Rotating Collapse: Exploring Poisson-Israel Model & Fate of Observer

[pervect]

This may be a repeat, but I don't think the earlier shorter post survived the recent PF upgrade, at least I couldn't find it. This is a better post anyway :-).

I've been trying to understand the Poisson-Israel model of a charged black hole recently. (The model is also expected to provide insights into rotating black holes as well as charged ones, as I understand it).

http://prola.aps.org/abstract/PRD/v41/i6/p1796_1

(the full text unfortunately requires one to visit a library with access)

I'd like to be able to answer simple questions such as "what is the fate of an observer who falls into such a hole?

Does he get spaghettified? http://en.wikipedia.org/wiki/Spaghettification

Does he get fried by infinite radiation, crushed by infinite pressures, or see the fate of the universe?

Some specificity may be needed to answer these questions, but at the moment I'm leaving the exact details of said black hole rather vague. Ultimately, I'm trying to generally understand the fate of someone falling into an actual rotating black hole like the one in the center of our galaxy, but I'm more than willing to consider simpler cases first.

I'd also like to better understand what the authors mean by "mass inflation".

So far I'm having a hard time even figuring out the metric. Using Schwarzschild coordinates (r,t) I gather that

$$g^{rr} = 1 - 2m(r,t) / r + e^2/r^2$$

m(r,t) appears to be the much-talked about "mass function", and it appears to become infinite (?!) near the inner horizon.

Because r is timelike inside the event horizon, this should be the most interesting metric coefficient, but I'm not sure how one goes about finding the other metric coefficients (other than by redoing the calculations).

Other related questions (this is probably already too long) - how does one justify the model of "backscattering" used to determine how some small component of infalling radiation gets "backscattered" into outwards going radiation?

 

[Chris Hillman]

Mass inflation

Hi, pervect,

This may be a repeat, but I don't think the earlier shorter post survived the recent PF upgrade, at least I couldn't find it. This is a better post anyway :-).

In addition to PF instability, I have been troubled by instability in the electric power grid, so I was forced to enter standby mode for a day or so. I did see your question about CTCs in Kerr, but had to log off just before answering. I'll try to find that again and answer at greater length, but the short answer is that the CTC's in the Kerr vacuum occur in the deep interior (inside the blocks of the Carter-Penrose diagrams which include the timelike curvature singularities). There is an ongoing debate in CQG between Bonnor and some others over whether or not CTCs occur in vacuum solutions in places where their effects could in principle be observed by distant physicists: my short answer is that while there is no doubt that many simple exact solutions suggest the disturbing answer "yes", in my view and that of the physicists arguing with Bonnor, these solutions, which exhibit "massless struts" and so on, are artifacts of inappropriate boundary conditions and thus cannot be regarded as physically acceptable.

I've been trying to understand the Poisson-Israel model of a charged black hole recently. (The model is also expected to provide insights into rotating black holes as well as charged ones, as I understand it).

Yes, based upon a mathematical analogy.

http://prola.aps.org/abstract/PRD/v41/i6/p1796_1
(the full text unfortunately requires one to visit a library with access)

Yes, but there used to be a website with an html version. I don't have that paper in front of me but I'll refer to Israel's article in the book Black Holes and Relativistic Stars, edited by Wald (aka the Chandrasekhar memorial volume). Another reference: the book Black Hole Physics by Frolov and Novikov extensively discussed mass inflation.

Those with access to none of these can refer to the crude ASCII sketch of the appropriate "Carter-Penrose conformal block diagram" offered toward the bottom of this webpage: http://www.math.ucr.edu/home/baez/RelWWW/history.html

I'd like to be able to answer simple questions such as "what is the fate of an observer who falls into such a hole?

Does he get spaghettified?

It depends on how he falls in. If he avoids the spacelike (not timelike!) curvature singularity, he must encounter a Cauchy horizon, and gtr can't predict what happens after that. Interestingly enough, as with some exact gravitational plane wave solutions discussed in an important paper by Ellis and Schmidt, such an observer might observe a divergence of tidal forces, but these might occur so quickly that his body, modeled an elastic solid, might not have time to distort enough to harm him before he passes through the danger region. This is the same place where one might worry about him being fried by EM radiation from starlight falling in from the external universe, etc.

Does he get fried by infinite radiation

That's the question asked by Penrose. This is a topic of current research interest in classical gravitation.

crushed by infinite pressures

Not if he avoids the spacelike curvature singularity (which is "crushing" in terms of volume as well as "sphaghetifying" in terms of shape).

or see the fate of the universe?

Well, the diagram I offered doesn't attempt to account for Lambda if that's what you mean. If you sketch the analogous picture starting from a Schwarzschild-de Sitter model, you can see the answer you probably want should be "no, he can't see any events lying above one of the cosmological horizons, hence he can't see the fate of most stuff in the universe".

Some specificity may be needed to answer these questions, but at the moment I'm leaving the exact details of said black hole rather vague. Ultimately, I'm trying to generally understand the fate of someone falling into an actual rotating black hole like the one in the center of our galaxy, but I'm more than willing to consider simpler cases first.

Well, this a topic of current research interest. As I mentioned in some other threads, Chandrasekhar and his students Valeria Ferrari and Basilis Xanthopolous (who was tragically murdered by a disgruntled student in Greece, thus cutting short a brilliant career) discovered a beautiful duality between certain colliding plane wave solutions and models of black hole interiors. Strictly speaking, this yields a local isometry of part of the black hole "interior region" (think of roughly $$m < r < 2 m$$ with the "interaction zone region" in a CPW model, but this suffices to study the question of whether the Cauchy horizon corresponds to a survivable geometric singularity or not. Several researchers have in fact constructed CPW models to model the effect of infalling incoherent radiation and so on. Some of these models suggest that the answer is "yes"; others, "no".

Another major goal is to try to understand how perturbations (due to infalling matter and radiation) can (probably) change the avoidable timelike singularities of the Kerr deep interior into a strong spacelike scalar singularity plus something at this Cauchy horizon.

I'd also like to better understand what the authors mean by "mass inflation".

So far I'm having a hard time even figuring out the metric. Using Schwarzschild coordinates (r,t) I gather that

$$g^{rr} = 1 - 2m(r,t) / r + e^2/r^2$$

m(r,t) appears to be the much-talked about "mass function", and it appears to become infinite (?!) near the inner horizon.

In the perturbed Reissner-Nordstrom model of an interior, the mass function is defined via the gradient of the "radial coordinate" (which we can treat as a monotonic scalar field):
$$| \nabla r |^2 = 1 - 2 m(t,r)/r + e^2/r^2$$
The field equations reduce to two coupled equations and lead to a nonlinear wave equation for m. For a null dust influx (e.g. incoherent EM radiation), Poisson and Israel concluded that m(t,r) diverges near the Cauchy horizon, which is known as "mass inflation". By a mathematical analogy of charge with angular momentum (compare Carter-Penrose diagrams for Kerr and RN with Schwarzschild), one expects something similar to happen with perturbations of Kerr due to influx of null dust.

Other related questions (this is probably already too long) - how does one justify the model of "backscattering" used to determine how some small component of infalling radiation gets "backscattered" into outwards going radiation?

The very extensive theory of perturbations of black holes (another major achievement largely due to Chandrasekhar, building upon work by Wheeler, Teukolsky, and others).

Chris Hillman

 

[Entropia]

Thank you for providing more information on the Poisson-Israel model and the fate of an observer falling into a charged black hole. It is indeed a complex and intriguing topic that requires a deep understanding of general relativity.

To answer your first question, the fate of an observer falling into a charged black hole depends on various factors such as the mass, charge, and rotation of the black hole, as well as the trajectory of the observer. In general, as the observer approaches the black hole's event horizon, they will experience extreme tidal forces, which can lead to spaghettification. This is because the gravitational pull of the black hole is much stronger at the observer's feet than at their head, causing them to be stretched and torn apart.

In the case of a rotating black hole, the situation is more complex due to the presence of the inner and outer horizons. As the observer crosses the outer horizon, they will enter the ergosphere, where the rotation of the black hole drags space-time around it. This can lead to extreme frame dragging effects, making it difficult for the observer to escape. As they approach the inner horizon, they will experience infinite tidal forces, causing them to be crushed and ultimately reaching the singularity at the center of the black hole.

Regarding the concept of "mass inflation", it refers to the increase in the mass function as the observer approaches the inner horizon. This phenomenon is a result of the backscattering of infalling radiation, which increases the mass of the black hole and leads to an increase in the gravitational pull near the inner horizon. This can also contribute to the crushing of the observer near the singularity.

As for the metric, it is indeed complex and requires a detailed understanding of the Einstein field equations. The metric coefficients are determined by solving these equations for a given distribution of mass and charge. The authors of the paper you mentioned have used a simplified model to study the effects of backscattering on the black hole's properties, but it is still a valid approach in understanding the overall behavior of a charged black hole.

I hope this helps in addressing your questions and providing a better understanding of the Poisson-Israel model. Keep exploring and learning about this fascinating topic!