Kruskal White hole

[Schrödinger's hat]

Where and when (in schwarzschild coordinates) is the white hole which appears in the eternal singularity model? Or if we want to avoid the mis-behavior of Schwarzschild coords, where and when is a region of comparatively low curvature outside of the white hole?

Hmm, this is harder to word than I thought.

I guess what I'm trying to ask is, were there to exist an eternal black hole like those dealt with in Schwarzschild geometry, where would I -- a roughly inertial observer in a very weak field -- look to try and find it?

 

In watching this Kruskal diagram:

It looks like it's at [math]t=-\infty[/math] and in the same place as the black hole.

Or would it only be at [math]t=-\infty[/math] if I was an observer who came out of it?

[timo]

Where and when (in schwarzschild coordinates) is the white hole which appears in the eternal singularity model?

I never heard the term "eternal singularity model", but the white hole region in Schwarzschild coordinates is the region where the radial distance from the center is negative, if I remember correctly (I leave it up to you to evaluate how sensible this is).

 

Or if we want to avoid the mis-behavior of Schwarzschild coords, where and when is a region of comparatively low curvature outside of the white hole?

Hmm, this is harder to word than I thought.

I guess what I'm trying to ask is, were there to exist an eternal black hole like those dealt with in Schwarzschild geometry, where would I -- a roughly inertial observer in a very weak field -- look to try and find it?

I'm not really sure I get your question. If I undertand you correctly then I would say guess you reached the while hole horizont once the radial distance equals minus the Schwarzschild radius (but that's just a guess based on symmetry).

 

In watching this Kruskal diagram: ...

It looks like it's at [math]t=-\infty[/math] and in the same place as the black hole.

Or would it only be at [math]t=-\infty[/math] if I was an observer who came out of it?

There is not axis labels on your diagram, particularly none called "t"

Edited September 16, 2011 by timo

[Schrödinger's hat]

I never heard the term "eternal singularity model", but the white hole region in Schwarzschild coordinates is the region where the radial distance from the center is negative, if I remember correctly (I leave it up to you to evaluate how sensible this is).

I made it up in trying to explain what I was talking about. Upon further reading, I think the correct term is just Schwarzschild geometry.

 

I'm not really sure I get your question. If I undertand you correctly then I would say guess you reached the while hole horizont once the radial distance equals minus the Schwarzschild radius (but that's just a guess based on symmetry).

This does not fit my understanding of what I have read, or is not quite the answer to the question I was trying to ask.

Here's another Kruskal diagram from the wiki page in a hideous shade of yellow:

I can see that region IV will clearly be at negative radius and so will one of the regions I or II, but the area I was trying to ask about is not (as far as my understanding goes). Let's say we're in the universe represented by region I.

The null line bordering region I and IV represents [math]t=-\infty[/math] for a Schwarzschild observer.

If I were an observer in an area of region I where space is approximately flat (Ie. I can deal with things in my immedeate vicinity using first order terms, or possibly use Schwarzschild metric if needed, it doesn't really matter).

Obviously this diagram does not represent my coordinate system.

But I know where the area near the border between I and II would be, I can look around to see an accretion disc and some lensing and say 'that's the event horizon of a black hole', and point to it. I can also make some predictions about the coordinates of a probe I send towards it in my frame (at least until the probe gets very close to the event horizon, then I get bored of waiting for new measurements/my Schwarzschild coordinates start playing up). In short I can put some bounds in the region containing those events in my coordinates, even in in a naive model where I assume space to be roughly flat.

 

What about the region between I and IV (a bit on the I side of the border so I can use my nice, comfy Schwarzschild coordinates)? If I am to believe my interpretation of the other diagram it is at the same r coordinate as the black hole, but at a time in the distant past, but I cannot reconcile that with my intuition.

 

My question is how do I go about looking for evidence of the event horizon of the white hole in this model?

 

 

There is not axis labels on your diagram, particularly none called "t"

I apologise. The axes for this diagram are the Kruskal coordinate (timelike) V or [math]u^1[/math] on the new diagram and (spacelike) U or [math]u^0[/math]. The blue lines are alledgedly constant r.

Edited September 16, 2011 by Schrödinger's hat

[timo]

If I understand your question correctly you are asking how one could ever hope to detect a white hole? Perhaps you would see traces of apparently some large amount of mass suddenly coming into appearance out of nowhere with a huge bang at some point in the past? That was a joke, of course. I spontaneously have no idea.

[Schrödinger's hat]

If I understand your question correctly you are asking how one could ever hope to detect a white hole? Perhaps you would see traces of apparently some large amount of mass suddenly coming into appearance out of nowhere with a huge bang at some point in the past? That was a joke, of course. I spontaneously have no idea.

 

Almost, except it was sort of the other way around.

Given that the object described by the Schwarzschild geometry exists (this is not the thing that we normally look for when we look for black holes), what would we be seeing?

 

I stumbled around the internet and found an answer to something similar to my question, sort of dual to it.

Seems it was wiki diagrams to the rescue again.

Still don't know where one would look for the outer event horizons, but the penrose diagram on the realistic collapsing star makes a lot of sense.

If I'm reading this right, the white-hole-ish bit only exists in the limit of a path of events which is approaching the angle of a null line, and thus an angle of infinitely far in the past. It only reaches the angle of a null line when this path intersects the in-going event horizon.

Trying to talk about it in terms of Schwarzschild coordinates says that point is 'where infinitely far in the past meets infinitely far in the future'. This further reinforces the idea that Schwarzschild coordinates are the wrong way to try and think about this, but unfortunately they're as far as my intuition will stretch at this point.

Earlier on in the star's collapse (given that we're some observer in the top-leftish area), it was finitely far in the past, and the only stuff we see come out is the stuff that isn't following the path towards the eventual event horizon.

This'd be all the supernova explody.

 

If we find situations closer and closer to a Schwarszchild black hole (maybe a slowly accreting neutron star until it collapsed) I think we may actually wind up with the situation you described.

The world lines of stuff that escaped during the accretion and collapse would be approximately the same as the white hole exit world lines, and then there'd be a black hole.

Edited September 16, 2011 by Schrödinger's hat

[Schrödinger's hat]

I think I get it now.

It represents the point any irrotational geodesics started. So anything that isn't accelerating in its own frame (and only travels radially) looks like it came from there.

 

Also would it be fair to say that:

Kruskal coordinates are to Schwarzschild as Minkowski are to Rindler?