If no singularity, what’s inside a big black hole?

[Bernie G]

On the other hand, most internet sources say radiation pressure equals (1/3)pc^2 and that gravitational potential energy for a neutron star (or radiation star) should equal (GM^2)/R. Using the viral theorem then also gives R = 0.75 SR for a radiation star. I'll try to get some authoritative opinion on this within a week.

 

[stevebd1]

On the other hand, most internet sources say radiation pressure equals (1/3)pc^2 and that gravitational potential energy for a neutron star (or radiation star) should equal (GM^2)/R. Using the viral theorem then also gives R = 0.75 SR for a radiation star. I'll try to get some authoritative opinion on this within a week.

The other thing to consider is that inside 2M, r is temporal (as t is temporal outside 2M) so you would also have to consider the spacetime metric which would have to switch from space-like to time-like again in order to maintain a stable radius, is there a solution/form synonymous with Schwarzschild metric that suits this and incorporates a radiation star? The switch back to time-like space does occur with a charged and/or rotating black hole, though the charged solution is considered not very realistic as the universe has a tendency to neutralise any object with a charge. In its own way, the Schwarzschild solution is also deemed unrealistic due to the fact that it is an absolutely static solution whereas it's almost certain that no matter how small, all celestial objects have some degree of spin. The event horizons for black hole with spin are-

$$r_\pm=M\pm\sqrt{M-a}$$

where $r_\pm$ represents the outer and inner horizon, spacetime becoming space-like in the radial at r₊ and reversing to time-like at r_-. The boundary of the radiation star (ring even) might occur within or at the inner horizon though the inner horizon (or Cauchy horizon) is sometimes described as the boundary of predictability, itself being a contender for a weak singularity.

 

[Bernie G]

To continue, to calculate the radius of a non-spinning, non-magnetic star in a black hole using the viral theorem, here’s the best formulas I’ve found so far for radiation pressure and gravitational potential energy of a gas star of density profile 1/r^2. If anybody can suggest better formulas, please do!

Most sources state the pressure exerted by radiation is one third of its average energy density. That sounds sensible. Even relativists would probably agree that as material collapses into a black hole, all or much of it becomes relativistic no matter what form it takes (radiation, neutrons, or exotic matter), and the maximum pressure it would exert should be one third of its average energy density if all the matter was converted into relativistic particles or radiation. Therefore (1/3)Mc^2 should be the maximum support energy of any form of star, and this should determine the minimum radius. (The radius would be larger if not all the mass was converted to relativistic form.)

Most sources say the gravitational potential energy of a gas star of density profile 1/r^2 is about (GM^2)/R , but I’m not satisfied with that formula and have estimated that the gravitational potential energy is 28% higher than that of a constant density profile star [ (0.6GM^2)/R ], or about (0.82GM^2)/R. If anybody wants to know how this estimate was done, or has a better estimate, please speak up.

Using the viral theorem, if (1/3)Mc^2 = (0.41GM^2)/R , then R = 1.23GM/c^2. This means, if you use this train of thought, that the minimum radius of a star inside a black hole should be at least 61.5% of the Schwarzschild radius. This very possibly is not large enough for a huge ejection to occur if 2 equal size small black holes merge. But again, the star should be larger than 61.5% of the SR if not all the mass is in relativistic form, which is very possible and probably likely. Hopfully the merger of 2 objects identified as nearly equal mass black holes will be observed in the next few decades. That’s about the best I can do at this time. If anybody has any suggestions or comments, fire away.

BTW, here’s an interesting tidbit, for what its worth. So far, of the 2000 observed neutron stars the largest have a mass of 1.97 solar mass, and this is probably near the upper limit. Also, of the 20 observed small black holes in the Milky Way, so far the smallest equals about 5 solar mass.

 

[pari777]

We recently had a lecture on black holes in the college. We were told about the new development going on in the field of theoretical physics on black holes. The prof. was telling that most of the singularities have been removed 2 a great extent but introducing another different set of co-ordinate system.

 

[tom.stoer]

The prof. was telling that most of the singularities have been removed 2 a great extent but introducing another different set of co-ordinate system.

Exactly, most of the singularities; this thread is about the singularity that cannot be removed by a clever joice of coordinates.

If you look at the Schwarzschild metric

http://en.wikipedia.org/wiki/Schwarzschild_metric

you find that it's singular at r=0 and r=2M. The latter singularity is due to the choice of the coordinates and can be removed, e.g. via Eddington-Finkelstein- and Kruskal-Szekeres- coordinates:

http://en.wikipedia.org/wiki/Schwarzschild_metric#Singularities_and_black_holes

The singularity at r=0 is not due to coordinates but is 'real'. This can be seen by looking at coordinate-independent scalars, e.g. the Kretschmann invariant

http://en.wikipedia.org/wiki/Curvature_invariant_(general_relativity )

which is obtained from a special contraction of the Riemann curvature tensor. The Kretschmann invariant scales as K(r) ~ 1/r⁶. Now you could use a different coordinate system; the function for K expressed in the new coordinates would look different, but at the space time point which corresponds to r=0 the Kretschmann invariant will again be singular.

 

[allintuition]

I've enjoyed this thread, but I have been missing some mention of the experimental record. No naked singularities have ever been detected so the black hole as a singularity is in the same company as monopoles and decaying protrons. If no singularities are produced at the LHC it might be time to give them up altogether.

There is recent data supporting a subsequent stage to a neutron star where matter flows without viscosity (http://www.nasa.gov/mission_pages/chandra/news/casa2011.html). This view receives interesting support from attempts to create a quark-gluon plasma (www.bnl.gov/rhic) which finds that at enormous temperatures protons appear to "melt" into a non-viscous state. The simple, classical way to explain what is happening is that when matter is sufficiently compressed a force arises that is powerful enough to resist gravity. We know that such short range powerful forces exist because the weak force behaves in this way.

Does anyone following this thread know of any attempts to explain why such quasi-superfluid states exist at enormous pressures and temperatures? Based on the present evidence, it seems possible that black holes may be superfluids.

 

[tom.stoer]

So you have in mind to identify a non-gravitational force that is able to resolve the singularity no matter how large the mass M of the object might be?

The first problem is that afaik no such force is known.

The second problem is that in order to understand GR (and QG) the singularity has to be resolved by gravity itself. You cannot expect that a theory X saves a theory Y that fails at a singuarity. It's up to Y (or an extension of Y) to cure itself.

 

[allintuition]

So you have in mind to identify a non-gravitational force that is able to resolve the singularity no matter how large the mass M of the object might be?

The first problem is that afaik no such force is known.

The second problem is that in order to understand GR (and QG) the singularity has to be resolved by gravity itself. You cannot expect that a theory X saves a theory Y that fails at a singuarity. It's up to Y (or an extension of Y) to cure itself.

You have answered my question as to whether or not you know of someone who is trying to explain the non-viscous states of matter seen recently at BNL and by NASA trying to explain the behavior of a spinning high mass neutron star. The answer appears to be no. I hope I can provide something of an answer to your questions.

Thinking classically, although I know that this may be inappropriate, I see that with helium superfluids, both He-4 (bosonic) and He-3 (fermionic) repel themselves after London forces are switched off due to extreme cooling. This possible repulsive force is of the same magnitude as that of gravity. If we imagine that it is an inverse square law force, compressing matter, whether in a neutron star or at the RHIC, would be able to reproduce the non-viscous behavior that we see with helium superfluids. The search for a fifth force that opposes gravity with roughly the same magnitude is ongoing with no clear result so far (http://en.wikipedia.org/wiki/Fifth_force). I don't know of any attempt to incorporate such a fifth force into our understanding of GR at this time. So the answer to your first question is that there are some who would very much like to see a fifth force, but results thus far are inconclusive due to the weakness that such a fifth force is expected to have at normal pressures and densities.

The second question has to do with theories, but I would like to consider only GR. The discovery of dark energy, though no such potential force had been identified during Einstein's life, did not upset GR because its behavior was consistent with the cosmological constant term. GR is well-defined only up to the Schwarzschild radius. A force which prevented a singularity (not the singularity at 2M which depends on the coordinate system which you have pointed out, but the singularity at 0) would not necessarily do any harm to GR just as dark energy has done no harm to GR. The question would be whether or not such a force would have an effect at lower matter densities where we do depend on GR, outside the Schwarzschild radius. Extremely careful measurements at the University of Washington (http://www.npl.washington.edu/eotwash/) so far indicate no additional forces at normal temperatures and densities.

I am suggesting the existence of a fifth force (sixth if we count dark energy). The possibility of a fifth force is not new. If I am suggesting anything new, it is that this fifth force will only be seen at extreme density or at very low temperature. Thanks for your response, by the way. I enjoyed thinking about the questions that you posed.

 

[Bernie G]

I think allintuition is on the right track by bringing up a quark-gluon plasma or other force in the core. One thing we can safely conclude about the core is that it is not neutrons. I no longer believe a star in a black hole would be a radiation ball. With a distributed mass star in a black hole (instead of a singularity), if all the matter was relativistic, pressure would be (pc^2)/3, and this pressure is so great it would force the mass far out beyond the Schwarzschild radius. A quark-gluon plasma in the core makes sense, and quarks have a higher collapse pressure than neutrons. But as for the upper layers and surface of the star, I think that could be neutrons since the pressure there is similar to pressures found in a neutron star.

 

[tom.stoer]

Even the QGP would not resist the collaps b/c it's not a specific interaction but simply the Fermi degeneration pressure that acts as a repulsive force. This is not sufficient to keep a massive neutron star stable and it would not change that much for a QGP- or a quark-star

Have a look at http://en.wikipedia.org/wiki/Degenerate_matter#Quark_degeneracy

 

[Bernie G]

But (pc^2)/3 is more than sufficient to prevent collapse.

 

[Bernie G]

"So you have in mind to identify a non-gravitational force that is able to resolve the singularity no matter how large the mass M of the object might be?"

Yes. For one model I think something roughly similar to a conventional neutron star could exist within the Schwarzschild radius. The upper layers and surface of the star could be neutrons since the pressure there is similar to pressures found in neutron stars. But pressures and densities in the core would be so great that the core material would go "relativistic" and generate a pressure of (pc^2)/3. It doesn't matter what the core is made of (quarks, etc), so long as it generates (pc^2)/3. The star might even have a radiation "atmosphere" - all located well within the Schwarzschild radius. But my earlier estimate of star size would be a little off because the upper layers would be supported mostly by neutron degeneracy pressure, which would be smaller than (pc^2)/3.

 

[tom.stoer]

what would be the equation of state?

afaik for a neutron star one uses p = ρ/3 which is ultra-relativistic and which does not prevent a collaps.

 

[Bernie G]

All that's needed is to prevent core collapse, since that's where collapse happens. I don't think P = (pc^2)/3 is used for a neutron star core prior to collapse. But I think it would apply after core collapse.

 

[Bernie G]

Also, at energies dramatically higher than that for quark production, where even quarks break up, it might be possible that P approaches or equals pc^2, but I don't think that much pressure is needed to support a star core in a black hole.

 

[Bernie G]

It doesn't change things much, but the gravitational potential energy of the proposed star is likely closer to (1.0)(GM^2)/R instead of my estimate of (0.82GM^2)/R.

 

[Bernie G]

"So you have in mind to identify a non-gravitational force that is able to resolve the singularity no matter how large the mass M of the object might be?"

Let me clarify. My estimates show (pc^2)/3 is more than sufficient to prevent core collapse up to 10 or 20 solar masses. Above that greater than (pc^2)/3 is needed. If you want I can show my estimates here for estimated core pressure due to gravity, and estimated core pressure generated by (pc^2)/3.

 

[Bernie G]

I might have made a mistake with the last statement. I never did core pressure estimates for larger black holes. If the star contained in a black hole is more or less a constant fraction of the Schwarzschild radius, then the star radius will be proportional to M. Since gravitational core pressure goes as (M^2)/(R^4), and if M is proportional to R, then larger black hole (neutron?) stars might have a smaller core pressure than that found in smaller black holes.

 

[Bernie G]

But I'd rather stick to discussing a star support mechanism inside smaller black holes. A star in a larger black hole could have a significantly different structure due to the weaker gravity in the outer layers of a large black hole.

 

[Bernie G]

Here are the pressure estimates of a hypothetical 3 solar mass star of 6 km radius with a relativistic core that generates pressure P = (pc^2)/3, where p is the density. Units used are kilogram/meter/seconds. These estimates show the core will not collapse even though the star radius is smaller than the Schwarzschild radius, ie.- the core pressure generated by (pc^2)/3 will exceed the core pressure due to gravitational compression. Please feel free to check these numbers and question the estimates:

(1) For the peak core pressure due to gravitational compression, which we can call CP, and assuming the star has a typical 1/r^2 density profile, CP should be given approximately as:

CP = (2GM^2)/(πR^4) = 1.3 X 10^36

(2) To calculate peak core density (and hence the generated pressure), assuming that half the star mass is contained within R/2, and assuming the peak density at the center is twice this average core density, the peak core density would equal 6 X 10^19. This is an astoundingly high number, more than 10 times the peak core density estimated in a neutron star, but that makes sense. With this peak core density, the pressure generated by (pc^2)/3 would equal 1.8 X 10^36, which is greater than CP (the pressure generated by gravitational compression).

 

[Bernie G]

Note that pi (3.14) shows up as typeface "π" in the above expression (2GM^2)/(πR^4).

 

[czes]

The Black Hole is an object with a maximum entropy for a given mass of the object.
Each particle is entangled with a number of non-local information on a longest distance = a circle of the object / Compton wavelength = [2 pi R / (h/mc)] in an object with a radius R and the whole mass M. How many bits of the information may contain M/m particles with an average mass "m" in a Black Hole where M=c^2 R /2G

(M/m) [2 pi R / (h/mc)] = pi R^2 / (hG/c^3 ) = A /4 lp^2

Where A is a surface of the Event Horizon of the Black Hole and lp^2 is Planck length squared.

It is interesting, that the information capacity of the Black Hole increases not with its volume but with its surface. Therefore the average density of the Black Hole Like Object decreases if the mass of the object increases.

http://en.wikipedia.org/wiki/Black_hole_thermodynamics

 

[Phrak]

It would be very satisfying, if at some time in the not too distant future, discussions of black holes did not contain meaningless temporal verbage. The word "is" is the current tense of the verb "to be". "Will be" is the future tense the verb "to be", ect.

The use of such key words in general relativity, and even special relativity, should be specified with clarity or a declarative statement about black holes is so perfectly ambiguous it could mean a number of different things to diffenent readers.

If you wish to discuss back holes, you-all should be aware that the verbs "is", "was", "will be", "never was", "never will be", etcetera are not physically meaningful without greater specification nor universally understood within your own personal gestalt. spacetime is not flat.

 

[Bernie G]

"In the beginning there was nothing, which exploded." — Terry Pratchet

Why there should be something is a very tough question.

 

[mesinik]

"Why there should be something is a very tough question.

With saturday night logic (where time and space are curved a lot) it might be postulated: when space becomes gravitationally curved enough, then you will observe time like one of "normal" dimensions, while some "ordinary" dimension behaves somewhat similar to "normal" time.
So, could we turn the question back like that: why should something be there in the future and/or past?
(? Après nous la ...)

 

[Bernie G]

To sum up my position, there’s a lot of confusion out there about black holes, mostly caused by so many people repeating the illogical argument that black holes are a point singularity. They use the incorrect argument that anything within the event horizon (Schwarzschild radius) must have energy greater than mc^2 if it is not to proceed to the center. This is getting facts backwards. The maximum gravitational energy of a star is (0.6GM^2)/R. If ALL a star’s energy goes into creating pressure, that energy would equal Mc^2 maximum. Setting Mc^2 = (0.6GM^2)/R results in the minumum radius R for anything, or any star, including the star in a black hole, of R(min) = (0.6GM)/(c^2). This is 30% of the Schwarzschild radius. Any smaller radius would mean the gravitational energy would have to exceed Mc^2.

Actual stars in nature have density profiles of about 1/(r^2), resulting in a gravitational energy of almost exactly (1.0GM^2)/R, or simply (GM^2)/R. And if all (or almost all) the mass inside a black hole were to go “relativistic” (I hate using that term), the total energy creating pressure would be (Mc^2)/3. The viral theorem, which is used to calculate the size of gravitational objects, says the energy creating pressure equals half the gravitational energy, or (Mc^2)/3 = (GM^2)/2R. This gives the radius of a star inside a black hole of R = (1.5GM)/(c^2), which is 75% of the Schwarzschild radius. It doesn’t matter what this star in a black hole is made of, quark matter, radiation, whatever; that's the size. Other basic math shows that the core density and core pressure of a star in a small black hole (of a few solar masses) is about 8 times the core density and core pressure of a neutron star of a few solar masses. Nothing profound or unrealistic about this. Also, if the star inside a black hole has a “atmosphere” of radiation, it would be small and would not affect the above calculations. This hypothetical radiation wouldn’t come anywhere near the Schwarzschild radius and would be contained in much the same way the Earth contains its atmosphere.

An interesting result of the above is if two EQUAL mass orbitting black holes merge, there can be a huge ejection from them or even annihilation of the 2 black holes. Hmmm. Its only a matter of time before black hole mergers will be observed. Let's hope some observed mergers are of equal sized ones.

Finally, I don’t know why so many people use the Tolmann Volkov equation for a black hole. Not only does it give the wrong answer (neutron star collapse at 0.7 solar mass), but its conclusion of infinate pressure at the Schwarzschild radius is kind of obvious nonsense. But I do agree with Tolmann Volkov that the contents of a black hole can be analyzed as a gas, but one where the "gas" pressure P = (rho/3)c^2. Sorry for the length of this. If anyone has any questions on the above email Berniepie at aol.com.

 

[Bernie G]

Minor correction to the above: The core density of a resulting black hole is about 20 times the core density of a neutron star, and the core pressure of the black hole is about 50 times the core pressure of a neutron star (of a few solar masses). Shouldn't do calculations in my head. Doesn't change anything; if you can accept the densities and pressures in a neutron star, these densities/pressures are also imaginable. The biggest weird thing is that if a neutron star collapses to about 3/8 its size (in terms of radius), the resulting star is 25% smaller than the Schwarzschild radius and nothing can escape unless 2 equal size black holes merge.

 

[Bernie G]

Final tweaking: I think a thin neutron crust is unlikely since that would probably require a temperature gradient. Also, if the core is very large (a more likely density profile) the gravitational energy energy could be as low as (4GM^2)/(5R), resulting in a star radius of only (1.2GM)/(c^2) instead of R = (1.5GM)/(c^2). Bottom line is a non-rotating star of relativistic material would have a radius between 60 - 75% of the Schwarzschild radius, which with the bulging effect should be enough for a massive ejection to occur if two approximately equal size orbiting black holes merge.

This also presents a different possible “origin” of our universe other than the big bang. Consider if two massive orbiting black holes merged, with each approximately half the mass of the universe. They would eject relativistic material for millions of years.

 

[mesinik]

One possible bet is, there might be just an indetermined state (like future) because "normal" time is meshed up with other dimensions. It is fun to think about this at least.
If so, it's probably even more fun to build a ring around an extra large black hole (large enough, so the gravitation on the edge is affordable). Now, let's sit on this ring and push a mirror inside with a stick, beyond the event horizon ... OK, OK, let's just look at the Zeldovich-Starobinsky-Bekenstein-Hawking radiation. Could we see and hear the future and/or past? Anyway, this scenario looks and feels somewhat like an oracle from any mythology...

Both string theory and loop quantum gravity claim possible elimination of the black hole singularities. If that is true, what do they predict the inside of a stellar size black hole contains? Is it some new ultra dense state of matter, or something else?

I will try to ask various authorities this question at the APS meeting in St. Louis next week. But what’s your opinion? Has anything been published?

The only concrete proposal I am aware of is the Mathur fuzzball (hep-th/0502050).

Jim Graber

 

[Bernie G]

I think your example of pushing a mirror inside the event horizon of a large black hole illustrates the sorry state of affairs of contemporary black hole analysis. The Tolman–Oppenheimer–Volkoff equation is normally quoted, and this equation results in infinite pressure inside the event horizon. So as an example, if we consider a 10 million solar mass black hole, the gravitational acceleration at the event horizon would be about one millionth that at the surface of a neutron star. The surface of a neutron star obviously doesn't have infinite pressure.

 

[Markus Hanke]

So as an example, if we consider a 10 million solar mass black hole, the gravitational acceleration at the event horizon would be about one millionth that at the surface of a neutron star.

Wrong.
How can the gravitational acceleration at an event horizon be smaller than at the surface of a neutron star ?

 

[Markus Hanke]

Now, let's sit on this ring and push a mirror inside with a stick, beyond the event horizon ...

Good luck to you, because I am not going to be the one sitting on your ring...it would be a decidedly uncomfortable position to be in, I can assure you, what with your brains being sucked out through your toes, all ten of which by the way would have been stretched to the length of a freight train...you get the picture.
Anyway, let's for argument's sake pretend it was possible to do such a thing - you wouldn't be able to see anything reflected off the part of the mirror which is at and inside the event horizon. Also, you would not be able to pull the mirror back out. Basically, this whole thing is a waste of time.

 

[suprised]

There is a recent review by Mathur that is very clearly witten and a pleasure to read:
http://arXiv.org/pdf/1201.2079
From his Fuzzball viepoint, these questions eg about a singularity are irrelevant.

 

[mesinik]

Thank you for positive feedback. I, too, would say: the points of view of avatar Bernie G might sometimes be a bit unanticipated, but they are fun to read and certainly on the positive side of this pleasant forum.

I think your example of pushing a mirror inside the event horizon of a large black hole illustrates the sorry state of affairs of contemporary black hole analysis. The Tolman–Oppenheimer–Volkoff equation is normally quoted, and this equation results in infinite pressure inside the event horizon. So as an example, if we consider a 10 million solar mass black hole, the gravitational acceleration at the event horizon would be about one millionth that at the surface of a neutron star. The surface of a neutron star obviously doesn't have infinite pressure.

 

[mesinik]

Dear person behind avatar Markus Hanke
Thank you for your attention.
I am pleased to see, my text was interesting for you.
But regrettably (probably my grammar was a bit too heavyish), there is some unnecessary misunderstanding here. I will try to use less grammar next time; but you, too, could you please next time consider reading a sentence from the beginning to the end (and if you don't get the point, then reading again and doing some thinktank work) ... before you try to make fun of it, OK?
Hint: compound sentences include often many parts and you should read all of these parts. You should not cut out 1 little piece and advertise this as the meaning of a compound sentence.

Good luck to you, because I am not going to be the one sitting on your ring...it would be a decidedly uncomfortable position to be in, I can assure you, what with your brains being sucked out through your toes, all ten of which by the way would have been stretched to the length of a freight train...you get the picture.