Do Black Holes End up as Quark Stars? and quantum gravity

[kodama]

this paper


Do Black Holes End up as Quark Stars ?

R.K.Thakur

(Submitted on 25 Feb 2007)

The possibility of the existence of quark stars has been discussed by several authors since 1970. Recently, it has been pointed out that two putative neutron stars, RXJ 1856.5 - 3754 in Corona Australis and 3C58 in Cassiopeia are too small and too dense to be neutron stars; they show evidence of being quark stars. Apart from these two objects, there are several other compact objects which fit neither in the category of neutron stars nor in that of black holes. It has been suggested that they may be quark stars.In this paper it is shown that a black hole cannot collapse to a singularity, instead it may end up as a quark star. In this context it is shown that a gravitationally collapsing black hole acts as an ultrahigh energy particle accelerator, hitherto inconceivable in any terrestrial laboratory, that continually accelerates particles comprising the matter in the black hole. When the energy \textit{E} of the particles in the black hole is ≥102GeV, or equivalently the temperature \textit{T} of the matter in the black holes is ≥1015K, the entire matter in the black hole will be converted into quark-gluon plasma permeated by leptons. Since quarks and leptons are spin 1/2 particles,they are governed by Pauli's exclusion principle. Consequently, one of the two possibilities will occur; either Pauli's exclusion principle would be violated and the black hole would collapse to a singularity, or the collapse of the black hole to a singularity would be inhibited by Pauli's exclusion principle, and the black hole would eventually explode with a mini bang of a sort. After explosion, the remnant core would stabilize as a quark star.
Comments: 6 pages

Subjects: Astrophysics (astro-ph)

Cite as: arXiv:astro-ph/0702671

(or arXiv:astro-ph/0702671v1 for this version) If this paper's conclusion is correct, and collapsing stars result in quark stars rather than black holes, that gravity cannot overcome fermion quark Pauli's exclusion principle astrophysical black holes are really quark stars. increasing its density simply results in excess energy being radiated away.

if the paper is correct, how would this affect black hole physics and quantum gravity theories? how would this effect black hole entropy, black hole information paradox, black hole firewalls, hawking radiation, event horizon, reconciling gravity with quantum mechanics, if black holes are actually quark stars as argued in the paper.

specifically, how would hawking calculation of hawking radiation, black hole entropy, structure of spacetime, holographic principle be modified if

every astrophysical "black hole" is actually a quark star, general relativity and gravity cannot over come Pauli exclusion principle of fermions, and that quark degenerate matter, quark-gluon plasma represents the upper limit possible for density as argued in paper above. above this density the quark gluon plasma simply radiates away excess energy.

string theory and lqg

 

[mathman]

If an astronomical object is small enough and massive enough that the escape velocity is greater than the speed of light, it is a black hole. Its physical composition is an open question, so it could be a quark star.

 

[kodama]

If an astronomical object is small enough and massive enough that the escape velocity is greater than the speed of light, it is a black hole. Its physical composition is an open question, so it could be a quark star.

i think he is arguing this

Since quarks and leptons are spin 1/2 particles,they are governed by Pauli's exclusion principle. Consequently, one of the two possibilities will occur; either Pauli's exclusion principle would be violated and the black hole would collapse to a singularity, or the collapse of the black hole to a singularity would be inhibited by Pauli's exclusion principle, and the black hole would eventually explode with a mini bang of a sort. After explosion, the remnant core would stabilize as a quark star.

the objects identified as black holes are actually quark stars. suppose for the sake of argument he is correct. what are the implications to string/LQG/bh physics on such issues as entropy thermodynamics holography hawking radiation firewalls information paradox if black holes are quark stars.

 

[ShayanJ]

Its strange that you guys are assuming that the paper is saying that a black hole is a quark star, it isn't!

Since quarks and leptons are spin 1/2 particles,they are governed by Pauli's exclusion principle. Consequently, one of the two possibilities will occur; either Pauli's exclusion principle would be violated and the black hole would collapse to a singularity, or the collapse of the black hole to a singularity would be inhibited by Pauli's exclusion principle, and the black hole would eventually explode with a mini bang of a sort. After explosion, the remnant core would stabilize as a quark star.

Its obvious from the above part, that the paper is suggesting that the scenarios that people usually don't assume to be a single class of scenarios with the same result of a black hole, should be divided into two classes, one that gives a black hole as a result and another that gives a quark star as a result. This paper is only suggesting its not "neutron star for this much mass, black hole for more massive" but " neutron star for this much mass, quark star for more massive, black hole for yet more massive". I don't see what part of this suggestion is new.

 

[PAllen]

Note that this paper was never published in a peer reviewed journal, and it has exactly zero citations (which means experts in the field believe it irrelevant, not even worth disputing).

On the other hand, broadly speaking, there is nothing wildly implausible about the speculations in this paper. It notes (correctly) that singularity theorems assume the correctness of GR at all energy scales, while most physicists doubt this is true. Is it the most likely candidate for what there is instead of singularity? Who knows, but probably not.

 

[Agave tequilana]

I would think that it might be a possibility after the black hole has went throuh Hawking radiation.

P.S. Highly advise to ignore me since I am an amateur compared to the rest on the site.

 

[kodama]

Note that this paper was never published in a peer reviewed journal, and it has exactly zero citations (which means experts in the field believe it irrelevant, not even worth disputing).

On the other hand, broadly speaking, there is nothing wildly implausible about the speculations in this paper. It notes (correctly) that singularity theorems assume the correctness of GR at all energy scales, while most physicists doubt this is true. Is it the most likely candidate for what there is instead of singularity? Who knows, but probably not.

ok can you peer review it?

what is the most likely candidate for what there is instead of singularity?

 

[PAllen]

ok can you peer review it?

what is the most likely candidate for what there is instead of singularity?

No one is going to do peer review on demand. Further, I am not a professional physicist and this is not my area of greatest expertise.

However, purely as a matter of opinion, my personal guess for the most likely alternative to a singularity is the fuzzball picture:

http://arxiv.org/abs/1312.4017
http://www.physics.ohio-state.edu/~mathur/faq2.pdf

 

[ohwilleke]

There are a several claims made in this paper which are only loosely related to each other that should each be evaluated on its own merits.

The possibility of the existence of quark stars has been discussed by several authors since 1970. Recently, it has been pointed out that two putative neutron stars, RXJ 1856.5 - 3754 in Corona Australis and 3C58 in Cassiopeia are too small and too dense to be neutron stars; they show evidence of being quark stars. Apart from these two objects, there are several other compact objects which fit neither in the category of neutron stars nor in that of black holes. It has been suggested that they may be quark stars.

The first claim is that a handful of objects that have been observed may be quark stars because they are too small and too dense to be neutron stars. The natural alternative to this claim would be that we have misunderstood something about the condensed matter physics of neutron stars (theoretical error), or that measurement error of some undetermined nature makes objects that are really neutron stars or small black holes look like something else. Given that our models of neutron stars are fairly crude and the precision of QCD calculations is not very great, particularly in complex systems, and that there objects are outliers in very large astronomy data sets (such that a several sigma deviation for the best fit value would be expected for a few data points out of all the data points measured), those explanations sound pretty convincing.

The big problem with this data point in relation to the claim that follows is that is the mechanism proposed below creates quark stars, then quark stars should be ubiquitous. Even if one in a ten thousand stars that collapse into quark stars when they would conventionally be expected to form neutron stars or black holes are in a sweet spot where a quark star trajectory is impossible, we would expect to see far more quark star candidates.

If the objects observed really are quark stars, one needs a mechanism that produces far fewer of them than the proposed mechanism would be expected to produce.

One plausible sort of mechanism that I can imagine that would produce the right number of observed events plus or minus, would be one in which a quark star is relatively short lived state that either goes nova, or transforms into some other better known state after some period of time from say, a few dozen years to a tens of millions of years. It needs to be long lived enough to be observed in repeated observations of an object from Earth during time periods when we had telescopes sufficient to see them, yet short lived enough to be very uncommon. If it were stable for time periods on the order of billions of years or even hundreds of millions of years, there would be far too many of them to match astronomy observations. For example, perhaps a quark star that subsequently gains additional mass by accretion becomes a black hole as soon as it acquires enough additional mass.

Another plausible mechanism that I can imagine would produce a quark star only in a truly tiny range of initial conditions (perhaps only in stars in a mass range of 0.001 solar masses between the neutron star and black hole threshold with virtually on outside gravitational forces preventing the mass distribution within the collapsing star from being anything other than perfectly spherically symmetrical) squeezed between those that produce a neutron star and those that produce a black hole, whose extreme rarity because the initial conditions requirements are so demanding, almost never happens but do happen in a handful of outlier cases. This might be compared by analogy to water that retains a phase outside the usual conditions of a phase diagram because it is so homogeneous that the phase change transition isn't triggered until it does so explosively one the slightest bit of anisotrophy is introduced into the system.

In either case, for the reasons discussed below, a trajectory of evolution that does not involve a black hole as an intermediate state would be more plausible.

In this paper it is shown that a black hole cannot collapse to a singularity, instead it may end up as a quark star. In this context it is shown that a gravitationally collapsing black hole acts as an ultrahigh energy particle accelerator, hitherto inconceivable in any terrestrial laboratory, that continually accelerates particles comprising the matter in the black hole. When the energy \textit{E} of the particles in the black hole is ≥102GeV, or equivalently the temperature \textit{T} of the matter in the black holes is ≥1015K, the entire matter in the black hole will be converted into quark-gluon plasma permeated by leptons. Since quarks and leptons are spin 1/2 particles,they are governed by Pauli's exclusion principle. Consequently, one of the two possibilities will occur; either Pauli's exclusion principle would be violated and the black hole would collapse to a singularity, or the collapse of the black hole to a singularity would be inhibited by Pauli's exclusion principle. . . .

Certainly, the claim that a black hole cannot collapse to a true singularity due to Pauli's exclusion principle is a common place one in pretty much any theory of quantum gravity. Almost nobody is lining up to propose theories that Pauli's exclusion principle is violated inside black holes to form true singularities. And, indeed, the density of neutron stars and black holes is such that violation's of Pauli's exclusion principle are not needed to explain any observable phenomena.

Most theorists decline to speculate on what is happening inside a black hole on the grounds that it is inherently not observable. But, there is nothing deeply troubling about following the laws of physics to their logical conclusion inside a black hole. The claim that the interior of a black hole collapses not to a singularity but towards a quark star as noted above, to the extent that it concerns the inner workings of a black hole is "not even wrong."

. . . and the black hole would eventually explode with a mini bang of a sort. After explosion, the remnant core would stabilize as a quark star.

Far more controversial is the further claim that these speculations about the interior of a black hole have an observable consequence at some point as this paper suggests because a quark star inside a black hole would "explode with a mini bang of a sort" and that after the explosion, the remnant core "would stabilize as a quark star", although if the circumstances in which a mini bang were triggered were rare enough, it would provide an alternate explanation for the rarity of quark stars.

One of the reasons that this analysis is suspect has to do with Black Hole entropy. The late Jacob Bekenstein showed that the notion of black holes in GR and the Second Law of Thermodynamics which holds that entropy always increases, are not in conflict if a black hole is a state of maximum entropy which he showed in a calculation establishing that entropy of a black hole based upon observable quantities on its event horizon and making a few other key assumptions.

For a black hole to "explode with a mini bang of a sort" that left a remnant core that would stabilize as a quark star, one would have to show that the quark star had more entropy than the black hole state that existed before the mini bang, so as not to violate the Second Law of Thermodynamics (or alternatively, that this circumstance was a singular exception to the Second Law of Thermodynamics). Neither of these possibilities seem very likely. Naively speaking, one would expect quark stars which are quite specific sets of microstates to have lower entropy than black holes, although perhaps higher entropy than neutron stars. There is also zero evidence for a black hole nova being directly observed.

* * *

As a final aside, unlike PAllen, I wouldn't automatically discount the ideas in a paper merely because it wasn't published and has no citations in an eight year period (even assuming that this is the case). First of all, "Sleeping Beauty papers" which go unnoticed for years and then suddenly see a surge of citations are a well known phenomena in science. http://washparkprophet.blogspot.com/2015/05/sleeping-beauty-papers.html

There are also a variety of reasons that a nearly finished paper wouldn't be published that have nothing to do with failing peer review. The investigator may be inept at following through on getting something published, may have seen another paper published around the same time that said exactly the same thing that may have been unknown to him until a peer reviewer pointed it out to him or he found it in a final polishing of the literature review pre-publication, the investigator may have simply gotten busy and never gotten around to resubmitting the paper, the investigator could have left academia for private industry and found that getting published was no longer a priority in his new position, or the ideas in the paper could have been incorporated into a new and better paper with a different name and different spin perhaps rendering the preprint obsolete. This author, for example, incorporated some of the same ideas into a 2011 preprint also incorporating the Pauli exclusion principle, this time in a Big Bang context: http://arxiv.org/abs/1103.3688 Or, the investigator could simply have gotten a bad peer reviewer who demanded too much to make it worth following through upon. These requests are sometimes reasonable and sometimes not. As physical anthropologist John Hawks has noted: "Peer review as practiced today is a form of hazing." http://johnhawks.net/weblog/topics/metascience/journals/peer-review-hazing-2015.html

A lack of publications and citations certainly don't provide automatic credibility to a paper that is present when publication and citations are present. But, their absence alone isn't a reason to discount a pre-print, particularly if the author is a professional physicist. This author has 11 other co-authorships from 2012-2015 as an investigator with a dark matter direct detection experiment's collaboration, as well as the 2011 paper mentioned above, a 2009 published paper http://arxiv.org/pdf/0901.1956.pdf, a published 1982 paper http://adsabs.harvard.edu/full/1982Ap&SS..84...99T, and so is clearly a professional physicist. http://arxiv.org/find/astro-ph/1/au:+Thakur_R/0/1/0/all/0/1 He appears to be a professor at a University in India. http://www.prsu.ac.in/departments/Physics/Physics.html While his paper didn't produce a lot of acclaim it may have been pivotal in getting hired for these jobs, suggesting his employers did not find it to be too off base.

In any case, if I'm not misreading the data, this particular paper actually does appear to have been published as a manuscript in the Journal "Astronomy and Astrophysics" on February 5, 2008 http://arxiv.org/pdf/astro-ph/0702671.pdf and to have received one citation in this paper: http://arxiv.org/abs/gr-qc/0512088 It also received a bit of discussion of several serious science blogs maintained and most read by scientists or well educated laymen.

While this still isn't terribly impressive, it is quite close to the median outcome and certainly suggests that it shouldn't be written off simply based upon "meta" analysis related to its authorship, publication status and citation history.

 

[kodama]

Hi thanks for replying. if there is a quark star inside every "black hole" how does it effect the bh issues I've identified like information paradox etc

 

[ohwilleke]

If there is a quark star in every black hole, this might reduce the entropy we expect in a black hole since the partition of its mass-energy into a quark star component and an non-quark star component would reduce the number of possible microstates and hence the entropy of the black hole, which would reduce the amount of information it could hold, albeit while structuring that information somewhat. I don't have the capacity to rigorously analyze that difference quantitatively however.

 

[kodama]

If there is a quark star in every black hole, this might reduce the entropy we expect in a black hole since the partition of its mass-energy into a quark star component and an non-quark star component would reduce the number of possible microstates and hence the entropy of the black hole, which would reduce the amount of information it could hold, albeit while structuring that information somewhat. I don't have the capacity to rigorously analyze that difference quantitatively however.

i'll post another paper on this soon i'd love to hear your take on it thanks.

 

[kodama]

[Mentor's note: Post merged with this thread]

in response to

The first claim is that a handful of objects that have been observed may be quark stars because they are too small and too dense to be neutron stars. The natural alternative to this claim would be that we have misunderstood something about the condensed matter physics of neutron stars (theoretical error), or that measurement error of some undetermined nature makes objects that are really neutron stars or small black holes look like something else. Given that our models of neutron stars are fairly crude and the precision of QCD calculations is not very great, particularly in complex systems, and that there objects are outliers in very large astronomy data sets (such that a several sigma deviation for the best fit value would be expected for a few data points out of all the data points measured), those explanations sound pretty convincing.

The big problem with this data point in relation to the claim that follows is that is the mechanism proposed below creates quark stars, then quark stars should be ubiquitous. Even if one in a ten thousand stars that collapse into quark stars when they would conventionally be expected to form neutron stars or black holes are in a sweet spot where a quark star trajectory is impossible, we would expect to see far more quark star candidates.

If the objects observed really are quark stars, one needs a mechanism that produces far fewer of them than the proposed mechanism would be expected to produce.

One plausible sort of mechanism that I can imagine that would produce the right number of observed events plus or minus, would be one in which a quark star is relatively short lived state that either goes nova, or transforms into some other better known state after some period of time from say, a few dozen years to a tens of millions of years. It needs to be long lived enough to be observed in repeated observations of an object from Earth during time periods when we had telescopes sufficient to see them, yet short lived enough to be very uncommon. If it were stable for time periods on the order of billions of years or even hundreds of millions of years, there would be far too many of them to match astronomy observations. For example, perhaps a quark star that subsequently gains additional mass by accretion becomes a black hole as soon as it acquires enough additional mass.

Another plausible mechanism that I can imagine would produce a quark star only in a truly tiny range of initial conditions (perhaps only in stars in a mass range of 0.001 solar masses between the neutron star and black hole threshold with virtually on outside gravitational forces preventing the mass distribution within the collapsing star from being anything other than perfectly spherically symmetrical) squeezed between those that produce a neutron star and those that produce a black hole, whose extreme rarity because the initial conditions requirements are so demanding, almost never happens but do happen in a handful of outlier cases. This might be compared by analogy to water that retains a phase outside the usual conditions of a phase diagram because it is so homogeneous that the phase change transition isn't triggered until it does so explosively one the slightest bit of anisotrophy is introduced into the system.

In either case, for the reasons discussed below, a trajectory of evolution that does not involve a black hole as an intermediate state would be more plausible.
Certainly, the claim that a black hole cannot collapse to a true singularity due to Pauli's exclusion principle is a common place one in pretty much any theory of quantum gravity. Almost nobody is lining up to propose theories that Pauli's exclusion principle is violated inside black holes to form true singularities. And, indeed, the density of neutron stars and black holes is such that violation's of Pauli's exclusion principle are not needed to explain any observable phenomena.

In any case, if I'm not misreading the data, this particular paper actually does appear to have been published as a manuscript in the Journal "Astronomy and Astrophysics" on February 5, 2008 http://arxiv.org/pdf/astro-ph/0702671.pdf and to have received one citation in this paper: http://arxiv.org/abs/gr-qc/0512088 It also received a bit of discussion of several serious science blogs maintained and most read by scientists or well educated laymen.

While this still isn't terribly impressive, it is quite close to the median outcome and certainly suggests that it shouldn't be written off simply based upon "meta" analysis related to its authorship, publication status and citation history.

what this paper is arguing is that many objects identified as "stellar mass black holes" are actually color-flavor locked quark stars.

a quark star that absorbs all electromagnetic radiation would appear to be identical to a black hole, except instead of an event horizon, you have the surface of the quark star. apparently the quark star radius is slightly above the schwarzschild radius

Can stellar mass black holes be quark stars?
Z. Kovacs, K. S. Cheng, T. Harko
(Submitted on 19 Aug 2009)
We investigate the possibility that stellar mass black holes, with masses in the range of 3.8M⊙ and 6M⊙, respectively, could be in fact quark stars in the Color-Flavor-Locked (CFL) phase. Depending on the value of the gap parameter, rapidly rotating CFL quark stars can achieve much higher masses than standard neutron stars, thus making them possible stellar mass black hole candidates. Moreover, quark stars have a very low luminosity and a completely absorbing surface - the infalling matter on the surface of the quark star is converted into quark matter. A possibility of distinguishing CFL quark stars from stellar mass black holes could be through the study of thin accretion disks around rapidly rotating quark stars and Kerr type black holes, respectively. Furthermore, we show that the radiation properties of accretion disks around black holes and CFL quark stars are also very similar. However, strange stars exhibit a low luminosity, but high temperature bremsstrahlung spectrum, which, in combination with the emission properties of the accretion disk, may be the key signature to differentiate massive strange stars from black hole.
Comments: 27 pages, 5 figures, accepted for publication in MNRAS
Subjects: High Energy Astrophysical Phenomena (astro-ph.HE); General Relativity and Quantum Cosmology (gr-qc); High Energy Physics - Theory (hep-th)
Journal reference: MNRAS, 400, pp. 1632-1642 (2009)
DOI: http://arxiv.org/ct?url=http%3A%2F%2Fdx.doi.org%2F10%252E1111%2Fj%252E1365-2966%252E2009%252E15571%252Ex&v=08e0698b
Cite as: arXiv:0908.2672 [astro-ph.HE]
(or arXiv:0908.2672v1 [astro-ph.HE] for this version)

if the above paper is correct that astrophysical astronmical objects identified as black holes are actually "stellar mass black holes" are actually color-flavor locked quark stars

how does this "quark star black hole" effect black hole issues from hawking radiation to black hole information paradox black hole entropy firewall holographic principle

 

[Bernie G]

If an astronomical object is small enough and massive enough that the escape velocity is greater than the speed of light, it is a black hole. Its physical composition is an open question, so it could be a quark star.

and "In this paper it is shown that a black hole cannot collapse to a singularity".

I agree. Ultra-relativistic pressure (like conventional pressure) increases as the inverse of r-cubed, faster than the forces of gravity increase. The formulas I use for calculating an ultra relativistic star’s size are: Ultra relativistic pressure P = (rho)(c^2)/3, implying a viral energy of M(c^2)/3. GPE = 1.1G(M^2)/R. Virial equation energy ratio = 2.0
This gives a star size = 1.65GM/(c^2), or 0.82 of the Schwarzschild radius.

 

[mfb]

Ultra-relativistic pressure (like conventional pressure) increases as the inverse of r-cubed, faster than the forces of gravity increase.

Gravity is not linear. Pressure actually increases the energy density, at some points it supports the collapse instead of slowing it.

 

[kodama]

Gravity is not linear. Pressure actually increases the energy density, at some points it supports the collapse instead of slowing it.

papers imply a lot of the energy is radiated away so it loses mass at same time

 

[Bernie G]

Gravity is not linear. Pressure actually increases the energy density, at some points it supports the collapse instead of slowing it.

Are you using the Tolman-Volkoff equation for that conclusion? This equation gives bad results for neutron star, let alone a black hole. If a star was significantly smaller than 1.0 SR, I think at 0.9 SR the gravitational acceleration would be about 0.8 that at 1.0 SR, a large number but still very finite. The core pressure in a neutron star is significantly less than (rho)(c^2)/3, which to me indicates that the next stage of support pressure after neutron collapse would be (rho)(c^2)/3.

 

[Bernie G]

Correction: I meant to write: Are you using the Tolman-Volkoff equation for that conclusion? This equation gives bad results for neutron star, let alone a black hole. If a star was significantly smaller than 1.0 SR, I think at 0.9 SR the gravitational acceleration would be about 1/0.8 that at 1.0 SR, a large number but still very finite. The core pressure in a neutron star is significantly less than (rho)(c^2)/3, which to me indicates that the next stage of support pressure after neutron collapse would be (rho)(c^2)/3.

 

[PAllen]

A few meta-points:

1) If you assume the dominant energy conditions holds (as it does microscopically, and exactly for classical theories like Maxwell EM), then horizon and singularity formation are inevitable for a wide range of initial conditions, as proven by Hawking and Ellis. No specific theory of matter need be assumed. Note that the dominant energy condition is also what is required to establish that the field equations of GR imply that small bodies move on timelike world lines. Violating it (macroscopically) implies bodies that can move FTL.

2) Almost no one believes (1) describes the what actually happens for collapsed states. There are two very different types of deviations expected:

a) Quantum theories violate the dominant energy conditions. There are bounds on the violation, but, to the best of my knowledge, there is no proof that these bounds recover the singularity theorems.

b) It is expected that GR is only a classical limit of a more accurate theory.

Given (2), irrespective of detailed matter models, there is no reason to believe a singularity forms. The majority view is that 'something macroscopically similar to a horizon' forms, but it is wide open what its microscopic structure is (and whether it acts like a firewall). The basis for believing that something like a horizon must form is that (especially for larger) BH, the collapse up to a horizon is not expected to have any meaningful deviations from the dominant energy condition, nor are quantum gravity corrections expected to be significant until well inside the horizon.

The more interesting, controversial claim for quark stars is not the possibility of such a state inside a macroscopic horizon, but the claim for such a possibility preventing an effective horizon forming. This requires deviation from GR with dominant energy condition under relatively non-extreme conditions (especially if supermassive BH are to be avoided).

 

[rootone]

If such a state as a quark star does exist it's an intermediate state between a neutron star and the traditional black hole.
A distinct final state of degenerate matter as a free quark soup is unknown but it's a definite maybe.
It would dispense with that annoying singularity (I think).
Could a hypothetical quark star form an event horizon?, if so then 'black hole' and 'quark star' could be just two names for the same thing.

 

[kodama]

A few meta-points:

1) If you assume the dominant energy conditions holds (as it does microscopically, and exactly for classical theories like Maxwell EM), then horizon and singularity formation are inevitable for a wide range of initial conditions, as proven by Hawking and Ellis. No specific theory of matter need be assumed. Note that the dominant energy condition is also what is required to establish that the field equations of GR imply that small bodies move on timelike world lines. Violating it (macroscopically) implies bodies that can move FTL.

2) Almost no one believes (1) describes the what actually happens for collapsed states. There are two very different types of deviations expected:

a) Quantum theories violate the dominant energy conditions. There are bounds on the violation, but, to the best of my knowledge, there is no proof that these bounds recover the singularity theorems.

b) It is expected that GR is only a classical limit of a more accurate theory.

Given (2), irrespective of detailed matter models, there is no reason to believe a singularity forms. The majority view is that 'something macroscopically similar to a horizon' forms, but it is wide open what its microscopic structure is (and whether it acts like a firewall). The basis for believing that something like a horizon must form is that (especially for larger) BH, the collapse up to a horizon is not expected to have any meaningful deviations from the dominant energy condition, nor are quantum gravity corrections expected to be significant until well inside the horizon.

The more interesting, controversial claim for quark stars is not the possibility of such a state inside a macroscopic horizon, but the claim for such a possibility preventing an effective horizon forming. This requires deviation from GR with dominant energy condition under relatively non-extreme conditions (especially if supermassive BH are to be avoided).

part of the claim is

onsequently, one of the two possibilities will occur; either Pauli's exclusion principle would be violated and the black hole would collapse to a singularity, or the collapse of the black hole to a singularity would be inhibited by Pauli's exclusion principle, and the black hole would eventually explode with a mini bang of a sort. After explosion, the remnant core would stabilize as a quark star. that a lot of mass gets shed

 

[PAllen]

part of the claim is

onsequently, one of the two possibilities will occur; either Pauli's exclusion principle would be violated and the black hole would collapse to a singularity, or the collapse of the black hole to a singularity would be inhibited by Pauli's exclusion principle, and the black hole would eventually explode with a mini bang of a sort. After explosion, the remnant core would stabilize as a quark star. that a lot of mass gets shed

Without major violation of energy conditions, such mass shedding is impossible per GR. The shed matter must follow a spacelike trajectory, which means locally FTL. Either that, or GR itself breaks down in a regime that is not expected. The original paper of this thread basically says "what if it were possible". It proposes no mechanism or plausibility argument. Its analogy with Big Bang is unsound, in that that does not involve a horizon or reversal of a collapse.

 

[PAllen]

If such a state as a quark star does exist it's an intermediate state between a neutron star and the traditional black hole.
A distinct final state of degenerate matter as a free quark soup is unknown but it's a definite maybe.
It would dispense with that annoying singularity (I think).
Could a hypothetical quark star form an event horizon?, if so then 'black hole' and 'quark star' could be just two names for the same thing.

If the quark star has an event horizon, that is the less interesting case I referred to, in that no one expects a singularity inside the horizon, and quark matter (with some new physics at the breakdown of GR at extremely high energy scales) is a plausible alternative. Also, not particularly controversial is the idea that there is mass range for quark stars slight more massive than neutron stars, without producing a BH and its horizon.

The interesting proposal (that I am skeptical of) is that collapses well into the range of mass expected to produce a BH, instead produce a quark star.

 

[kodama]

Without major violation of energy conditions, such mass shedding is impossible per GR. The shed matter must follow a spacelike trajectory, which means locally FTL. Either that, or GR itself breaks down in a regime that is not expected. The original paper of this thread basically says "what if it were possible". It proposes no mechanism or plausibility argument. Its analogy with Big Bang is unsound, in that that does not involve a horizon or reversal of a collapse.

GR, but does that take into account the quantum properties of matter? one variation of the idea is that stellar masses cannot collapse into black holes to begin with since the quantum properties of matter results in shedding of mass

 

[kodama]

If the quark star has an event horizon, that is the less interesting case I referred to, in that no one expects a singularity inside the horizon, and quark matter (with some new physics at the breakdown of GR at extremely high energy scales) is a plausible alternative. Also, not particularly controversial is the idea that there is mass range for quark stars slight more massive than neutron stars, without producing a BH and its horizon.

The interesting proposal (that I am skeptical of) is that collapses well into the range of mass expected to produce a BH, instead produce a quark star.

how would a quark star inside every black hole effect issues like entropy holographic principle etc. yes the interesting proposal is tied to the idea of mass loss and that quark stars themselves can be perfect black bodies

 

[PAllen]

GR, but does that take into account the quantum properties of matter? one variation of the idea is that stellar masses cannot collapse into black holes to begin with since the quantum properties of matter results in shedding of mass

That is not what the paper in post #1 argues. It discusses escape from collapse, with no justification or plausible model. It literally says "what if", in fancy language.

It is known from observation that mass shedding does not prevent neutron stars from forming, nor BH candidates from forming. Clearly, for BH candidates, they have a mass and properties such that they must be a BH per GR. Mass loss is already moot. Note, also, observational evidence is steadily increasing that BH candidates do not have any type of surface, as expected per GR.

 

[PAllen]

how would a quark star inside every black hole effect issues like entropy holographic principle etc. yes the interesting proposal is tied to the idea of mass loss and that quark stars themselves can be perfect black bodies

Most such models would say the interior state has no discernible effect on the quasi-horizon entropy or properties.

 

[Bernie G]

If such a state as a quark star does exist it's an intermediate state between a neutron star and the traditional black hole.
A distinct final state of degenerate matter as a free quark soup is unknown but it's a definite maybe.
It would dispense with that annoying singularity (I think).
Could a hypothetical quark star form an event horizon?, if so then 'black hole' and 'quark star' could be just two names for the same thing.

Collider experiments show the break up of an atomic nucleus results in ultra relativistic quark soup, so that's a reasonable explanation for the contents of a star after neutron disintegration. Any mass smaller than its Schwarzschild radius would contain ultra relativistic matter and light, so a hypothetical star of ultra relativistic quark soup should have to be smaller than its Schwarzschild radius or its surface contents would escape.

 

[stevebd1]

This gives a star size = 1.65GM/(c^2), or 0.82 of the Schwarzschild radius.

There's a few issues with this. According to Schwarzschild metric, world lines within 2M (or the Schwarzschild radius) are spacelike which means r is temporal, just as t is temporal outside 2M, we cannot stop the ticking of seconds, just as inside 2M we cannot stop the collapse of r towards the singularity, regardless of pressure. To hold a stable r within the event horizon would be like holding at 3pm outside the EH. Secondly, according to the the Schwarzschild interior metric (which looks at the curvature of spacetime within a spherical mass), the minimum stable radius for a spherical mass is 9/4M (or 2.25M), this is the point where the time dilation reaches zero at the centre of the mass, then as the spherical mass collapses beyond 2.25M, the event horizon begins to move outwards from the centre until the EH and the star surface meet at 2M. See this post and this diagram (the pink line represents the event horizon, the blue lines the collapsing star). If there is a quark star just below the surface of an event horizon, then both the Schwarzschild interior and vacuum solutions would need to be modified.

 

[Bernie G]

There's a few issues with this. According to Schwarzschild metric, world lines within 2M (or the Schwarzschild radius) are spacelike which means r is temporal, just as t is temporal outside 2M, we cannot stop the ticking of seconds, just as inside 2M we cannot stop the collapse of r towards the singularity, regardless of pressure. To hold a stable r within the event horizon would be like holding at 3pm outside the EH. Secondly, according to the the Schwarzschild interior metric (which looks at the curvature of spacetime within a spherical mass), the minimum stable radius for a spherical mass is 9/4M (or 2.25M), this is the point where the time dilation reaches zero at the centre of the mass, then as the spherical mass collapses beyond 2.25M, the event horizon begins to move outwards from the centre until the EH and the star surface meet at 2M. See this post and this diagram (the pink line represents the event horizon, the blue lines the collapsing star). If there is a quark star just below the surface of an event horizon, then both the Schwarzschild interior and vacuum solutions would need to be modified.

The support mechanism in a star is pressure, not energy.

 

[stevebd1]

The support mechanism in a star is pressure, not energy.

The gravitational field in GR is a product of two properties, the stress-energy tensor and the metric tensor. This can be represented by the following equation-

g=\frac{Gm}{r^2}\frac{1}{\sqrt{1-\frac{2Gm}{rc^2}}}

where the first part of the equation (Gm/r^2) is an approximation of the stress energy tensor and the second part is the radial component from the Schwarzschild metric tensor. As you can see, gravity becomes infinite at 2M (the event horizon or the Schwarzschild radius). The stress energy tensor '..describes the density and flux of energy and momentum in spacetime' and the metric tensor '..captures all the geometric and causal structure of spacetime'. You can't look at gravity in GR without considering both. Looking at the Schwarzschild Metric-

c^2 {d \tau}^{2} =-\left(1-\frac{2Gm}{rc^2} \right) c^2 dt^2 + \left(1-\frac{2Gm}{rc^2}\right)^{-1} dr^2

In this form, the time component is negative and temporal for r>2M but when r<2M, the signs flip and r component becomes negative and temporal and there is no stable r, irrespective of pressure, ticking down towards r=0 hence the singularity. This is explained in more and better detail here in the The Schwarzschild Metric and Inside the Black Hole sections-

Spacetime Geometry Inside a Black Hole

Again, if we look at the time component from the Schwarzschild interior metric tensor (note- interior in this case means interior spacetime of an object of mass)-

c\ d\tau=\left( \frac{3}{2}\sqrt{1-\frac{2M}{r_0}}-\frac{1}{2}\sqrt{1-\frac{2Mr^{2}}{r_0^{3}}}\right)c\ dt

where r_0 is the radius of the star, M=Gm/c^2 and r is the radius of the star where you want to calculate the time dilation. Irrespective of pressure, type of mass or density, if r₀ collapses to 2.25M, then \tau will equal zero at r=0 (i.e. the centre), there will be a runaway effect and a black hole will form. When considering the halting of collapse within a black hole, you need to consider both the mass and its effect on spacetime.

For the full Schwarzschild interior metric, see this post

 

[Bernie G]

“As you can see, gravity becomes infinite at 2M (the event horizon or the Schwarzschild radius).”

I think the Tolman-Volkoff equation and the equations above are wrong and result in the mis-analysis of black holes. For example, at 0.9 SR the gravitational acceleration would be about 1.23 that at 1.0 SR.

 

[PAllen]

“As you can see, gravity becomes infinite at 2M (the event horizon or the Schwarzschild radius).”

I think the Tolman-Volkoff equation and the equations above are wrong and result in the mis-analysis of black holes. For example, at 0.9 SR the gravitational acceleration would be about 1.23 that at 1.0 SR.

The Tolman-Volkoff equation is an exact consequence of GR given spherical symmetry and stasis. It is obviously inaccurate to the extent that a real world body is rotating and not strictly static - but only to that extent, unless you reject GR. The other statements you make (about gravitational acceleration inside the SR) are trivially wrong. If you can provide a reference for some way your claims can be considered correct, please do so, or desist from stating personal theories at odds with accepted physics as fact.

 

[PeterDonis]

Are you using the Tolman-Volkoff equation for that conclusion? This equation gives bad results for neutron star, let alone a black hole. If a star was significantly smaller than 1.0 SR, I think at 0.9 SR the gravitational acceleration would be about 1/0.8 that at 1.0 SR, a large number but still very finite.

The TOV equation assumes static equilibrium. But static equilibrium is not possible for an object with a surface radius less than 9/8 SR = 1.125 SR, by Buchdahl's Theorem. Any object with a surface radius smaller than this must be collapsing; it cannot be static. So the TOV equation does not apply. (Also, the "gravitational acceleration" goes to infinity at 1.0 SR; the TOV equation is not the equation that determines the gravitational acceleration.)

 

[PeterDonis]

According to Schwarzschild metric, world lines within 2M (or the Schwarzschild radius) are spacelike

More precisely, curves of constant $r$ in the region $r < 2M$ are spacelike. This makes it impossible, as you say, for any object to "hover" at fixed $r$ in the region $r < 2M$.

which means r is temporal, just as t is temporal outside 2M

More precisely, $r$ becomes a timelike coordinate for $r < 2M$ in Schwarzschild coordinates. There are other coordinate charts describing the same geometry (such as Painleve or Eddington-Finkelstein) for which $r$ is spacelike everywhere. A better way of stating the coordinate-independent feature of this spacetime for $r < 2M$ is that all timelike worldlines must have decreasing $r$.

according to the the Schwarzschild interior metric (which looks at the curvature of spacetime within a spherical mass)

The metric you refer to is only valid for the interior of a spherically symmetric mass of uniform density. There is no known closed-form solution for the (much more realistic) case where the density varies with radius; that case has to be solved numerically. The uniform density metric is still useful as an idealized example to illustrate qualitative features, but it's good to be aware of its limitations.

the minimum stable radius for a spherical mass is 9/4M (or 2.25M)

This is actually true regardless of the behavior of the density as a function of radius; the general theorem that demonstrates this for any spherically symmetric, static mass distribution, as I noted in my previous post, is Buchdahl's Theorem.

 

[Bernie G]

Probably the sentence should have been "I think at 0.9 SR the gravitational acceleration would be about 1.23 that at 1.0 SR."

 

[Bernie G]

The TOV equation assumes static equilibrium. But static equilibrium is not possible for an object with a surface radius less than 9/8 SR = 1.125 SR, by Buchdahl's Theorem. Any object with a surface radius smaller than this must be collapsing; it cannot be static. So the TOV equation does not apply. (Also, the "gravitational acceleration" goes to infinity at 1.0 SR; the TOV equation is not the equation that determines the gravitational acceleration.)

I think "the gravitational acceleration goes to infinity at 1.0 SR" is just plain wrong. Gravitational acceleration = force/mass = F/m. Why should a contracting object exert infinite force at some point? That would be like creating energy for free.

 

[PeterDonis]

I think "the gravitational acceleration goes to infinity at 1.0 SR" is just plain wrong.

Have you actually done the math? This is a common homework problem for physics students studying GR; it is not at all contentious.

Gravitational acceleration = force/mass = F/m.

In GR, gravity is not a "force", and you can't use Newtonian reasoning like this. The proper acceleration required to maintain a constant altitude goes to infinity at the horizon because of the geometry of spacetime, not because of any "force" becoming infinite.

 

[Bernie G]

Have you actually done the math? This is a common homework problem for physics students studying GR; it is not at all contentious.

Please give an internet source with the solution.

 

[PAllen]

Please give an internet source with the solution.

With your rude tone, you don't deserve this, however here is such a link:

http://en.wikipedia.org/wiki/Proper_acceleration#Four-vector_derivations

Note the formula for acceleration needed maintain hovering positions, measured locally by an accelerometer:

This becomes infinite as r decreases to r_s.

 

[PeterDonis]

Please give an internet source with the solution.

You are the one making the extraordinary claim (that GR is wrong), so it's up to you to give references. However, PAllen was nice and gave you the formula. Please bear in mind the PF rules about personal theories and making extraordinary claims without references to back them up. If you continue to repeat erroneous claims, you will receive a warning.

 

[Bernie G]

"With your rude tone, you don't deserve this, however here is such a link"

Please accept my apologies if I am coming across as rude. Perhaps you are interpreting tenacity for rudeness. I'll be posting one last question about black hole spin rates shortly. Do not feel obligated to answer it if I'm trying your patience.

 

[Bernie G]

Don't the rather slow and finite spin rates of the fastest spinning black holes imply a large object within the black hole? Its logical that the equatorial velocity of the fastest spinning black holes would be a large fraction of the speed of light. If an equator has a radius of 10 - 25 km and a spin rate of 1000 revolutions per second, its tangental velocity would be 0.2c - 0.5c, which seem reasonable. But if the equator only had a radius of 1 km and a spin rate of 1000 RPS the equatorial velocity would only be 0.02c, which seems illogical for the fastest spinning black holes.

 

[PAllen]

Don't the rather slow and finite spin rates of the fastest spinning black holes imply a large object within the black hole? Its logical that the equatorial velocity of the fastest spinning black holes would be a large fraction of the speed of light. If an equator has a radius of 10 - 25 km and a spin rate of 1000 revolutions per second, its tangental velocity would be 0.2c - 0.5c, which seem reasonable. But if the equator only had a radius of 1 km and a spin rate of 1000 RPS the equatorial velocity would only be 0.02c, which seems illogical for the fastest spinning black holes.

A black hole with spin can have any spin, and this is unrelated to any surface or object inside the horizon. The BH would end up with roughly the same spin as just before final collapse.The GR solution describing such objects is the Kerr solution:

See:

https://en.wikipedia.org/wiki/Rotating_black_hole
and https://en.wikipedia.org/wiki/Kerr_metric

 

[PeterDonis]

Don't the rather slow and finite spin rates of the fastest spinning black holes imply a large object within the black hole?

No. I don't understand why you think it would.

Its logical that the equatorial velocity of the fastest spinning black holes would be a large fraction of the speed of light.

A rotating black hole doesn't have an "equatorial velocity". It has an angular momentum, and you can compute an angular velocity from this that is sometimes loosely interpreted as the "angular velocity of the horizon". But the hole doesn't have a surface, so the angular velocity doesn't translate into an ordinary speed. An object rotating around the hole with the angular velocity of the horizon, at the horizon, would have to be moving at the speed of light; no actual object can do this (just as no actual object can "hover" at the horizon of a non-rotating black hole).

In short, you are thinking of the hole as an ordinary rotating object. It isn't, and trying to reason as though it is will lead you to incorrect conclusions.

 

[stevebd1]

Don't the rather slow and finite spin rates of the fastest spinning black holes imply a large object within the black hole? Its logical that the equatorial velocity of the fastest spinning black holes would be a large fraction of the speed of light. If an equator has a radius of 10 - 25 km and a spin rate of 1000 revolutions per second, its tangental velocity would be 0.2c - 0.5c, which seem reasonable. But if the equator only had a radius of 1 km and a spin rate of 1000 RPS the equatorial velocity would only be 0.02c, which seems illogical for the fastest spinning black holes.

A couple of things here. firstly black holes are observed to spin slower than they are actually spinning due to gravitational and relativistic redshift. For instance, when an article says '..the black hole is spinning at close to the speed of light..' they should add 'as observed from infinity'. Also what is being measured isn't strictly speaking the BH itself, but the frame dragging rate of the spacetime being dragged around the BH. The tangential velocity of the frame dragging rate for a black hole (as observed from infinity) at various r is-

v=\omega R

where \omega is the frame dragging rate (or angular velocity) and R is the reduced circumference, but the actual local tangential velocity is-

v=(\omega R)/\alpha

where \alpha is the reduction factor or redshift. For the full equations for \omega, R and \alpha, see this old library post- What is frame dragging.

Secondly, if the BH has any kind of spin, no matter how small, it will have an ergosphere (r_{e+}) just outside the horizon (r_+), and if it has an ergosphere, there will be a region where spacetime is rotating faster than c relative to infinity so technically, all BH's locally spin at c at the ergosphere boundary. For the equations for r_{e+} and r_+, see this old library post- https://www.physicsforums.com/threads/radius-of-a-black-hole.762981/

When observed from infinity, this rapid rotating of the ergosphere will not be detected, and the event horizon will appear to spin at anything up to c (with c being maximal, i.e. a/M=1), whereas locally, the boundary of the ergosphere will always be spinning at c which will increase exponentially as you proceed through the ergosphere to the event horizon (it's also worth noting that a BH's spin is normally measured by the location of the marginally stable orbit (MSO) which is at 6M for a static BH (a/m=0) and at M for a BH at maximum spin (a/m=1).

 

[PeterDonis]

black holes are observed to spin slower than they are actually spinning due to gravitational and relativistic redshift.

Actually, you can't observe a black hole's spin directly at all. You can only observe the motion of objects close to its horizon, and note that, the closer to the horizon they get, the more constrained their angular velocity is. In the limit of an object at the horizon (which is not actually physically possible, as I've said before), the angular velocity of the object can only have one value, which is often (somewhat sloppily) called "the angular velocity of the horizon".

As I said before, translating this angular velocity into a "speed" has a number of issues involved. See below.

what is being measured isn't strictly speaking the BH itself, but the frame dragging rate of the spacetime being dragged around the BH.

No, what is being observed, as above, is the motion of objects close to the horizon. The possible worldlines for such objects are affected by the frame dragging rate, so by measuring them, you can indirectly measure frame dragging. But what you measure, even indirectly, is not a velocity; at best, it's an angular velocity. See further comments below.

The tangential velocity of the frame dragging rate for a black hole (as observed from infinity) at various r is-

$$
v=\omega R$$

where $\omega$ is the frame dragging rate (or angular velocity) and $R$ is the reduced circumference, but the actual local tangential velocity is-

$v=(\omega R)/\alpha$

where $\alpha$ is the reduction factor or redshift.

What is the physical meaning of these velocities? How would you measure them? I think that, if you try to answer these questions, you will find a number of issues lurking that you haven't considered.

if the BH has any kind of spin, no matter how small, it will have an ergosphere $(r_{e+})$ just outside the horizon $(r_+)$, and if it has an ergosphere, there will be a region where spacetime is rotating faster than c relative to infinity

This is, at best, extremely sloppily phrased, and at worst, simply wrong. I strongly recommend taking some time to think through the physical meaning of what's going on. Spacetime does not "rotate", and you can't observe spacetime directly; you can only observe the motion of objects. An object inside the ergosphere cannot be static; that is, it cannot remain at constant spatial coordinates. In other words, it cannot have zero angular velocity. But that in no way translates to "rotating faster than c relative to infinity"; I don't know where you're getting that from.

locally, the boundary of the ergosphere will always be spinning at c

I don't know where you're getting this from either. What references are you using?

 

[stevebd1]

'At a point r_e it becomes impossible to counteract the rotational sweeping force. The particle is in a kind of space-time maelstrom. The surface determined by r_e is the static limit: from there in, you cannot avoid rotating. Space-time is rotating here in such a way that you cannot do anything in order to not co-rotate with it.'
Source- Introduction to Black Hole Astrophysics - Page 51

'..the (outer) event horizon of a Kerr black hole is surrounded by a second critical surface, the static limit, which has the shape of an oblate spheroid and touches the event horizon at the poles. The space between these two surfaces is called the ergosphere. Within the ergosphere the spacetime is dragged in the direction of the spinning black hole at a speed greater than c with respect to the outside universe at rest, while at the static limit this speed equals c.'
Source- Extreme Environment Astrophysics - Page 17

'In a rotating black hole, the ergosphere is associated with the stationary limit, the location at which space-time is flowing at the speed of light'
Source- http://www.astro.cornell.edu/academics/courses/astro201/ergosphere.htm (Cornell Centre for Astrophysics and planetary science)

'All objects in the ergosphere become dragged by a rotating spacetime.'
(the phrase rotating spacetime is highlighted and actually links to the frame-dragging page)
Source- Penrose process (Wikipedia)

While I understand that frame dragging cannot be measured directly at the event horizon or at the boundary of the ergosphere, it can be predicted by establishing where the marginally stable orbit is (which is what I state at the end of my post). A number of recent articles have stated that '..the black hole is spinning at half the speed of light.' or '..the black hole is spinning at close to the speed of light making it almost maximal'. I imagine these statements are based on where the MSO is which is normally defined by the inner edge of the accretion disk where there's plenty of matter sending pulses of radiation. For example, if a black hole has an MSO at 4M, then working backwards using the equations (eq 32, http://relativity.livingreviews.org/Articles/lrr-2013-1/articlese2.html ) to establish the MSO, the rate of spin can be established, in this case, the spin would be a/M=~0.56. Based on this, using \omega R, it can be predicted that the frame-dragging rate at the event horizon is 0.305c as observed from infinity, (the black hole is spinning at nearly a third the speed of light)

One source of \omega R/\alpha is-

v_s=(\Omega_s-\omega)\frac{R}{\alpha}

where v_s is the local velocity required for a stable orbit and \Omega_s is the angular velocity required for stable orbit. This equation can be rewritten as-

v_s=(\Omega_s-\omega)\frac{\Sigma^2\sin\theta}{\rho^2\sqrt{\Delta}}

which can be seen in some form in this paper, equation at the top of page three. Remove \Omega_s and you have the local tangential velocity of the frame-dragging rate. Remove the \alpha component and you have the tangential velocity of the frame-dragging rate as observed from infinity. I've also seen the equation in various forms elsewhere.

This is backed up to some extent when you look at the boundary for the ergosphere (or the static limit). If we look at the ergosphere of a 3 solar mass BH with a spin parameter of 0.95 (numbers rounded up). Note that r is the coordinate radius and R is the reduced circumference in the azimuth plane-

r_e=M+ \sqrt{M^2-a^2 \cos^2\theta}

See what is frame dragging to calculate \omega, R and \alpha

For \theta=90, frame-dragging rate at r_e (8861.099m) is 3.6937e-05 rads/s, R=10,674.7743m, \alpha=0.394296 which means v=1c

For \theta=45, frame-dragging rate at r_e (7712.59695m) is 4.8909e-05 rads/s, R=6559.6475m, \alpha=0.320827 which means v=1c

For \theta=5, frame-dragging rate at r_e (5861.7987m) is 8.06154e-05 rads/s, R=629.7636m, \alpha=0.050769 which means v=1c

which collaborates with the previous statements that spacetime is being dragged at 1c at the ergosphere boundary (or the static limit).

I'll admit, there appears to be a coordinate singularity at \theta=0 though that just might be the way I've got things set up.

\omega R also features in this link (in this case, R is written as \varpi), albeit in an algebraic sense. Wheeler also mentions 'tangential velocity Rd\phi/dt as recorded by the Kerr bookkeeper' in Exploring Black Holes (where d\phi/dt=\omega).

 

[PeterDonis]

One source of ωR/α\omega R/\alpha

What are $R$ and $\alpha$?

$v_s$ is the local velocity required for a stable orbit

I assume that by "local velocity" you mean "velocity relative to a static observer at the same spatial coordinates". But in the ergosphere there are no static observers--that is, there are no timelike worldlines with constant spatial coordinates/zero angular velocity; all timelike worldlines have positive angular velocity. So "local velocity" as you are using the term has no meaning in the ergosphere. (Your second expression for $v_s$ makes this clear, since $\sqrt{\Delta} = 0$ at the static limit, and becomes imaginary inside the static limit since $\Delta$ itself is negative.)

As you get closer and closer to the horizon, the range of allowed angular velocities for timelike worldlines gets smaller and smaller; in the limit at the horizon, there are no timelike worldlines at all that do not fall into the hole, and the outgoing null worldlines that stay at the horizon all have one fixed positive angular velocity, which is sometimes referred to as the "angular velocity of the hole" (a more precise term would be "angular velocity of the horizon"). This is the correct way of saying what the various sources you quote say sloppily.

Remove $\Omega_s$ and you have the local tangential velocity of the frame-dragging rate.

What do you mean by "remove $\Omega_s$? And what do you mean by "the local tangential velocity of the frame-dragging rate"? Remember that, as above, there are no static observers inside the static limit, so there is no meaning to the concept of "local velocity" relative to such observers.

If you are not familiar with the concept of Zero Angular Momentum Observers (ZAMOs) in Kerr spacetime, I strongly recommend learning about it. This family of observers gives a much better way of physically realizing things like what you are calling the "frame dragging rate" without having to use the sort of sloppy terminology that the sources you refer to use. This paper has a good discussion of ZAMOs:

http://arxiv.org/abs/1408.6316

Another very good source is Visser's paper on Kerr spacetime:

http://arxiv.org/abs/0706.0622

Remove the $\alpha$ component and you have the tangential velocity of the frame-dragging rate as observed from infinity.

I'm not sure what you mean by "remove the $\alpha$ component", but if you mean removing the "time dilation factor" from the formula to adjust for the "rate of time flow" at infinity, it's important to recognize that the numbers you get that way have no physical meaning; they are just "bookkeeper" numbers, as the Wheeler reference you mention indicates. Calling such a number "the velocity observed at infinity" is, IMO, very misleading; it's the sort of thing that leads to misconceptions like thinking that objects "freeze" at a black hole's horizon and never fall inside.

 

[stevebd1]

\sqrt{\Delta}=0 occurs at the event horizon (r_+), the static limit is defined by g_{tt}=0 where
g_{tt}=1-2Mr/\rho^2
v_s=(\Omega_s-\omega)R/\alpha becomes meaningless at the photon sphere where v_s=1 which ranges from 3M for a static BH to 1M for a/M=1 (eq 30, http://relativity.livingreviews.org/Articles/lrr-2013-1/articlese2.html ) (though for an object of mass, the last stable orbit (MSO) ranges from 6M for a/M=0 to 1M for a/M=1). \ v_s reduces to the static solution for a/M=0.

R is the reduced circumference where R=(\Sigma/\rho)\sin\theta, \ \alpha is the reduction factor (or redshift) where \alpha=(\rho/\Sigma)\sqrt{\Delta}. \ R is equal to coordinate r in schwarzschild metric but in Kerr metric, R becomes slightly greater than coordinate r. \ \alpha reduces to \sqrt{1-2M/r} when a/M=0.

(For the definitions of \Sigma, \rho and \Delta see this link).