Wednesday, July 07, 2010

In Praise of Black Holes

First time I came to work on black holes was a funny story. I went to one of the senior profs at the institute asking for a topic for my master's thesis. The focus area of the institute was heavy ion and nuclear physics, so the prof suggested two topics from this field. Since I wasn't too excited about that, he added a third that was to show that black holes can't exist by working out an argument that he briefly sketched. Faced with that selection, I went away with the task of showing black holes can't exist.

I was an undergraduate at that time, I had a bachelor's degree in math, not physics, and knew next to nothing about black holes and quantum effects in their vicinity. Still, it didn't take me more than 3 weeks to figure out that his idea wouldn't work for what I thought were quite obvious reasons. I went to some other prof asking for advice, upon which I learned that the topic had previously been given to two other students who, unfortunate for the prof, came to the same conclusion as I. I was then luckily handed over to a very bright postdoc who worked in astrophysics with whom I deviated the topic to one in which we could arrive at results and not start with the result.

Knowing what we know today, the topic suggested might seem odd to you. However, one has to see this in historical context. This anecdote happened in the mid 90s and by then it had been discussed for decades whether black holes are merely mathematically possible solutions of Einstein's theory of General Relativity, or if they are physical reality. It was only in the mid to late 90s that observational evidence became accurate enough to convince most scientists of the existence of black holes. At this time, evidence meant that a black hole is the most straightforward and most plausible, minimalistic, and generally accepted explanation of the data. As I recall one seminar speaker saying: If it looks like a duck and quacks like a duck, then it probably is a black hole. Though I might have garbled that up. The senior prof's argument against their existence was that god would not disconnect himself from part of his universe. Can't make that stuff up.

In any case, for some while there have been more or less artificial alternative explanations around for super-compact objects that resemble black holes but have no horizon. These alternative solutions typically are very implausible on the theoretical side. In addition, since a few years we also have experimental evidence that allows us to distinguish a solid surface from a horizon, see earlier post on Evidence for the Black Hole Horizon. So that duck clearly is a black hole. If you need any more evidence for the existence of black holes: they've made it into pop music. Muse, Supermassive Black Hole. Love that song, sorry, urgent need to embed:


Oohm, back to the physics, let me be very clear here: a black hole is characterized by its horizon, not by its quack. The full solution of Einstein's field equations does not only have a horizon, it also has a singularity at the center of the black hole. First thing to note is that we can't see the singularity because it's surrounded by the horizon and thus hidden from all observational efforts. (That's known as Cosmic Censorship.) Maybe more important, close by the singularity we come into a regime of very strong curvature and it is doubtful that General Relativity is still the correct theory to apply. Close by the singularity, quantum effects of gravity should become important. We do not yet know what the right theory is, but the correct solution in this regime looks probably nothing like the classical solution.

If a singularity appears in your theory, it usually means you've used your theory out of its range of applicability. A nice example for that are singularities that appear in hydrodynamics. Stefan wrote about this in his post Singularities in your Kitchen. Needless to say, these singularities are an artifact of still using a theory on scales where instead a more precise theory of the microscopic structure of the fluid would have become relevant. Most people working on quantum gravity believe that the singularity in the black hole solution is a similar artifact of applying a theory out of its range of applicability.

What is so fascinating about black holes is how many different areas of physics they combine. There is, first of all, of course the astrophysical aspects, both in experiment and theory. These differ greatly between supermassive black holes (like at the center of galaxies) and stellar black holes (formed from collapse of stars). An interesting topic is also the transition from the latter to the former. There's further so-called 'primordial black holes,' that might have been formed from density fluctuations in the early universe. These are usually thought to be of much smaller mass than stellar black holes. These primordial black holes come back every now and then as dark matter candidates or as explanation for gamma ray sources etc, but so far none of the models, both for their formation and signature, has been really convincing. And then there's the quantum black holes that might be possible to produce in colliders in scenarios where the Planck scale is lowered; for more details see my post on Micro Black Holes.

Since black holes inevitably imply the existence of regimes where quantum gravitational effects are relevant, it is not surprising that they have received a lot of attention in the quantum gravity community. The study of black holes has thus lead us to discover and explore the relation between General Relativity and thermodynamics, a topic that's been going strong since almost 30 years now and most recently resulted in Erik Verlinde's claim that General Relativity actually is thermodynamics. Black hole physics has given rise to 't Hooft's conjecture of the Holographic Principle, probably one of the most surprising and also intriguing thoughts that physics has heard of since Einstein, and a thought that has spurred a rather dramatic amount of research. A lot of this research has been dedicated to find a solution to the Black Hole Information Loss Problem. The need to connect black hole physics (both emission and accretion) to the standard model further brings in particle physics and there are also interesting analogies between black holes and some condensed matter systems. You can find a nice review of the latter in Ralf Schützhold's recent paper Emergent Horizons in the Laboratory.

The workshop on black holes that I was attending 2 weeks ago in Bonn covered a good part of these topics, though the astrophysics was clearly dominant and other aspects like holography were entirely missing. In any case, it reminded me of my initial fascination with the topic. If you are a student, I can warmly recommend to work on a topic related to black hole physics because it gives you the opportunity to learn about a broad range of interesting concepts.


  1. Brilliant! That's as nice a summation of the subject as I've ever seen, and I look forward to exploring your links in detail.

    I'm a student, as in a lifelong one. I've studied, at least at snorkeling level, most of current mathematical physics, saving the best, Black hole science, for last.

    Musser sums it up nicely in his book: At the event horizon, Quantum Mechanics breaks down, and at the singularity, General Relativity breaks down.

    Questions for Professor Bee:

    1) Would you recommend Susskind's book on the subject, the one where he wins a bet with Hawking?

    2) Didn't Smolin in one of his books say that in the real world, singularities probably don't exist, although they do make good mathematical approximations? I think it was TTwP but it may have been 3RtQG. In any event, Infinity is the darling of Mathematicians. Engineers know that in reality, there's no such thing.

    3) Where is the closet black hole we know of? Cygnus X-1? 6000 light years away, right? That's close enough for me. Also, out of 200 billion stars in our galaxy, what is the best current estimate of black holes per cubic light year, or parsec or something?

  2. Hi Steven,

    Thanks for the kind words. Not sure why quantum mechanics should breaks down at the event horizon, not sure what is meant by that.

    1) Haven't read Susskind's book and since I wasn't too enchanted by his previous book, I'm not going to buy the new one.

    2) You can ask questions... It is very well possible Lee wrote something like this, as I said it's the most plausible and thus most wide spread interpretation of the singularities in GR. But I can't really recall, it's been years back I read Lee's books.

    3) V4641 Sgr?




  3. First thing to note is that we can't see the singularity because it's surrounded by the horizon and thus hidden from all observational efforts.

    Fans of the new Battlestar Galactica series may have noticed that the base station of the cylons was positioned near a "naked singularity" that does not have a horizon (season finale), see e.g. Reissner-Nordström metric.
    (Sorry for the nitpicking :-)

    Most people working on quantum gravity believe that the singularity in the black hole solution is a similar artifact of applying a theory out of its range of applicability.

    Then there are black hole solutions like the (slowly rotating, i.e. a < M) Kerr black hole, where some regions of the spacetime are "very" acausal in the sense that any two points can be joined by a timelike curve. I think it is safe to say that most physicists would consider this as unphyisical, too, besides the singularity.

  4. Hi Tim,

    Yes, there's been some recurring talk about the possibility of naked singularities. I believe SciAm had an article about that either last year or earlier this year. As you can guess, I've been sticking here to the most conservative statements, a question of how many details can one possibly cram in a blog post. The general sense seems to be that if naked singularities (or one should maybe say naked quantum gravitational regions) exist, they would be tremendously interesting but probably also very rare.

    And well, eternal solutions are unphysical anyway. But I don't consider timelike curves a priori as unphysical. It's just that they often lead to problems with stability. Best,



  5. The general sense seems to be that if naked singularities (or one should maybe say naked quantum gravitational regions) exist, they would be tremendously interesting but probably also very rare.

    Interesting, I had no idea that those were not ruled out completly.

    ...eternal solutions are unphysical anyway.

    Is "eternal" the same as "asymptotically flat"? (I mean, if we had a realistic GR spacetime model of the Universe, it would necessarily describe the whole time evolution from and until the breakdown of GR, and be "eternal" in this sense, right?).

    But I don't consider timelike curves a priori as unphysical.

    That's supposed to be closed timelike curves, correct?
    Hm, in the Boyer-Lindquist block III (hope that's the correct buzzword) of slow Kerr spacetime every point is both in the future and the past of every other point. You think such a situation could occur in our universe?

  6. Hi Tim,

    Well, if naked singularities had been ruled out completely, then the cosmic censorship hypothesis was probably a cosmic censorship theorem. The SciAm article that I had in mind was this one. Maybe better, check the papers of the authors for example this one. Can't say that topic really flies in the community, but i think it's interesting.

    With "eternal" I mean they have not been formed at any finite time. Doesn't have to be asymptotically flat, neither is it necessarily static.

    I neither consider closed nor unclosed timelike curves a priori unphysical, except for the problems with stability that they tend to come with.


  7. Please stop talking about NAKED Singularities! You're making this Puritan-American blush!

    :-) Jus' kidding.

    I remember seeing that SciAm article. I remember it depressing me too, that scientists think such a thing is possible. Will have to re-explore. (But even if they don't exist, they're still fascinating, as useful in setting bounds).

    Hi Tim van Beek:

    1) Who will win the Germany-Netherlands World Cup Finals IYO? I'm of partial Dutch and German heritage, so I won't know who to root for! Wait, Espana still has a say later today. The World Cup: almost as exciting as Physics. ;-)

    2) THANK you, Tim, for introducing us to the amazing Hans Reissner (Nordstrom I knew about). As an eEgineer, I did find the following passage of the Wikipedia author's biography of Reissner a bit disturbing:

    Curiously, it was this engineer, rather than a physicist or mathematician, who first solved Einstein's equation for the metric of a charged point mass. His Reissner-Nordstrom metric demonstrated that an electron has a naked singularity rather that an event horizon.

    Why so curious?

    Plenty of MathPhys theoreticians began as Engineers. Dirac for example, and Einstein would have been a damned good one.

  8. Yes, singularities are purely mathematical concepts not applicable to physical reality so I agree they signal the breakdown of the theory. Cosmic censorship doesn't help as certain initial conditions lead to a naked singularities after collapse.

    All this makes me quite skeptical of black holes, I think it's quite possible an unified theory will do away not only with singularities but also with event horizons as we understand them today.

  9. Hi Tspin,

    It is exceedingly unlikely that a unified theory will do away with event horizons as they can be formed at arbitrarily small curvature and densities, ie in regimes we know GR applies to excellent precision, and besides that we have experimental evidence for their existence. Best,


  10. @Bee: thanks for the links, will take a look.

    Who will win the Germany-Netherlands World Cup Finals IYO?

    Ugh, despite the great attraction of the German team so far this is still a little bit off topic. What's really important is that the German fans will be very pleased because the team made it this far and did it's best in every game (in the next two they will, too, I am sure). That means the party will go on here right to the end of the world cup, no matter who wins.

  11. Watching Stephen Hawking's talk at PI, I learned of the distinction between those black-holes that you can see and those you can't.

    The idea is that if something is perfectly spherical and collapses there is no way to see this, while considering this spherical collapse in a geometrical sense again "not spherical collapse symmetrically inward" initiated. These we can see cosmologically?

    Is my summation correct?


  12. "Compactified dimensions" are a nice way of packaging empirically defective mathematics. A black hole pinches down one or more of the Fab Four extant. One then wonders about the reverse process presumably revealing the richness of unrealized physics.

  13. Ugh, despite the great attraction of the German team so far this is still a little bit off topic

    I take it you're a European? If so, pls be advised that THIS World Cup attracts the interest of Americans, still, even though our team is eliminated. Since we're mainly of German-English descent with quite a bit of Dutch and Spanish as well, the remaining teams (all European, heh) definitely hold our interest. South Africa '10 will likely go down in "Soccer" History as the World Cup when Americans finally ===> "Got it." Is that bad? Well I guess it's bad if one "gets off" on Americans not getting. Not to worry, we have other faults. Take ur pick.

    Football ===> "Association" Football (UK English term, to differentiate "Football" from Rugby Football) ===> asSOCiaton football ===> SOCcer (UK English slang), for those who still don't get slang's source (Yup, it's the Brits fault). It's a fine and exhausting athletic sport. Our little girls have been playing it for years. We got better! :-)

    You're right with being off topic, but as long as some ON-topic stuff is included, is there a problem? For example:

    Plato, Hawking is really OUT there. He has the right to be, given his and Penrose's (and Russians'/Soviets')contributions to this field in the 60's and 70's. That doesn't necessarily mean he is RIGHT! However, Hawking Radiation is amazing! What a great one-two punch he and Bekenstein gave to this VERY important field of inquiry!

    Personally, I find myself in disagreement with 80% of everything "new" by Hawking, i.e., his latest speculations when first I hear them, only, upon further study, in agreement with 60%, with the remaining 40% being more in the "prove it!" category. Amazing man, no question, and quite the coup for Perimeter, for sure.

    But at the end of the day, I miss Karl Schwarzschild. What an amazing man; an amazing loss. Gone too soon.

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  16. Bee, paper on the event horizon of Sgr. A*:

  17. Bee,

    I'd also recommend reading Susskind's book, The Black Hole Wars one, because it's umm, illuminating. Shame on you Bee for not wanting to read it, but by all means borrow it from the library.

    My vague understanding of Black Hole Complimentarity is that somehow quantum mechanics breaks down at the horizon, or is modified, or the quantum xerox principle at least is somehow defly avoided. This seems to be what Susskind and t'Hooft are going on about.

    I'm also going to get very annoyed with people calling any type of horizon an "event horizon". We should stick to Hawking's technical definition in my opinion. I notice that wikipedia has already begun to waver and astronomers have been lost for a long time.

  18. Hi Eujin,

    Oh, that was what the remark was aimed at, thanks for the clarification. Not sure I'd call that a breakdown though, seems to be somewhat of a misnomer. In any case, what good would it be to borrow the book from the library if I wouldn't read it? ;-) Should it happen that I run out of reading material, I'll consider it. Ie maybe next century or so. Best,


  19. Hi Bee,

    First I’d just like to echo the others as to say thanks for this interesting piece. I must tell you that ever since I found out about the initial research that those like Hawkings and Penrose did in respect to understanding the physical implications of Schwarzchild’s solution to GR it’s captivated my imagination. I’ve always found it curious that despite knowing of this solution to his equations Einstein never gave it a second thought, as considering Schwartzchild’s solution more as to have demonstrated what stands as being a limit of physical reality and therefore nothing to have taken seriously.

    I for one do hope that despite what are considered slim odds that the energy densities attainable with the LHC be enough to produce a sub atomic black hole, for in that time which would have that of a blink of an eye seem to be a lifetime we woul then be more assured such things actually exist. As for the seeming impossibility for the existence of singularities I think it comes down to having trouble believing something can exist with no space in which to have accommodated. I would say however it’s more a problem of where a singularity can exist in relation to what we consider to be a place. That’s to say if space itself is an emergent construct the problem we have with such notions are ours alone and not that of reality’s.



  20. Hi Steven,

    As for Susskind’s book ‘The Black Hole Wars’, although having it I find it difficult to finish reading. Not that it exceeds my ability to comprehend as being too technical or having trouble with the research it advocates, yet more to do with the tone of the book as being mostly a self promotional work. So in respect to its value despite it having lots of good science sprinkled in, yet for my taste it has a tad too much of the author’s self admiration you are forced to endure in having it communicated. Then again perhaps it’s strictly a problem I have as I’ve always thought the science in the end to hold more significance than the scientist.



  21. Haha, that's exactly why I didn't like Susskind's previous book, so definitely not a problem only you have :-) Best,


  22. Hi Phil,

    My memory might fail me, but I thought that Einstein in his late years became quite interested in black holes. I believe he thought they might represent elementary particles (he was looking for a unifying theory), an idea that keeps coming back every now and then and never really lead anywhere.



    PS: Schwarzschild, not Schwarzchild.

  23. Hi Bee,

    That was a bold admission particularly since you find having to meet him in person every so often, so I hope his bark be worse than his bit:-) The title of the book however I would say is being true with what I perceive as his nature, as hjm taking the proliferation and acceptance of his ideas from the perspective of it being a personal battle.



  24. Well, I happened to sit in a few of Susskind's lectures at PI (on, guess what, black holes) and have to say they were great. I mean, really, I'm not making this up, he's an excellent teacher. (All recorded, check PIRSA). I just don't like his writing style. Call it a matter of taste.

  25. Hi Bee,

    I will check out those lectures you mentioned and believe you when you say he knows his stuff. Unfortunately the only exposure I’ve had to him have been his public lectures and his book in which he maintains the posture I expressed here. I would say the counter to him would be t’Hooft in such regard that walks more softly yet whose thoughts penetrate as to be impossible to have ignored.



  26. There's a paper Einstein wrote about why the r=2M "singularity" cannot occur in the Schwarzschild metric. I can't remember if it's written before the war or after but I'm guessing sometime in the 1930's. He shows that there are no stable orbits for particles in multi-body systems to source such a vacuum solution and hence it cannot exist.
    I guess this is the main evidence people have when they say Einstein didn't believe in black holes.

  27. Well, it's correct, there is no singularity at r=2M. I'll see if I find the reference about Einstein and the black holes. I vaguely recall it was actually an arXiv paper where I read that.

  28. The Einstein paper I'm talking about is "On a stationary system with spherical symmetry consisting of many gravitating masses" which is on JSTOR for those who want to take a look.

    From the conclusion: "The essential result of this investigation is a clear understanding as to why the "Schwarzschild singularities" do not exist in physical reality. Although the theory given here treats only clusters whose particles move along circular paths it does not seem to be subject to reasonable doubt that more general cases will have analogous results. The "Schwarzschild singularity" does not appear for the reason that matter cannot be concentrated arbitrarily. And this is due to the fact that otherwise the constituting particles would reach the velocity of light."

    The "Schwarzschild singularity" he is talking about is the r=2M surface. But I think his argument is similar to the naked singularity conjectures; that there's no way of forming such a thing from regular initial data (or particles in his case). He doesn't seem to have considered the possibility that all the matter just falls into the true singularity and disappears.

  29. Leonard Susskind's writing style probably wouldn't bother me, since NYC where he grew up is my local city and having been there about 1000 times, I guess I've become numb to New Yorkers' strong egos and brusqueness. I can however see how polite Canadians or efficient organized properly behaved Germans would find his writing style annoying. I did too when I first encountered them. However, when you get to know them one-on-one they are actually warm and friendly and more often than not quite extraordinary people.

    Lee Smolin, Brian Greene, and Lisa Randall are also from NYC, yet seem quite wonderful and quite lacking in Lenny's attitude.

    Why? Because, Susskind grew up in the South Bronx, one of the toughest places to raise children on Earth.

    But my question referred to whether or not his conclusions are correct. I hate reading opinions presented as fact. I understand every author does that to a certain extent, my fear though (and possibly unfounded) is that Susskind may (I don't know!) be piling it on thick and deep, and that's why I asked.

    If so, I won't read it. I already reject his Anthropic Landscape nonsense with 99.99999% of my being, and find its promotion nothing less than a ginormous sales job (albeit with good maths, I'm sure), and my fear is that whatever he concludes in BHWar may suffer from the same inherently dishonest promotion (with good maths).

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  31. I hate reading opinions presented as fact. I understand every author does that to a certain extent ...

    Roger Penrose least of all, or not at all. The thing I admire most about Sir Roger is that before he speculates, he gives fair warning that he is about to do so.

    How adorable is that? What a terrific guy. Then again I'm a sucker for honest people. If only more professionals had that quality.

  32. Hey Stefan, check this out. The proton is 4 percent smaller than we thought it was. How will that shake up Physics?

  33. OK, but is there anything MORE stable than a proton, Bee?

    Which blows my mind, because the simplest hadron is more complicated than people know. The whole ... virtual particle atmosphere thing. Wow.

    Having said that, I wouldn't be surprised if this turns out to be wrong. Some Scientists rush to judgment a bit too soon, Science Journalists ten to the octillionly so.

  34. Photons, electrons, neutrinos, etc - light elementary particles are for all we know stable. Yes, there's many things we don't know about bound states in QCD. And let's not even get started about nuclei. That's why I get a little cynical when people start talking about a theory of everything. So far, reductionism is a lofty ideal, not practical reality. Best,


  35. Hi Steven,

    Despite all the praise attributed to the standard model it’s more of a patchwork quilt with all its free parameters and forced in by hand values then many would imagine. To have a four percent difference now discovered doesn’t come as much of a shock, that is no more significant then what was discovered with the revelations of quark plasma more being a soup. They managed to fiddle with the model in that case enough to have people satisfied and i don’t suspect this will be any different. Actually this was the initial reason for searching for a so called TOE as to have eliminated or more limited the free parameters which the current one permits and relies on. Problem is however all they have managed to do thus far as Bee eludes to is make the situation worse rather the better. However I still hold out hope that we are nor at the end of the usefulness of reductionism, with the study of Black Holes being an indicator, as how could things be more reduced then it is with a singularity.

    As for the Susskind book like I said there is lots of good physics to be found within it and yet the focus of the book is more to tell the story of how he arrived at his conclusions about the holographic principle where he uses scientists such as Hawkings as his sparing partners and bunching bags to have shown how he succeeded where they had failed. Like I said for my tastes it resembles more of a pissing contest which takes some of the enjoyment out it for me. Come to think of it this may give reason to have more female researchers as they seem less inclined in such regard as perhaps resultant of lacking the equipment required:-)



  36. Bee - there was also Soundgarden's early 90's hit BlackHole Sun

  37. Hi Snowboarder,

    Oh yes, of course! There's quite a lot physics-themed songs actually, if one looks around a bit, though few ever make it mainstream. Best,


    -- James Ph. Kotsybar

    Containing nothing that’s ambivalent,
    more than dark, which would only be dreary,
    death’s non-spiritual equivalent
    crushes our intellect to theory.

    Passage through is most certainly one way,
    and thus it incites our speculation.
    What would occur, if we wandered astray
    into this singular aberration?

    It’s relative to where you’ve placed your clocks.
    From outside we’d seem to fall forever.
    Beyond that, it’s puzzling paradox.
    We only know that we’d leave it never.

    A downward orbit is how it begins,
    and nothing’s jolly when gravity wins.

  39. That's a nice poem, but I have to disagree with the tone of the last two lines. Black holes are where both Quantum Gravity and General Relativity fail, at least theoretically, which to my mind, makes black holes the singularly most interesting topic in all of physics.

    As practical people who use equations but are not enslaved by them as theoreticians are, Engineers know that "Infinity is just a concept." Infinity doesn't exist in the real world, therefore: "singularity" is a concept, not a reality. Which is GREAT for General Relativists, because it is at the dead center of a black hole, in Mordor where the alleged "singularity" lies, that GR allegedly "fails."

    Phew, once again, Engineers save the world from going down the Lady Gaga extremes of speculation like those espoused by Science Journalists mis-quoting Susskind or Andrei Linde. At least, I hope they're being misquoted. :-)

    Case in point about misquoting, or at least Scientific Journalistic confusing metaphors. Bee provided us in her tweets with the following fine recent article from Villanova about Erik Verlinde and his notion of Gravity as an "Entropic Force":

    That was nice and enjoyable reading, until about 1-third of the way thru I read about the Newtonian view of Gravity as "spooky action at a distance."

    Spooky?! No, just plain old action at a distance. SPOOKY action at a distance was Einstein's word for Quantum Entanglement, a completely different subject altogether.

  40. As a followup, Andrew Zimmerman Jones here links to a paper here about Torsion as an alternative to Inflation, including black hole theory and opening Lee Smolin's Fecundity theory that we're living inside a black hole in another universe. Sean Carroll, noted dissident of such thinking (and of so much more), is away doing research atm, but it will interesting to see his reaction.

  41. I don't see why singularities need indicate a breakdown in theory. I'm a grass farmer who lives on a sphere. Whenever I try to smoothly comb my grass down on the sphere, it develops at least one singularity. But those are just the facts of life, given the geometry of my grass farm. Why are curvature singularities any different?

  42. The singularities that indicate a breakdown of the theory are those that imply a divergence of physical quantities. Singularities certainly exist in a mathematical sense and you can write down theories that have singularities of one sort or the other. If these theories are arbitrarily detached from reality (like, say, a farmer on a sphere) what use it is asking for their physical meaning? Show me a farmer on a sphere who has measured a singularity in a physical observable and I will believe that singularities can be physically meaningful. Best,


  43. Hello Bee!
    I stumbled upon your blog by accident and I appreciated the post about black holes.
    Just I was a bit puzzled reading the comments where you said "Can't say that topic really flies in the community" about naked singularities. I'm not sure what you mean by fly but I'd still like to point out a couple of things.

    Naked singularities are a subject within analytical general relativity (so no computer simulations and no experiments). The whole area is not really vibrant with new ideas lately (say in the last 20 years) and I'd say that naked singularities are one of the most interesting topics in the field.

    If by 'flying' you mean that they're not exactly well accepted I'd say that it's not the case anymore. The 'community' now sort of agrees that such solutions are possible... but maybe it can be useful to better define the problem in order to understand...

    Naked singularities are a general feature of exact solutions of Einstein's field equations. The question is wether they can happen in the real universe. In this sense let's just distinguish 2 types of naked singularities:

    1- Static Axyally symmetric space-times: Like Schwarzschild is the exterior field of a sphere there's plenty manifolds that represent the exterior field of something not spherical. If you remove the source (which bytheway we don't know how to describe) and extend the spacetime just as you let r->0 in Schwarzschild you obtain a naked singularity (see the no hair theorem). The question is: When an object that is not spherical collapses what happens? Does it become spherical and thus a black hole or does it remain prolate (or oblate) and thus form a naked singularity. The answer is 'we have no clue'.
    So regarding these singularities (that stay there 'forever') you can think what you like since we have absolutely no idea.

    2- Naked singularities arise as the endstate of spherical gravitational collapse in a wide variety of situations. What happens here? The cloud of matter collapses and depending on the initial configuration 2 scenarios are possible:
    a- the horizon forms before the singularity and keeps it covered => black hole
    b- the horizon forms at a later stage than the singularity => naked singularity. In this case though the singularity is visible just for a limited amount of time, then the horizon forms and everything settles to a black hole. The set of initial data that can lead to this kind of solution is very wide and these solutions are now widely accepted by the 'community'. There's no way in General Relativity to get rid of them and they're physically viable. Therefore if you want to rule them out you need something else (quantum gravity?).
    Still these solutions are harmless. No causality violations or weird things happening. And the singularity (as many have noted) just points out the breakdown of the theory.

    Sorry if I was too long, I got a bit carried away. Hope this will help whoever is interested in understanding a bit better the situation with naked singularities.

    All the best


  44. A very nice post Daniele, thank you.

    I don't believe naked singularities
    exist, because I don't believe singularities exist. They do however, make for EXCELLENT teaching instruction as a "special case", as does DeBroglie-Bohm Pilot Wave Theory in Quantum Hydrodynamics.

    Pure Mathematicians LOVE: Infinity, Daniele. They love it perhaps, a bit too much.

    Applied Mathematicians and Physicists, on the other hand, know what Infinity actually IS: an approximation, and a darned good one.

    Everything dies, therefore everything is finite. Well, some people say "THE Universe" is eternal. Well, if so ... we have an exception! :-)

  45. Hi Daniele,

    What I meant with "it doesn't fly" is that it's not a vibrant area as you put it, it doesn't attract a lot of interest and doesn't cause a lot of excitement. (Anybody wants to tell me that's not actually what the phrase means, go ahead, you all know I'm not a native speaker). Please don't mistake that being my opinion, it's simply a matter of fact. I know there's recently been renewed interest in the topic, I read one or the other paper (see my replies to Tim above), and on our last week's workshop we had a talk about the topic. If you look at small enough sub-communities of sub-communities every single topic in physics is hot, because the person working on it says so. What I meant is simply that the biggest part of the community could care less. Possibly that will change at some point, but I think the essential step on the way is observational evidence for their existence. Best,


  46. Hello,

    thanks for the reply! I'd say that the whole field of exact solutions of Einstein field's equation is not really 'flying' at the moment...
    I'll leave you with just one comment. As you correctly point out it's possible "that (it) will change at some point, but I think the essential step on the way is observational evidence".
    Well, to have observational evidence we need to know what we are looking at. And analytically we can't produce a spectrum of how a 'realistic' naked singularity might look like. We need computer simulations (just as in black hole formation for astrophysicists) and the numerical 'sub-community' is not very interested (I'd say they were more interested 10 years ago).
    But maybe it will change... we'll see.
    Thanks for reading. Hope someone found that a bit interesting...

    All the best


  47. This comment has been removed by the author.

  48. "If you look at small enough sub-communities of sub-communities every single topic in physics is hot, because the person working on it says so."

    That's funny! Nicely put, Bee.

    Also, as one increases the level of one's specialization, fewer and fewer people are working on the problem.

    (I'm reminded of Arthur Eddington's response to the question: "Is it true only 3 people on Earth understand Relativity?" to which he responded, "Give me a moment, I'm trying to think of who the 3rd person is.")

    This current state may be part of the expression: "Physics is a victim of its own success."

    Problem is, there are only so many Physicists, and sooner or later the number of problems worthy of study will exceed the number of available Physicists! Are we there yet?

    Each answer seems to spawn more than one question. So the good news is that at this rate ... we'll all be specialists!

    (But who will check our work if the checkers are off doing their own thing?)

  49. Two followups:

    First, Karl Schwarzchild was probably the 3rd person, but no not him, because he passed away of a disease he contracted fighting WWI, in WWI.

    (And in a gratuitous on-topic point, I think Schwarzchild's mathematical furtherfication (is that even a word?!) of Gen Rev was astounding in a beautifully simple Mathematical way. In Mathematics, "simple" is a good thing. The "simple" part was that the black hole NOT rotate. A great start. A great teaching aid to "Reality" (a.k.a.: Physics), because we can be pretty darn sure everything is rotating.)

    Second, the "are we there yet?" question I posed may or may not be true in Physics, but I think it happened in Mathematics a LONG time ago. There are so few Mathematicians in the world compared to the number of Physicists, yes Bee? What's the ratio, because I haven't a clue. 10:1? 100:1? And I'm talking about ALL Mathematicians, not just the Applied ones, and I bet the Pure ones outnumber the Applieds. True statements?

    Upshot, there are TONS of unsolved problems that Mathematicians know exist, but few if any are working on. Job security, much?

    Eh, maybe not. I think it comes down to that 2nd most evil "f" word on the planet: "Funding."

  50. Hi Steven,

    Depends on who you count as mathematician. In my experience most math departments are smaller than the physics departments. But then you find mathematics in so many other sciences (think economics, statistics in the social sciences & biology, physics, etc) that it's hard to get a good impression merely from that.

    I would agree that physics is a victim of its own success but for the exactly opposite reason. Today, there's too many people for not enough problems. Consequently, increasingly more people create their own over-specialized niches most of which will inevitably turn out to be entirely irrelevant. Of course there's tons of unsolved problems, but most of them don't matter. Among the problems that matters there's not enough for all those physicists. Just look at the arxiv, they're all beating around on the same bush, and the bush has hardly any leaves left. (That's of course to some extent dependent on the field, so excuse the generalization.)



    PS: Schwarzschild, not Schwarzchild.

  51. Hi Daniele,

    Yes, I agree with you. I think it is plausible that the field will attract more attention in the soon future, but hard to tell. Best,


  52. Hi Bee,

    Depends on who you count as mathematician.

    PhD's in Mathematics, either Pure or Applied.

    In my experience most Math departments are smaller than the physics departments.

    Thanks, you confirmed my worst fears. I was actually hoping the two groups were equal in number. I am not surprised.

    But then you find Mathematics in so many other Sciences (think Economics, Statistics in the Social Sciences & Biology, Physics, etc) that it's hard to get a good impression merely from that.

    Good points, Bee. There's even Linear Algebra in Social Anthropology now, did you know that? Humanity marches on.

    Other than Euclid, I think the MAIN reason Math has caught fire was the introduction of the idea of Zero as an integer, to Europeans, from India, and probably though Arabia, about 500 years ago. That co-incided with the discovery that 2 of the 7 continents existed, by said Europeans. Two great things happening at the same time. We've been stoned ever since.

    (John "Penguin" Ellis and Carl Sagan, doubly so.)

    I would agree that physics is a victim of its own success but for the exactly opposite reason. Today, there's too many people for not enough problems.

    That's the impression. We both read Not Even Wrong, and that is definitely what Woit says there. "Dark Matter Phenomenology" is the TOP sought-after specialty amongst the top American research institutions for newly minted PhD's in Physics, yes, where only 1 of very 100 said newlies may expect a tenure-track offer at SAID institutions.

    Fine, but is that situation the same in Europe? The other Americas? Asia? Africa?

    I'm sorry Bee, but there are TONS better problems to work on than DMP, and I really don't give a rat's right rear butt cheek if Haaavid or Staaanford disagree. Wake UP Ivy Leaguers, there are many institutes of advanced study now! Age matters, but sometimes .... not very well.

    Consequently, increasingly more people create their own over-specialized niches most of which will inevitably turn out to be entirely irrelevant.

    Yes, it reminds me of going down too much in detail of a single curclicue of the Mandlebrot Set, when every now and then it wouldn't hurt a brother or sister to back up and see the whole picture.

    Of course there's tons of unsolved problems, but most of them don't matter. Among the problems that matters there's not enough for all those physicists.

    I believe this is why we have institutes of advanced study, Bee. If Academia were actually doing their job, we wouldn't need them. But they're not, and so ... blowback.

    Just look at the arXiv, they're all beating around on the same bush, and the bush has hardly any leaves left. (That's of course to some extent dependent on the field, so excuse the generalization.)

    Cornell has utterly screwed itself, and the sad fact is, the leaders of Cornell don't know it.

    Then again, they DO have the highest suicide rate amongst the Ivys, but Ithaca is so lovely! There must be a "mean" virus going around.


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