Monday, August 04, 2014

What is a singularity?

Not Von Neumann's urinal, but a
model of an essential singularity.
[Source: Wikipedia Commons.]
I recently read a bit around about the technological singularity, but it’s hard. It’s hard because I have to endure sentences like this:
“Singularity is a term derived from physics, where it means the point at the unknowable centre of a black hole where the laws of physics break down.”
Ouch. Or this:
“[W]e cannot see beyond the [technological] singularity, just as we cannot see beyond a black hole's event horizon.”
Aargh. Then I thought certainly they must have looked up the word in a dictionary, how difficult can it be? In the dictionary, I found this:
noun, plural sin-gu-lar-i-ties for 2–4.

1. the state, fact, or quality of being singular.
2. a singular, unusual, or unique quality; peculiarity.
3. Mathematics, singular point.
4. Astronomy (in general relativity) the mathematical representation of a black hole.”
I don’t even know where to start complaining. Yes, I did realize that black holes and event horizons made it into pop culture, but little did I realize that something as seemingly simple as the word “singularity” is surrounded by such misunderstanding.

Von Neumann.

Let me start with some history. Contrary to what you read in many places, it was not Vernor Vinge who first used the word “singularity” to describe a possible breakdown of predictability in technological development, it was von Neumann.

Von Neumann may be known to you as the man behind the Von Neumann entropy. He was a multiple talented genius, one of a now almost extinct breed, who contributed to many disciplines in math and physics, and what are now interdisciplinary fields like game theory or quantum information.

In Chapter 16 (p 157) of Stanislav Ulam’s biography of Von Neumann, published in 1958, one reads:
“One conversation centered on the ever accelerating progress of technology and changes in the mode of human life, which gives the appearance of approaching some essential singularity in the history of the race beyond which human affairs, as we know them, could not continue.”
The term “singularity” was then picked up in 1993 by Vinge who coined the expression “technological singularity”. But let us dwell for a moment on the above Von Neumann quote. Ulam speaks of an “essential singularity”. You may be forgiven mistaking the adjective “essential” as a filler, but “essential singularity” is a technical expression, typically found in the field of complex analysis.

A singularity in mathematics is basically a point in which a function is undefined. Now it might be undefined just because you didn’t define it, but it is possible to continue the function through that point. In this case the singularity is said to be removable and, in some sense, just isn’t an interesting singularity, so let us leave this aside.

What one typically means with a singularity is a point where a function behaves badly, so that one or several of its derivatives diverge, that is they go to infinity. The ubiquitous example in school math is the poles of inverse powers of x, which diverge with x to zero.

However, such poles are not malign, you can remove them easily enough by multiplying the function with the respective positive power. Of course this gives you a different function, but this function still carries much of the information of the original function, notably all the coefficients in a series expansion. This procedure of removing poles (or creating poles) is very important in complex analysis where it is necessary to obtain the “residuals” of a function.

Some singularities however cannot be removed by multiplication with any positive power. These are those cases in which the function contains an infinite number of negative powers, the most commonly used example is exp(-1/x) at x=0. Such a singularity is said to be “essential”. Please appreciate the remarkable fact that the function itself does not diverge for x to zero, but neatly goes to zero! So do all its derivatives!!

So what did von Neumann mean with referring to an essential singularity?

From the context it seems he referred to the breakdown of predictability at this point. If all derivatives of a function are zero, you cannot make a series expansion (neither Taylor nor Laurent) around that point. If you hit that point, you don’t know what happens next, basically. This is a characteristic feature of essential singularities. (The radius of convergence cannot be pushed through the singular point.)

However, predictability of the laws of nature that we have (so far) never breaks down in this very sense. It breaks down because the measurement in quantum theory is non-deterministic, but that has for all we know nothing to do with essential singularites. (Yes, I’ve tried to make this connection. I’ve always been fond of essential singularities. Alas, not even the Templeton Foundation wanted anything to do with my great idea. So much about the reality of research.)

Geodesic incompleteness.
Artist's impression.
The other breakdown of predictability that we know of are singularities in general relativity. These are not technically essential singularities if you ask for the behavior of certain observables – they are typically poles or conical singularities. But they bear a resemblance to essential singularities by a concept known as “geodesic incompleteness”. It basically means that there are curves in space-time which end at finite proper time and cannot be continued. It’s like hitting the wall at 32km.

The reason for the continuation being impossible is that a singularity is a singularity is a singularity, no matter how you got there. You lose all information about your past when you hit it. (This is why, incidentally, the Maldacena-Horowitz proposal to resolve the black hole information loss by putting initial conditions on the singularity makes a lot of sense to me. Imho a totally under-appreciated idea.)

A common confusion about black holes concerns the nature of the event horizon. You can construct certain quantities of the black hole spacetime that diverge at the event horizon. In the mathematical sense they are singular, and that did confuse many people after the black hole space-time was first derived, in the middle of the last century. But it was quickly understood that these quantities do not correspond to physical observables. The physically relevant singularity is where geodesics end, at the center of the black hole. It corresponds to an infinitely large curvature. (This is an observer independent statement.) Nothing special happens upon horizon crossing, except that one can never get out again.

The singularity inside black holes is widely believed not to exist though, exactly because it implies a breakdown of predictability and causes the so paradoxical loss of information. The singularity is expected to be removed by quantum gravitational effects. The defining property of the black hole is the horizon, not the singularity. A black hole with the singularity removed is still a black hole. A singularity with the horizon removed is a naked singularity, no longer a black hole.

What has all of this to do with the technological singularity?

Nothing, really.

To begin with, there are like 17 different definitions for the technological singularity (no kidding). None of them has anything to do with an actual singularity, neither in the mathematical nor in the physical sense, and we have absolutely no reason to believe that the laws of physics or predictability in general breaks down within the next decades or so. In principle.

In practice, on some emergent level of an effective theory, I can see predictability becoming impossible. How do you want to predict what an artificial intelligence will do without having something more powerful than that artificial intelligence already? Not that anybody has been able to predict what averagely intelligent humans will do. Indeed one could say that predictability becomes more difficult with absence of intelligence, not the other way round, but I digress.

Having said all that, let us go back to these scary quotes from the beginning:
“Singularity is a term derived from physics, where it means the point at the unknowable centre of a black hole where the laws of physics break down.”
The term singularity comes from mathematics. It does not mean “at the center of the black hole”, but it can be “like the center of a black hole”. Provided you are talking about the classical black hole solution, which is however believed to not be realized in nature.
“[W]e cannot see beyond the [technological] singularity, just as we cannot see beyond a black hole's event horizon.”
There is no singularity at the black hole horizon, and predictability does not break down at the black hole horizon. You cannot see beyond a black hole horizon as long as you stay outside the black hole. If you jump in, you will see - and then die. But I don’t know what this has to do with technological development, or maybe I just didn’t read the facebook fineprint closely enough.

And finally there’s this amazing piece of nonsense:
“Singularity: Astronomy. (in general relativity) the mathematical representation of a black hole.”
To begin with General Relativity is not a field of astronomy. But worse, the “mathematical representation of a black hole” is certainly not a singularity. The mathematical representation of a (classical) black hole is the black hole spacetime and it contains a singularity.

And just in case you wondered, singularities have absolutely nothing to do with singing, except that you find both on my blog.


  1. How the singularity looks like at the water surface. How similar singularities look like in our universe (event horizon and particle horizon). More detailed look into black hole singularity. For me therefore even the event horizon is a singularity from perspective of 3D space, it's still continuous environment from 4D perspective. The singularity is therefore an emergent concept low-dimensional observational perspective of higher-dimensional environment. For hypothetical very dense observer even the event horizon of black hole or surface of atom nuclei would look like rather homogeneous continuous environment. Analogously for stupid people of low-dimensional thinking many problems are singular tasks, whereas these smarter ones can still see some causal continuum (i.e. solution) inside of them.

  2. I think you're being unnecessarily uncharitable to an interesting idea. The technological singularity is singular in couple senses.

    First, imagine we have some kind of metric for technological progress (let me call this "tech"). It's evident that the rate with which we make tech increases as our tech increases, because improved tools allow us to make the next generation of tools faster, for example. The theory of the tech singularity is that the differential equation defined in this was diverges (has a mathematical singularity) in finite time. Perhaps this is because we create an artificial intelligence ("AI") that can create improved AIs faster and faster such that the inter-generation time goes to zero. So that's sense #1.

    Second, we know that divergent tech is not really possible; what would that even mean? So the divergence of our tech function indicates the breakdown of our effective theory of technological progress. This is analogous to the breakdown of physical effective theory (GR+SM) at the singularity of a black hole. (I'm not saying it's a great analogy, but that's what is intended.)

    The concept is meant to highlight the interesting implications of self-improving technology.

  3. Observe a medium's refractive index as it is frequency swept through an electronic transition. Expected infinite refractive index does not obtain a singularity at the transition. Given two of these puppies closely frequency spaced, it can obtain, arXiv:0012060, 0805.2993, or

    The universe remains uncollapsed, somewhere between relief and disappointment. Technological singularity is phlegmatized by smartless social policy grinding its face with a boot, forever. Elon Musk spends $billions to create the future. Bill Gates redirects tens of $billions to prevent the future. The Beltway steals $trillions to invert it.

  4. In all likelihood actual physical spinning black holes in a turbulent environment (normal space) will have no singularity.

    "Thus we reach the conclusion that at timeline or null geodesic or orbit cannot reach the singularity under any circumstances except in the case where it is confined to the equator, cos() = 0.....Thus as symmetry is progressively reduced, starting from the Schwarchild solution, the extent of the class of geodesics reaching the singularity is steadily reduced likewise, ... which suggests that after further reduction in symmetry, incomplete geodesics may cease to exist altogether"

    Kerr Fields, Brandon Carter 1968.

    Not cosmic censorship, but almost the opposite - singularities can't exist in an GR universe (one with bumps) because there are no paths to them.

  5. I would say a singularly good article, but that would be just for the sake of the pun; you have so many good articles that only a bad one would be singular.


  6. I know I'm fighting a losing battle here, but the singularity inside a Schwarzschild black hole *isn't* at the center. Take a look at the Kruskal diagram again. Actually, that makes the analogy with the "technological singularity" much better: a disaster waiting for us in the future. But I don't want to dwell on that too much, because the "technogical singularity" people are such bullshitters.

  7. That was a very good post.

    Just a remark:

    As I see it I wouldn't describe the singularity as a point in space time, the singularity is not somewhere in space time. It is more like the absence of space-time, or a hole of the manifold if you prefer.

    Also you say:

    “This is why, incidentally, the Maldacena-Horowitz proposal to resolve the black hole information loss by putting initial conditions on the singularity makes a lot of sense to me. Imho a totally under-appreciated idea.”

    Maybe you mean final conditions?

  8. Sabine, the term singularity certainly needs to be better defined. I myself use the term "spacious singularity" which I now imagine makes little sense to readers. I need to follow your links to know what your take on it is as I do not find it apparent as in "No one these days imagines there is a singularity inside a black hole."
    A simple analogy is the Earth with poles. To ask what came before the North Pole is a meaningless question. The archaic word Loxidrome where a map projection path never reaches it but spirals into it expresses one part of the problem. Do we not say the same thing for an ideal absolute zero? That a vague law as is what not that long ago the idea of Projective geometry was once thought the generalization of everything.
    The Agyptians thought that one could go beyond the North pole into some idea of higher space, a spirital journey to pass thru it at the top of the world. The built the archetecture and aimed it to the North star, although those stars shifted in cycle. In the astrology it seemed as clear time of birth, position, and which star repeated like history between spirals set the word patterns from the fixed stars of the heavens.
    The Greeks also knew the Earth as round and measured its curvature. Their careful technology concluded the stars were fixed, a false conclusion similar to Gauss who instruments of the time could not match the velocity of light with his concepts.
    An ideal point in the landscape, regardless if there is spin and that corresponds to some classical radius speaks for an apparent or illusion of a difference in a limiting velocity or fixed light frame reference. Yes, the singularity if an isolated object may be outside the black hole or something of illusions if our universe stands outside it may seem reasonable.
    It is especially not clear that we can compactify things or make them dense and that the only description of how we might move from A to B in space by the illusions or changes to the velocity of light. It could be a combination as in the speculation of code reading DNA with several genome functions.

  9. Zephir, one does not need to say high or low dimensions for the illusion of event horizons and singularies. It can be a shifting of adjacent dimensions in these matters of continuity and discreteness, for Newton defined continuity as continuous, contiguous, and consecutive. So things can be superimposed or totally separate or "touch" in a sense. Not all of the connections may be realized and all present of paths. I had a sailor friend who saw Star Trek as a sci fi analogy to a submarine under water where there is a wider freedom of motion as if gravity is not the main sense. Of course in earlier times crossing the equator was a big ritual deal for we discover the opposite change of seasons and do not fall off the flat earth.
    Xerxes, to put GR+SM in the same bundle is not the only way to view possible overall structures of a vacuum which may have physical laws at some moving scale zero point. We can arrange a programming of matching dominoes but there are no equations that work unless we see a difference in the numbers. So condensing or expanding beyond a certain perceptiual scale may not have a total end or omega point that tells us a difference in AI or something vaguely conscious as something else. Such programming may have diminishing return and the fire walls or boundaries if reached may begin again if transparent and neutral. The Omega concept has been used for spiritual eras predicted also, or for some end of knowing or for a state of the world in some ideological history. It is possible to orgainize the sub parts of a hypercube varying Euler's formula as to what adds up to get the numbers involved where we distinguish rest and motion abstractly as GR or SM.
    This issue as the adjacent dimensions of surface and volume a holographic idea where the paths on the surface are fractal and all is seen more or less as linear (non-linear is a vague term as is uncertainty, especially if what emerges or is given of sentience matches the combinations and computations of intellect with what is applied singularity theory.)

  10. Giotis,

    Well, to me a final condition is the same as an initial condition. I would prefer calling them boundary conditions, but people might mistake it for being about space only. Yes, you're right of course that one better not speak about the black hole singularity as a point, but for the present purpose and without wanting to draw diagrams I think it's good enough. Best,


  11. Xerxes:

    If your theory of technological progress diverges, it just means you've left its regime of applicability. You're not seriously trying to tell me you believe there is indeed anything diverging at finite time? Besides this, I'm not "uncharitable" to the idea, but to the word. Best,


  12. Rastus,

    Yeah, you lost that battle. The singularity is at R=0, which is as good as you can make a definition of "center" in spherical coordinates. Best,


    1. Maybe you should make it clear that you're joking....

  13. Hi Sabine,
    does the word singularity originates from von neumann ? I thought the concept comes from calculus of complex variable, as you mention in the post. In that case, may be Cauchy or Reimann used it ? It is interesting to note that the concept of singularity is central to mathematics of Navier-Stokes equations and also for blast waves. Two subjects, von neumann made fundamental contributions to.

  14. Ghonada,

    No, the word itself does not come from Von Neumann. I wrote he was the first to use it to describe the (alleged) breakdown of predictability in technological development.

  15. I dunno about technological singularity, but the current evolution in physics already exhibits many signs of conceptual singularity in similar way, like the physics before one hundred of years. The situation, when the originally coherent description of nature becomes fragmented into many mutually inconsistent phenomenological descriptions is just what the energy does at the proximity of singularity. The real singularity isn't just some abstract point, where only formal models diverge, but it actually diverges the physical energy spreading (Hamiltonian, Lagrangian) like the light at the turbulent surface of black hole. The regular circular ripples at the water surface are doing the same, when they get fragmented into turbulent (extradimensions of) underwater at small or large scales.

  16. Yes, nice blog post. Complex analysis is fascinating. Also, for an essential singularity, the function takes on all values in the complex plane as it approaches the singularity.

  17. Zephir etc al,
    perhaps a little humility is in order for scientists still learning, especially where physics comes to the frontier of issues we call non-linearity.
    For me questions of arithmetic and visualization of geometries seem all important but even this may be too simple for new generalizations. Praise then to whom may stumble on new deep yet simple foundational principles by speculation.

    Zephir, there are ideas and formulas that on water surfaces tending flat and ignoring parameters that fade exponentially into depths. If there is something to your aether vision it escapes me or is not conveyed clearly. Are you making an analogy to something like string tension in the sense "gravity waves " in there well measured circularity relate to "capillary " waves? Einstein remarked that physics in 4D is in a sense simpler than in 3D, but evidently he considered going beyond this case.
    QM issues aside fluids and magnetism as in MHD have interesting zero points that fix the fields to the flow something pre-Maxwellian.

    An old visualization analogy to a hypercube asks in effect does a half of a 4D hypersphere give us half a worm if we bite into it. For it would eat the volume of the 3D projected down apple, come fi the surface then eat as much volume again, then the surface itself.
    Is there a central singularity, in which volume?

    Complex analysis can be but part of a bigger picture. Bounderies can be quadratic at least. These 2D x 2D as in recent visualizing H bonds by microwaves.

    We can simulate naked singularities as at the poles of a spinning star.

    A 2D snapshot of equations of unity by powers of absolute values without a moving picture misses point exchanges between curves and isolated regions.

    Sabine is right that initial and end points are essential to the big picture.

    Best, humbly yours, this singular lifetime at least.

  18. "Singularity is a term derived from physics, where it means the point at the unknowable centre of a black hole where the laws of physics break down."
    A very strong statement to my taste.
    When you put water into your freezer you also go through a singularity in a certain sense (phase transition).

  19. For me at least and for space-like (cosmological) singularities in the context of GR the term “Singularity” is used more as a way to parameterize our ignorance than to describe any Physics in a meaningful way.

  20. or better to define our ignorance:-)

  21. Also the idea of chaos where the laws of algebraic operations break down. Where is the chaos in QM theory? Can we observe or explain plasma pinching when the helicity is neutral and successively circular.
    MarkusM perhaps the ambiguous statement is not strong enough. Stars and planets have shell structures as if the crystalline structures have successive total phase changes.
    A BH in a sense a hyperbolic crystal where the center is not a point as Newton for forces spacious or at a point for outside finite measuring, but a limit boundary of some hyperbolic grid like an incompressible liquid.
    So what of the air above the water? Riemann 101 Removes thee singularity and the negative real axis that paths cannot cross over. Controversial.
    Can a QM cat riding a wormhole path inside? A black hole notice no change or half alive vanishes into the singularity?

  22. Dear Alice,

    The Wannabee of Physics is probability.

    All the information you or anyone will ever need to create a universe (or, if you will, to describe all phenomenon) will have a label everyone agrees to: 1.

    Sounds singular to me.

    Your truly,
    simple-minded Bob.

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  24. Sabine, if you believe it is not approriate, feel free to remove my comment.

    There is a strange relationship purely geometric relative to black holes. The mass of a black hole can be determined in function of its Schwarzschild radius (r_s) as follows:

    [1] M = 1/2 DP r_s lp ^ 2
    dp = density of Planck
    lp=Planck length

    It’s trivial to prove that the equation [1] is equivalent to the classic equation M = r_s c ^ 2 / 2G.

    The advantage to rewrite the equation as in [1] is that it can easily be seen as the product of lp^2 r_s has the size of a volume and therefore we can reduce the equation to a sphere placing 2M / dp = 4/3 π r ^ 3

    solving r (r_n) we get:

    r_n=sqrt3(3M / dp 2π) = sqrt3 (3M ħ G^2 / c^5 2π)

    From the analysis of the equation we see that the size of the radius calculated is really tiny. the value of the ratio (constant) ħ G^2 / c^5 2π is in fact equal 9.258^-98 m^3 kg^-1.

    Being the mass below the cube root, the function grows very slowly with increasing of M.

    To realize how tiny it is, if we calculate r_n of a black hole with a mass equal to the mass of our universe that we get r_n=10^-14 meters.

    The equations r_s = 2GM / c^2 and r_n = sqrt3 (3M ħ G^2 / c^5 2π) are thus both a mathematically correct representation of a black hole, but only for first one we know how to give a precise physical meaning (Schwarzschild radius).

    However, since the size of r_n become so tiny, we can only assume that it represents the minimum size mathematically possible of a black hole singularity.

    In other words, we could say that the equation seems to suggest that the maximum density that matter can take in nature is equal to half the density of Planck, since by [1] we obtain that the relationship between black hole mass and the volume lp^2 X r_s is:

    M / r_s lp^2 = 1/2 dp.

    As there is no theoretical limit on the maximum size of a black hole, then becomes interesting to study the functions for very small values ​​of M and see how r_s and r_n vary depending on the size.

    The most immediate and easy to be calculated is for what values ​​of M is equal r_s to r_n.

    Let us then

    2GM / c^2 = sqrt3(3M ħ G^2 / c^5 2π) and solving for M we get: M = + / - sqrt (3 ħc / 16πG).

    The mass must be equal to 5,32x10^-9 kg (approximately 1/4 of the Planck mass).

    To get r_s and r_n then multiply sqrt (3 ħc / 16πG) x (2G /c^2) = sqrt (3 ħ g / 4πc^3) = 7,90x10^-36 meters.

    The solution tells us two interesting things:

    The value of r_s and r_n are less than the Planck length;

    For mass values ​​< sqrt (3 ħc / 16πG) and that is about 5,32x10^-9 kg r_n becomes larger than r_s! In other words, r_s that is the event horizon of a black hole, an ideal surface purely mathematical, it becomes smaller component of the physics of matter collapsed into a singularity. The center of mass of the sphere ideal of a black hole passes outside the event horizon, the 'black hole' becomes a 'white hole', the matter explodes outside. Internal and external exchange of role!

  25. @nemo: The similar situation happens with photons at the radius = 2 x schwarzschild radius (the relativistic aberration is double of this classical one), i.e. at the surface of photon sphere. It's probable, for heavier particles additional event horizons for another bosons exist inside of black hole beneath the schwarzschild horizon.

  26. Zephir, nemo has an interesting idea and I would suggest h can be understood as a middle scale rather than a minimum. The flatland is the sphereland up close, yes? maybe these also are interchanged. So at 8 dimensions the sphere fills the hypercube exactly thence the volume goes the other way in comparison. What are you suggesting but something applying to electrons too, that is the description of an atom with shells of the 120 elements. Now do we need vastly higher dimensions or resonances of string vibrations or just find the laws that average or limit these as physical properties (up to subshells in the the nucleus) that are adjacent low dimensions of variable scale? Perhaps this is what Michio Kaku means that dark matter will confirm string theory... but I see that as most likely unphysical speculation even if we find such evidence by experiment. There is a next level but the type of matter would not be the SUSY variety. It may of course exist, and quantized inside black hole like objects. Nature sees or projects dimensions as we do, sort of smashed down shadows rather than voids extending to some infinitely distant direction. Why can we not imagine analogs to galaxies as molecules on the next adjacent scale makes matter as well as energy materialize in the vacuum?

  27. If only journalism students knew a little complex analysis and GR they wouldn't make these mistakes;)

    Of course the root meaning of singular is unique, and complex analysis did considerable violence to this term by using it for poles and essential singularities. Von Neumann was similarly making an analogy between the so-called technological singularity and the essential singularities of complex analysis.

    Any analogy only makes sense if you think about its domain of applicability, and Von Neumann's point was, as you say, breakdown of predictability. Not physical predictability, of course, but social predictability.

    At any point during the past 50,000 years people could speculate about the future and confidently predict that it would be a lot like the present: people would eat, drink, talk to each other, associate in groups, pair up and have babies, get diseases and die.

    The technological singularity means that we can't be confident of the future of any of these behaviors 50 years from now. Humans may be replaced by cyborgs or robots - more likely the latter. In a real sense, social predictability is lost.

  28. If we assign to ħ, G and c value = 1
    the eq. r_n=sqrt3(3M ħ G^2 / c^5 2π) becomes r_n=sqrt3(3M/2π) while r_s=2M.

    So we may rewrite as x=sqrt3(3y/2π) and x=2y


    these the plot:^%281%2F3%29&a=^_Real

  29. The problem is not so much the physics as it is the tools which are currently used to study singularities. General relativity is based on mathematics from 1867, and is not suitable to study quantum gravity. The more proper treatment, these days, is to dispense with the classical manifold concept and study dg categories.

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  31. Interpretation of the physical meaning of r_n

    r_n per se is only a limit geometric, able to enclose the whole mass of the black hole in a volume that always has a constant density equal to half of the density of Planck. However, if we consider a stellar-mass black hole (of any mass), it turns out that this can not happen, and that is that it's not ... 'geometrically' possible that all the masses to be concentrated in a single volume. The reason is simple and implicit in the very nature of matter. The matter is made of particles, and then you can put a lot of stuff in a small volume in order to reach the limit of half the density of Planck, but as soon as you try to turn a particle, or a set of particles in a black hole , it forms a horizon of events that is smaller than the volume in which are enclosed and explode outside. it is like to reverse a sphere and bring the outside into the inside. This is because for blacks holes less than about 1/4 of the Planck mass (5,32x10^-9 kg) the event horizon is always smaller than r_n!
    It follows that to create a black hole, it must necessarily have n particles very dense, but never a singularity, where n> = | 2M √(G / ħ c) |


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