Naked singularities

[Norman Albers]

It is a subtle mathematic process by which we lay out considerations of metric solutions of the Einstein field equations. The free-space constraints may be stated: [math]R_{ab}=0[/math], and in the Schwarzschild solution we apply reasonable symmetry considerations to get to a form: [math]ds^2=A(cdt)^2 - Bdr^2 -Cr^2d\Omega^2 [/math]. At this point crucial decisions are made, and they have deeply physical ramifications! <A,B,C> are functions of r alone. By a simple linear rescaling of r, we can produce: [math] ds^2=A(cdt)^2 - Bdr^2 -r^2d\Omega^2[/math]. This is the chosen form of the Schwarschild metric, where: [math]ds^2=(1-2m/r)(cdt)^2 - (1-2m/r)^{-1} dr^2 -r^2 d\Omega^2 [/math]. My textbook goes on to a very curious move. Citing a mathematic motivation to have a line element which "agrees most closely with our intuitive notion of space" they develop isotropic coordinate representation, where the same metric coefficient multiplies all three space differentials. This yields a form: [math] ds^2= \frac{(1-m/2\rho)^2} {(1+m/2\rho)^2}(cdt)^2 -(1+m/2\rho)^4 d\sigma^2 [/math] Now look at the radial transform: [math] r-m= \rho+m^2/4\rho[/math], and see there is some funny stuff going on. We may write equivalently: [math]\sqrt{r^2-2mr}\,+r-m=2\rho[/math]. The argument in the \sqrt goes through zero at r=2m, where [math]\rho/m=1/2[/math]. Then what??? If we wish to continue to smaller r we must allow negative \sqrt arguments. Put in r=m and you get [math]\rho/m = \frac 1 2 i [/math]. Go ahead and solve for r=0 and you get: [math]\rho/m=-1/2[/math]. No problem, eh??? We have just witnessed a rotation in the complex plane. [To be continued...]

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Merged post follows:

Consecutive posts merged

For six months in the start of 2007 I enjoyed exchange with H.Puthoff, who likes the term "dark gray holes". He solves for the isotropic form as described above, but simply does it with that assumption of metric form from the start. This produces singularities with no event horizons, as if that EH was contracted to zero (degenerate). For various reasons he champions this; I suspect it is erroneous. It is mistake to seek coordinates that "make us comfortable". To understand physics from one reference frame to another you must include multipication by the metric coefficients acknowledging the coordinate transforms used to get the expressions. We can see that the radial transform getting us from the first Schwarzschild form, to the isotropic form, gets us into a most curious mathematic position. If we dare to think of [math]r< 2m[/math] we must allow complex values of the radial coordinate [math]\rho[/math]. There is simply an assumption of possible mathematic form of spacetime here. Sure there is nothing "less than [math]\rho=0[/math]" but you won't even get here from there! I mean if you consider only real [math] \rho[/math] you won't even get to zero, or below 1/2. This is the naked singularity and I think it is conceived on shaky assumptions. I have the same attitude about Kruskal coordinates and BH interiors but this is for another night's fishfry. . . . . . . . . .To elaborate hopefully clearly, Puthoff et.al. use these two radial representations in what I think is a confused manner. However this stems from my disposition of having demonstrated a charge singularity as an integrable field of vacuum polarization, also. I can show that a randomly offered "percolating vacuum with little electrons and positrons popping in and out" (and I really do dislike such language, though it serves purpose) will show dynamics of overall densification, but with depletion of radially oppositely oriented pairs. This constitutes, in the small, the tensor differences in the Schwarzschild GR form, between radial and tangential changes. I am saying it is not helpful to mathematically "dial away" these things, if you seek the mathematics of physics.

Edited January 26, 2009 by Norman Albers
Consecutive posts merged.

[Norman Albers]

In Roger Penrose's book The Road to Reality, p.425, I read of "...Lambert's sphere of imaginary radius." When I complete reading this tome I will be on a different level of discussion. I am meditating on the 3-vector in real space which characterizes the degenerate metric form, and also on the complex 3-vector characterizing nullspeeds. This is true inside an event horizon, if you look at tangential propagation, and also in the AM case. It is the Eddington term which complicates involvement of dr and so only in such differentials do we face an imaginary nullspeed.

[npts2020]

Norman, have you seen the article in February's Scientific American about naked singularities by P. Joshi? I was wondering your opinion (or anyones) of the explanation given.

[Norman Albers]

Yes Pankaj Joshi is excited for good reasons. He and others have shown the possibility of an inhomogeneous, or non-uniform, stellar distribution collapsing to a singular state with no horizon. He speaks of models with no pressure but with density gradient thinning outwards; he also speaks of a prolate spheroid collapsing to singular "spindle" ends. Impressively when pressure is included the behavior persists. The limit of my knowledge is expressed in the second paragraph in the section "A Lab for Quantum Gravity". This all is exciting, and the article details how a future satellite, the Square Kilometer Array, will be able to resolve the very near details which would show evidence possibly supporting Joshi, or maybe Puthoff, or maybe the rest of us. Stay tuned. I should say, or/and's.

Edited February 1, 2009 by Norman Albers

[npts2020]

I agree. The most exciting part to me is that much of this may be (dis)proven in the next two to five years.

[scalbers]

Indeed a wonderful article in SciAm with some good evidence that naked singularities may be common with

stellar collapse. I did wonder about how realistic the prolate spheroid (spindle) case was, otherwise it seemed

fairly reasonable in their discussion. A rotating star would be an oblate spheroid, unless some sort of magnetic effect would squeeze it otherwise? Quite the term to contemplate: "quantum star", or should it be Planck star? What would a quantum star look like close up? I have long thought something would intervene to prevent an infinite collapse. I had heard previously of quark stars, though this obviously takes things to a new level. Just get out your telescope to see quantum gravity in action.

 

A clarification that the SKA is a ground based continental scale telescope and it might take nearly 10 years to get the full results from it. See http://en.wikipedia.org/wiki/Square_Kilometre_Array#Description. The Square Kilometer array reminds me of the Allen or "One Hectare" array at http://en.wikipedia.org/wiki/Allen_Telescope_Array.

Edited February 1, 2009 by scalbers

[npts2020]

He may have been thinking of the Extreme Universe Space Observatory to be launched from ISS in 2013.

[Norman Albers]

Maybe the prolate spheroid had enough density, and just sufficient AM, to be collapsing as per this 'spindle points' singularity.

[scalbers]

Perhaps, though I wonder how the star would become a prolate spheroid in the first place?

[Norman Albers]

Maybe some dynamic collision/ejection scenario, in the z-direction?¿?¿

[Klaynos]

Perhaps, though I wonder how the star would become a prolate spheroid in the first place?

 

Alot of spinning maybe?

[moth]

Perhaps, though I wonder how the star would become a prolate spheroid in the first place?

 

maybe a parasitic companion?

[Norman Albers]

Spinning and collapsing as per the model of 'increasing density'.

[moth]

sorry i was thinking of the shape not the collapse.

[Norman Albers]

Don't be sorry, this is a cool discussion. Perhaps the oblate spheroid we usually picture of a rotating mass, can collapse in this fashion?

Edited February 14, 2009 by Norman Albers

[moth]

i'm in way over my head but are you saying it will have quantum spin like an electron?

[Norman Albers]

I'm in way over my head too, babe!!! No I am not in the realm of quantum here, this is great globs of mass-energy, think JERRY LEE LEWIS.

Edited February 14, 2009 by Norman Albers

[moth]

and see there is some funny stuff going on. We may write equivalently: [math]\sqrt{r^2-2mr}\,+r-m=2\rho[/math]. The argument in the \sqrt goes through zero at r=2m, where [math]\rho/m=1/2[/math]. Then what??? If we wish to continue to smaller r we must allow negative \sqrt arguments. Put in r=m and you get [math]\rho/m = \frac 1 2 i [/math]. Go ahead and solve for r=0 and you get: [math]\rho/m=-1/2[/math]. No problem, eh??? We have just witnessed a rotation in the complex plane. [To be continued...]

 

what is rotating in the complex plane? a star? anyway naked singularities won't be black so what do you think would be a good name

Edited February 14, 2009 by moth

[Norman Albers]

What is rotating in the complex plane is the value of [math]\rho[/math]. This is another way of saying that we admitted at the start of this process that there is an imaginary term added to z, so the expression for what was simply r now has a real and an imaginary component. There is still the imaginary part, and only that, when r=m. The larger point here is that you must acknowledge the nature of the coordinate system you are using. We go through these math transforms to get a metric form we can sort of understand, but it must be interpreted with care.

Edited February 14, 2009 by Norman Albers

[moth]

thank you for taking the time to answer my question.if only everyone were like you we could all be as informed as we are curious by simply asking a question.

i think it's unlikely naked singularities exist for the same reason cosmic censorship was created, we would have detected them or been destroyed by them before now.

[Norman Albers]

Why do you think we would be destroyed by them?

[moth]

i thought the sci-am article said naked singularities would "leak" quantum fluctuations like gravity waves or exotic particles.

[Norman Albers]

Yes that's one of the first statements. Go on to read of the possible variations and confusions. I am not so concerned about quantum foam somewhere making this my last 24 hours of existence. Yes, moth, we are in over our heads and that's good. I have just started really investigating the spin AM field in the "low mass" case, and the astronomic realm is not the same. Joshi does describe some of his scenarios as time-dependent, unstable. I am trying to understand the AM singularity (angular momentum) for the general relativity solution outside of "something spinning intensely" but with more attitude than mass. We are perhaps seeing a connecting range here from Joshi. . . . . . . . . . . Here is my brother's observation to me on what we all can observe: "On the topic of radio telescopes and black holes, the present limit of measurement can actually now start to provide some interferometric measures of the diameter of the event horizon for the black hole at the center of the Milky Way, as mentioned in a recent issue of Physics Today magazine. VLBI imaging is now about a factor of three away from showing us pictures of this black hole. The planned ALMA array (the other major new radiotelescope array contemplated besides the SKA), may in around 5 years actually provide images of the black hole silhouette and such."

Edited February 15, 2009 by Norman Albers

[moth]

i'm looking at the online version of sci-am(no 'script right now) but i read "But if singularities can be naked, their unpredictability would infect the rest of the universe. " and my mind wandered as i read the next paragraph or so. maybe destroyed is too much,but they should be conspicuous, so it seems like we should "see" them.

i did a quick look around for what happens to all that angular momentum from nuclear spin during black hole formation but i can't find anything (anything i can understand). do you know any overviews for newby's that include this?

[Norman Albers]

No. This is rock and roll. Dig it.