Big trouble with Kerr solution

[Dmitry67]

'Ideal' schwarzschild solution is for an eternal black hole. It also includes a 'white hole' part, which is excluded, or course, from a 'realistic' collapse solution.

What's about the kerr solution? Inside kerr black hole there is ring with enougmous frame dragging around it, so strong, that it creates CTL around it. In fact, the ring itself is not a ring in space, but a ring in time - a perfect example of CTL.

However, such ring must be also eternal - as it is looped in time, in can't be formed in a first place. So apparently that beautiful ring is also a part of an 'idealistic' kerr Black hole. I wonder, what is inside a realistic rotating black hole?

 

[Bill_K]

Dmitry67, the closed timelike lines in the vacuum Kerr solution are restricted to a particular region r < 0.

As you say correctly, a collapsing object will replace the 'white hole' part of the Schwarzschild solution with an interior (non-vacuum) solution. The same thing happens to Kerr. In the real world, the anomalous portion of vacuum Kerr will be replaced by a collapsing non-vacuum solution, and so is not present within any rotating black hole that was formed by collapse.

 

[Dmitry67]

Hm...
So what is inside a second horizon of a realistic Kerr Black hole?

 

[George Jones]

'Ideal' schwarzschild solution is for an eternal black hole. It also includes a 'white hole' part, which is excluded, or course, from a 'realistic' collapse solution.

What's about the kerr solution? Inside kerr black hole there is ring with enougmous frame dragging around it, so strong, that it creates CTL around it. In fact, the ring itself is not a ring in space, but a ring in time - a perfect example of CTL.

However, such ring must be also eternal - as it is looped in time, in can't be formed in a first place. So apparently that beautiful ring is also a part of an 'idealistic' kerr Black hole. I wonder, what is inside a realistic rotating black hole?

Dmitry67, the closed timelike lines in the vacuum Kerr solution are restricted to a particular region r < 0.

As you say correctly, a collapsing object will replace the 'white hole' part of the Schwarzschild solution with an interior (non-vacuum) solution. The same thing happens to Kerr. In the real world, the anomalous portion of vacuum Kerr will be replaced by a collapsing non-vacuum solution, and so is not present within any rotating black hole that was formed by collapse.

See the figure between pages 14 and 15 from the link below.

Hm...
So what is inside a second horizon of a realistic Kerr Black hole?

There is a weak curvature singularity at the inner (Cauchy) horizon of a rotating black hole. Seminal work on this was done by Poisson and Israel, and this work was continued by Ori. See

http://physics.technion.ac.il/~school/Amos_Ori.pdf ,

particularly pages 15, starting at "Consequence to the curvature singularity at the IH: (IH = Inner Horizon), 16, and 24. On page 24, Ori says that classical general relativity cannot predict what happens inside the inner horizon,

For Novikov's take on this, see

http://arxiv.org/abs/gr-qc/0304052.

Roughly, if components of g (the metric) are continuous but "pointy" (like the absolute value function), then first derivatives of g have step diiscontinuities (like the Heaviside step function), and second derivatives of g (used in the curvature tensor) are like Dirac delta functions. If a curvature singularity blows up like a Dirac delta function, then integration produces only a finite contribution to the tidal deformation of an object, which, if the object is robust enough, it can withstand.

 

[Dmitry67]

Wow. Thank you.
So there is no answer.

But I failed to understand why GR can't make predictions in that case. Numerical approach works when everything else fails... What am I missing?

 

[George Jones]

Wow. Thank you.
So there is no answer.

But I failed to understand why GR can't make predictions in that case. Numerical approach works when everything else fails... What am I missing?

Ori says that numerical work might be sufficient to determine the evolution of the inner horizon, not that numerical work will predict what happens inside the inner horizon.

 

[Dmitry67]

Thank you.
Is there any progress in related problem - formation of superextreme black holes? I remember, there were some numerical simulations showing that it was possible.
Interesting how realistic naked singularity would look like.

 

[Dmitry67]

Well, I think this is much worse than I expected.

It is not about the ability to solve the equations - the claim in the article should be taken literally: GR can't predict what happens inside BH - even if you try numerical approach.

At first, I have to admit that I was wrong: CTLs can be ‘created’ and even can be limited in 'external' time - eternal inside, they can be created and they can dissipate. However, it is possible that there are objects inside these CTLs with circular trajectories. Such objects are eternal: they are not created, they just exists ‘forever’ inside CTLs.

But when CTL is ‘created’, how many eternal objects are inside? Say, there is a blob of iron… or may be plumber? Or plasma? Hot or cold? A spectrum of options can be compatible with the boundary conditions of CTL. Usually, we predict future by calculating the trajectories of particles in curved spacetime. In case of CTLs. We can't do that. We get an area of spacetime, which has no past. Hence, we can’t predict what is inside.

It is not a problem of getting analytics solutions, all numerical methods will fail miserably.