Rotating black holes in Penrose diagrams

[relativityfan]

hello,

in Penrose diagrams it says that once you have crossed a first Cauchy horizon(of a rotating black hole) , then, with the repulsive singularity, it is possible to cross another Cauchy horizon, and then another event horizon to escape the black hole and go into another universe.

so I wonder:

-since the event horizon and the Cauchy horizon of the classic black holes are similar and symmetrical, why is it possible to cross the Cauchy horizon since it is not possible to escape an event horizon?

-then, how is it possible to cross the second Cauchy horizon and the second event horizon since no matter can escape the event horizon of a black hole?

this seems a total non sense for me...
can anyone give it some sense to me?

thank you!

 

[JesseM]

hello,

in Penrose diagrams it says that once you have crossed a first Cauchy horizon(of a rotating black hole) , then, with the repulsive singularity, it is possible to cross another Cauchy horizon, and then another event horizon to escape the black hole and go into another universe.

so I wonder:

-since the event horizon and the Cauchy horizon of the classic black holes are similar and symmetrical, why is it possible to cross the Cauchy horizon since it is not possible to escape an event horizon?

-then, how is it possible to cross the second Cauchy horizon and the second event horizon since no matter can escape the event horizon of a black hole?

this seems a total non sense for me...
can anyone give it some sense to me?

thank you!

In the solution for an eternal rotating black hole, the rotating black hole is connected to a rotating white hole in another "universe", with both the black hole and the white hole having the same interior region inside their Cauchy horizons. An observer who enters this interior region by crossing the Cauchy horizon of the black hole can only exit by crossing the Cauchy horizon of the white hole, and anything between the Cauchy horizon and the event horizon of the white hole must inevitably exit the white hole event horizon. Compare this with a Penrose diagram or Kruskal-Szekeres diagram for an eternal nonrotating black hole, where there is also a white hole interior region (region IV on the diagram) and two different "universes" outside the black hole and white hole (regions I and III) but it's impossible to travel from region I to region III. Of course all these solutions involving "eternal" black holes/white holes are unrealistic, and with a nonrotating black hole that forms at some finite time from a collapsing sphere there wouldn't be a white hole region, but I'm not sure if anyone knows how to analyze a spacetime where a rotating black hole forms from a rotating object. The rotating black hole solution is also unrealistic in that any infalling waves (electromagnetic or gravitational) would be infinitely blueshifted at the Cauchy horizon, which means such waves would cause the energy density to go to infinity at the Cauchy horizon in GR, suggesting we need a theory of quantum gravity to figure out what's really going on (quantum gravity effects are supposed to become significant whenever the energy density approaches the Planck density)

 

[relativityfan]

thank you, but why could it be possible to cross the Cauchy horizon since the singularity is repulsive and the Cauchy horizon is like the event horizon. I mean it is like an event horizon for someone between the singularity and the Cauchy horizon, so matter should remain between the event horizon and the Cauchy horizon, or I am wrong?

 

[JesseM]

thank you, but why could it be possible to cross the Cauchy horizon since the singularity is repulsive and the Cauchy horizon is like the event horizon. I mean it is like an event horizon for someone between the singularity and the Cauchy horizon, so matter should remain between the event horizon and the Cauchy horizon, or I am wrong?

I believe the gravity from the ring singularity is only "repulsive" if you pass through the center of the ring singularity into the "antiverse" depicted in the bottom Penrose diagram on this page--for example, p. 21 of this book says "the shape of the singularity is a ring within the equatorial plane, so that some trajectories (A) can pass through the ring and reach an asymptotically flat space-time inside the black hole where gravity is repulsive". And note that whether a singularity is repulsive or attractive has nothing to do with whether a horizon is uncrossable from the inside (like a black hole horizon) or uncrossable from the outside (like a white hole horizon)--both a black hole and a white hole are attractive from the perspective of observers on the outside, for example. Penrose diagrams and Kruskal-Szekeres diagrams are designed with the property that all worldlines of light rays are at 45 degrees and all worldlines of slower-than-light objects have slopes closer to vertical than 45 degrees, just like a Minkowski diagram in SR, so from this perspective the reason it's impossible to enter a white hole horizon from the outside is just the same as the reason it's impossible to enter the past light cone of a given event from the outside in SR (because it's a surface moving inward at c so you can't catch up with it to cross it).

 

[relativityfan]

thank you for your reply, but i don't believe it is only repulsive where you think, because there is no ring singularity in a Reissner Nordstrom black hole but it is also repulsive with also a very similar structure. by lookling just a the metric, I deduce that time becomes contracted and length becomes dilated near the singularity (the opposite of the classic time dilation) which explains why it is repulsive

 

[pervect]

The Kerr solution, like the Schwarzschild solution, assumes a perfectly symmetrical collapse.

Unfortunately, this is unlikely to actually happen, making both solutions rather doubtful in the interior region. The Penrose diagram of the interior an actual rotating black hole caused by gravitational collapse is still a matter of some speculation and debate.

See for instance http://prl.aps.org/abstract/PRL/v63/i16/p1663_1 for an old papers on the topic, I don't know what the current status of the debate is.

 

[JesseM]

thank you for your reply, but i don't believe it is only repulsive where you think, because there is no ring singularity in a Reissner Nordstrom black hole but it is also repulsive with also a very similar structure. by lookling just a the metric, I deduce that time becomes contracted and length becomes dilated near the singularity (the opposite of the classic time dilation) which explains why it is repulsive

Well, look at the waterfall coordinate animations for both a Reissner-Nordstrom black hole and a Kerr black hole...the animation for the Reissner-Nordstrom black hole shows that the speed of infalling space (represented by the magnitude of the light blue arrows) does reach zero at a finite radius from the singularity, whereas the two waterfall animations for the Kerr black hole don't show the light blue arrows reaching zero until they reach the plane of the inside of the ring singularity.

I don't see why "repulsive gravity" should be equivalent to time contraction and length dilation, perhaps someone more knowledgeable could comment...

 

[JesseM]

The Kerr solution, like the Schwarzschild solution, assumes a perfectly symmetrical collapse.

Unfortunately, this is unlikely to actually happen, making both solutions rather doubtful in the interior region. The Penrose diagram of the interior an actual rotating black hole caused by gravitational collapse is still a matter of some speculation and debate.

See for instance http://prl.aps.org/abstract/PRL/v63/i16/p1663_1 for an old papers on the topic, I don't know what the current status of the debate is.

Thanks pervect, that paper isn't available freely online but I did find this more recent paper discussing the "mass inflation" which Poisson and Israel first derived in that earlier paper:

http://arxiv.org/abs/0811.1926

 

[relativityfan]

thank you for this link.
after having read this (ingoing and outgoing fluids) I do believe that even without the mass inflation, there is no white hole.
the structure of the "white hole" if we look at the metric should be exactly the same as the structure of the black hole, and matter could excape because it can be accelerated faster than the speed of light(like matter inside the event horizon) . So this would not be a white hole but a black hole, because the metric would be the same.
Does anyone disagree with this?

 

[JesseM]

thank you for this link.
after having read this (ingoing and outgoing fluids) I do believe that even without the mass inflation, there is no white hole.
the structure of the "white hole" if we look at the metric should be exactly the same as the structure of the black hole, and matter could excape because it can be accelerated faster than the speed of light(like matter inside the event horizon) . So this would not be a white hole but a black hole, because the metric would be the same.
Does anyone disagree with this?

Yes, it is incorrect according to GR, which is a time-symmetric theory so that the time-reverse of any solution (like a black hole that forms at some time from a collapsing star, then exists eternally afterwards, the time-reverse of which would be a white hole that has existed eternally and then finally 'explodes' into an expanding star) must also be a valid GR solution. And if one wants to find the "maximally extended" spacetime for an eternal black hole (i.e. a spacetime where geodesics can be extended arbitrarily far in both directions of their proper time unless they hit a singularity at some finite proper time), one must include a white hole interior region that's separate from a black hole interior region. The reason this is needed is that there are possible worldlines of particles outside the event horizon which in exterior Schwarzschild coordinates or ingoing Eddington-Finkelstein coordinates (see the bottom half of this page for an intro. to different coordinate systems for a Schwarzschild black hole) have been rising away from the event horizon for an infinite coordinate time--in the limit as coordinate time goes to negative infinity, the distance of these particles from the horizon approaches zero but their proper time approaches some finite value, so a "maximally extended" spacetime must include a region where they came from before crossing the event horizon in the outward direction. On the page I linked to above, look at the orange worldlines to the right of the vertical red line representing the event horizon, which are moving away from the event horizon as time increases but which approach it in the limit as time goes to negative infinity:

Schwarzschild coordinates

Eddington-Finkelstein coordinates

By means of a coordinate transformation one can transform to Kruskal-Szekeres coordinates where these outgoing worldlines crossed the event horizon in the outward direction at finite coordinate time rather than at negative infinity (in this coordinate system the event horizon is split into two different lines at 45 degrees from vertical, one shown as pink and the other shown as dark red):

But this diagram does not illustrate a "maximally extended" spacetime because those outgoing worldlines just end at the event horizon (this diagram only shows the worldlines of outgoing light rays which don't have a 'proper time' although one can define an 'affine parameter' for them that's similar to proper time, but in this diagram one could also draw on the worldlines of outgoing slower-than-light particles which would also cross the event horizon at finite coordinate time and at a finite value of their proper time, whereas once again 'maximally extended' implies that both proper time and other affine parameters should extend to arbitrary values unless the worldline runs into a singularity). To have a maximally extended spacetime you have to continue those outgoing worldlines by having them come from a "white hole interior region" at the bottom, and you also have to add a second "exterior region" (a sort of parallel universe) on the left:

Also, this statement of yours doesn't really make sense to me:

the structure of the "white hole" if we look at the metric should be exactly the same as the structure of the black hole, and matter could excape because it can be accelerated faster than the speed of light(like matter inside the event horizon) .

No matter moves "faster than the speed of light" in a local sense, whether inside or outside the event horizon. We can choose a coordinate system where light always moves at the same coordinate speed throughout the black hole spacetime, and where massive objects always travel slower than light--this is true of the coordinates used to draw a Penrose diagram and also of Kruskal Szekeres coordinates which can be obtained by a transformation from Schwarzschild coordinates, I recommend reading the nontechnical section of the wikipedia article on KS coordinates to get an idea of how things work in this coordinate system and how they relate to Schwarzschild coordinates. Note that region I on the diagram on that page corresponds to "our" region of spacetime outside the black hole, region II corresponds to the region in the interior of the black hole where infalling particles end up, and region IV on the bottom corresponds to the white hole interior region where outgoing particles must have come from (region III is the 'parallel universe' which is not entirely disconnected from our region since particles that have fallen into the black hole event horizon from region I may meet up with particles that have fallen in from region III).

Anyway, once you've reviewed the basic features of the Kruskal-Szekeres diagram (most of which are duplicated in the Penrose diagram), consider again what I said in an earlier post:

Penrose diagrams and Kruskal-Szekeres diagrams are designed with the property that all worldlines of light rays are at 45 degrees and all worldlines of slower-than-light objects have slopes closer to vertical than 45 degrees, just like a Minkowski diagram in SR, so from this perspective the reason it's impossible to enter a white hole horizon from the outside is just the same as the reason it's impossible to enter the past light cone of a given event from the outside in SR (because it's a surface moving inward at c so you can't catch up with it to cross it).

Do you see that in terms of these coordinate systems, as coordinate time progresses the white hole event horizon contracts inwards at the speed of light, until at the center of the diagram it becomes a black hole event horizon which expands outward at the speed of light? Can you see why this means that nothing can cross the white hole event horizon starting from the outside going in, while nothing can cross outward from the interior to the exterior of the black hole event horizon?

 

[relativityfan]

yes I understand what you mean

 

[Yog-Sothoth]

I've been trying to figure out the Penrose-Carter diagrams of the Kerr black hole or actually the way they are made. I've read e.g the article of Carter: "Complete Analytic Extension of the Symmetry Axis of Kerr's Solution" http://www.luth2.obspm.fr/~luthier/carter/trav/Carter66.pdf" , and the same thing from Chandrasekhar's "Mathematical Theory of Black Holes". But I find hard to understand how they actually manage to represent the diagrams. I understand (on some level) how they extend the Kerr spacetime but the part where P-C-diagrams step into the picture is unclear.

It seems to me that they don't use any exact transformation for the metric. At least Carter says "..since the precise transformations need not to be specified the diagrams do not have any scale on them". It also seems to me that the conformal factor in this case isn't that clear. Or maybe I've spent too many hours and it turns out to be something trivial (I doubt it)

Could somebody enlighten me how they manage to represent the diagrams? Thank you!