is a pair of events connected only by mixed paths (no pure spacelike, timelike or lightlike path - forget geodesic). In 1+1 d this is trivial to achieve with a connected but not simply connected manifold. However, for 3+1 d I am baffled; I can't see a construction to achieve 'no non-mixed paths' between two events, without also achieving no paths at all between them. But I really don't know.

To close the loop on this side discussion, I have succeeded in constructing a 2+1 d metrically flats Minkowski space, that is connected but not simply connected, such that for two particular events, every smooth path connecting them is mixed (neither timelike, spacelike, or null over the whole path). My guess would then be that it is possible for 3+1 d, but I don't intend to work that one out.