Can Black Holes Truly Exist for Earth Observers?

[PeterDonis]

Certainly, in a traversible wormhole spacetime, you have events with both spacelike and timelike geodesics connecting them.

Yes, because these spacetimes contain closed timelike curves. But the original question was about whether there can be a pair of events that are *not* connected by *any* geodesics, spacelike, timelike, *or* null. That seems to me to require much more pathological conditions than having multiple geodesics connecting the same pair of events.

 

[PAllen]

Question: if we exclude black holes, does the GR definition of "space-like separated" coincide with the definition I gave?

No, for all the reasons I gave in my response. You can't talk about frames at all in GR, irrespective of BH. You can talk about light cones, and impossibility of a timelike path.

 

[rjbeery]

The proper distance between any external observer and the portion of the interior of the event horizon which is not like-like separated is infinite, which is not a real number value.

False.

Yes, I erred but it is against my personal rules to edit a post after someone has replied to it. I was referring to the proper length to the singularity which is a prerequisite for the event horizon. If it's true that space-like separation and concepts of simultaneity must be reworded in order to work within the context of GR then so be it, but I don't see this as much different from the complaints to my original OP which claimed that any conclusion is possible if we adjust our definitions.

 

[PeterDonis]

If it's true that space-like separation and concepts of simultaneity must be reworded in order to work within the context of GR then so be it

It isn't true. The correct concepts in SR are the same as in GR; it's just that in SR, since there are global inertial frames, you can get away with rewording the concepts in terms of reference frames instead of light cones and still get the right answers.

 

[PAllen]

Yes, because these spacetimes contain closed timelike curves. But the original question was about whether there can be a pair of events that are *not* connected by *any* geodesics, spacelike, timelike, *or* null. That seems to me to require much more pathological conditions than having multiple geodesics connecting the same pair of events.

Multiple geodesics of the same type between events is routine, of course. Of different types requires, wormholes, warp bubbles, Godel, or something like that is necessary.

Your no geodesic possibility is interesting, and may depend on how you define geodesic. For example, consider a non-simply connected SR topology where the neighborhood (world tube) of the spacelike geodesic between two events is removed, leaving the points (and maybe tiny open ball around them. Then there is no geodesic satisfying the Euler-Lagrange equation between them; however there may be one or more spacelike paths with less proper length than any other paths (or not, depending on how you do the cut - you can do it so no path realizes GLB of proper lengths; or allow for such a path to exist). So is this a geodesic, if it exists?

Anyway, the above recipe, generalized to AF flat regions of a GR solution: just cut out tube with open boundary such that no path realizes the GLB. You now have no geodesic of any type between the events, even with the loose definition.

 

[PeterDonis]

Then there is no geodesic satisfying the Euler-Lagrange equation between them; however there may be one or more spacelike paths with less proper length than any other paths (or not, depending on how you do the cut - you can do it so no path realizes GLB of proper lengths; or allow for such a path to exist). So is this a geodesic, if it exists?

I would say no, since such a curve would not satisfy the geodesic equation. Consider the corresponding timelike case: cut out a world tube around a timelike geodesic but leave its endpoints still in the manifold. Then there would (given the appropriate type of cut) be a timelike curve of maximal proper time between the two events, but an observer following such a curve would not be in free fall; such an observer would feel a nonzero proper acceleration. So calling such a curve a "geodesic" would go against the key physical fact that observers following geodesic worldlines should be in free fall.

 

[PAllen]

I would say no, since such a curve would not satisfy the geodesic equation. Consider the corresponding timelike case: cut out a world tube around a timelike geodesic but leave its endpoints still in the manifold. Then there would (given the appropriate type of cut) be a timelike curve of maximal proper time between the two events, but an observer following such a curve would not be in free fall; such an observer would feel a nonzero proper acceleration. So calling such a curve a "geodesic" would go against the key physical fact that observers following geodesic worldlines should be in free fall.

From a physics point of view I would agree. The locally extremal (or parallel transport straight) property is the important property, not global properties.

 

[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.

 

[Dale]

I was referring to the proper length to the singularity which is a prerequisite for the event horizon.

No, the singularity is not a prerequisite for an event horizon. However, even on the singularity there are events which are timelike, lightlike, and spacelike separated from any event outside the horizon. This is easiest to see on a Kruskal-Szekeres diagram.

The black hole spacetime is well behaved, except for right at the singularity. Take any event in the entire manifold. From that event you can define 3 regions:
-) All events inside the light cone
0) All events on light cone
+) All events outside the light cone

Region 0 is lightlike separated, region - is timelike separated, and region + is spacelike separated from the event. Regions - and 0 can be further subdivided into future (-f, 0f) directed and past (-p, 0p) directed lightlike and timelike intervals respectively. These regions cover the entire spacetime.

For any event outside the horizon there are events inside the horizon in regions +, 0f, -f, but not in regions 0p, -p. For any event inside the horizon there are events outside the horizon in regions +, 0p, -p, but not in regions 0f, -f. For any event anywhere in the spacetime the singularity is in regions +, 0f, -f, but not in regions 0p, -p.

 

[atyy]

They're using the one I gave in post #9: a black hole "exists" if the spacetime contains an event horizon.

Contains "when"?

There is no when. The spacetime is a 4-dimensional geometric object; the event horizon is an invariant geometric feature of that geometric object. The statement "the spacetime contains an event horizon" is therefore an invariant geometric statement; it's not associated with any "time". It's just a geometric fact, like the fact that the Earth's equator is a great circle.

http://arxiv.org/abs/1006.0064
http://arxiv.org/abs/astro-ph/0512211

The first is a review which cites the second for providing the strongest evidence for an event horizon. They seem to say, well let's suppose there's a surface that exists at a certain time on which matter comes to rest, and look for it. Assuming GR and the absence of exotic phenomena, they can't find such a surface, which they say leads to the conclusion of an event horizon.

Edit: Reading PAllen's reference http://www.aei.mpg.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?

 

[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.