Saturday, December 12, 2020

Are Singularities Real?

Last week we discussed whether infinity is real, and came to the conclusion it is not. This week I want to talk about a closely related topic, singularities. What are singularities, where do they appear in physics, and are they real?

A singularity is a place beyond which you cannot continue. But singularities in mathematics can be rather dull. In mathematics a singularity may just be a location where an object, for example a function, is not defined. But it may not be defined just because you didn’t define it.

If I define for example a piecewise function that has the value one for x strictly smaller and strictly larger than zero, then that’s not defined exactly at zero. You can’t go from left to right. So, that’s a singularity. It is however a singularity that is easy to remove, just by filling in the missing point. Correspondingly, this is also called a removable singularity.

But many functions have singularities that are more interesting than that. The simplest example that’s still interesting is the function one over x, which has a singularity at zero. This singularity cannot be removed. There is no point you could fit in at zero that would make this function continuous. You won’t get from the left to the right.

For the function one over x that’s because the function diverges when x gets close to zero, so the value of the function becomes infinitely large. However, and this is a really important point, a singularly does not necessarily have to come with anything infinite.

Take for example the function sine of one over x, which I have plotted here. This function has a singularity at x equals zero, but that’s not because the value of the function becomes infinitely large. It’s because there’s no such thing as the value of the sine function at one over zero.

For a mathematician, a function doesn’t even have to look odd to have a singularity. The best example is the function e to the minus one over x square. This looks perfectly fine if you plot it. But this function has a really weird property. If you calculate the value of the function and the derivatives of the function at zero, you will find that they are all exactly zero.

What this means is that if you reach zero from one side, you don’t know how to continue the function. There are infinitely many ways to continue from there, all of which will perfectly fit to the other side. For example, you could continue with a function that’s zero everywhere and glue this onto the other side. Or you could take the function e to the minus pi over x square. This type of singularity is called an “essential singularity”.

Okay, so singularities are arguably a thing in mathematics. But do singularities appear in reality? Not for all we currently know. Physicists use mathematics to describe nature, and, yes, sometimes this mathematics contains singularities. But these singularities are in all known cases a signal that the theory has been applied in a range where it’s no longer valid.

Take for example water dripping off a tap. The surface of the droplet has a singularity where the drop pinches off. At this point the curvature becomes infinitely large. However, this happens only if you describe water as a fluid, which is an approximation in which you ignore that really the water is made of atoms. If you look closely at the pinch-off point of the droplet, then there is no singularity, there are just atoms.

There are other examples where singularities appear in physics. For example, in phase transitions, like the transition from a liquid to a solid, some quantities can become infinitely large. But again this is really a consequence of using certain bulk descriptions in approximation. If you actually look closely, there isn’t really anything singular at a phase transition.

There is one exception to this, and that’s black holes.

Black holes are solutions of Einstein’s theories of general relativity. They have a singularity in the center. Because it’s a common misunderstanding, let me emphasize that there is no singularity at the black hole horizon. There is actually nothing particularly interesting happening at the horizon. It’s just the boundary of a region from which you cannot get out. Instead, once you cross the horizon, you will inevitably fall into the singularity.

And in a nutshell, this is pretty much what Hawking and Penrose’s singularity theorems are about. That in general relativity you can get situations where singularities are unavoidable because all possible paths lead there.

But what happens if you fall into a black hole singularity? Well, you die before you reach the singularity because tidal forces rip you to pieces. But if your remainders reach the singularity, then that’s just the end. There’s no more space or time beyond this. There’s just nothing.

At least that’s what the mathematics says. So what does the physics say? Is the black hole singularity “real”? No one knows. Because we cannot see what happens inside of a black hole. Whatever happens there is really just pure speculation.

Most physicists believe that the singularity in black holes is not real, but that it is instead of the same type as the other singularities in physics. That is, it just signals that the theory, in this case general relativity, breaks down and to make meaningful predictions, one needs a better theory. For the black hole singularity, that better theory would be a theory for the quantum behavior of space and time, a theory of “quantum gravity” as it’s called.

Some of you may wonder now what’s with the technological singularity. The technological singularity usually refers to a point in time where machines become intelligent enough to improve themselves, creating a runaway effect which is supposedly impossible to predict. It’s called a singularity because of this impossibility to make a prediction beyond it, which is indeed very similar to the mathematical definition of a singularity.

But of course the technological singularity is not a real singularity. It may be in practice impossible to predict what happens afterwards but lots of things are in practice impossible to predict. There is nothing specifically unpredictable about the laws of nature at a technological singularity, if that ever happens in the first place.

In summary, singularities exist in mathematics, but we have no evidence that singularities also exist in nature. And given that, as we saw earlier, certain types of singularities do not even require any quantity to become infinite, it is not impossible that one day we may discover an actual singularity in nature. In contrast to infinity, singularities are not a priori unscientific.

You can join the chat about this week's topic on Saturday, Dec 12, 12PM EST / 6PM CET.


  1. The singularity is as old as Newtonian mechanics. The gravitational potential Φ = -GM/r blows up as r → 0. In classical physics this was not so much a problem with gravity because bodies have a finite extent. However, with the electric field and a point charge this became a problem. Also Laplace pondered what would happen if a gravitating body were at a point and came up with some early ideas of black holes. In complex variables we have a sort of taming of poles, 1/(z – z_0) for instance, where the pole happens at z → z_0. The integration of this with the Cauchy integral formula leads to residues that are the relevant aspect of the pole.

    With general relativity things get a bit wooly because there is this signature change in a metric. The Schwarzschild metric as a singularity at r = 0, but because this metric flip between radius and time there is a sort of Poincare duality. The singularity is not an infinity at r = 0, but rather the dual that is a 3-dimensional spatial surface at a finite proper time where curvature diverges. For an observer there this singularity is in their future within this spacetime region.

    The Schwarzschild singularity is split and is a spacelike region that bounds the upper black hole and lower white hole region. This is curiously similar to the lowering and raising operators of quantum mechanics. A quantum black hole generating lots of Hawking radiation is really in effect more of a white hole. The singularity is a sign I think there is a quantum principle relating the black and white hole is a superposition.

    A Kerr black hole has a further timelike region, timelike in the sense spacetime is timelike outside a black hole, containing a singularity. Orbiting this are closed timelike curves. What is relevant are these orbits, and how they link spaces that are not diffeomorphic; they do not have the same Cauchy data. These are similar to Riemann's sheets in complex variable theory. This is related to how a spatial surface outside a black hole can be continued into another world or the black hole interior time like region. This is a form of monodromy that defines the physically relevant aspect of a singularity.

    A rotating black hole may though have another singularity. For the eternal black hole the inner singularity r_- = m - √(m^2 - a^2) for a = J/mc the angular momentum parameter, is continuous with the I^∞. This means there is potentially an infinite number of null rays near this inner horizon that one must cross. This is a sort of Cauchy horizon that has singularity properties. The black hole though emits Hawking radiation, and this does break the continuity with r_- and I^∞. However, an observer who reaches I^∞ may do so in the final moments of the black holes quantum evaporation. In effect you are quantum scattered into the Hawking radiation. Whether the internal timelike region with the ring singularity actually exists or not is unclear. This is a form of the so called mass-inflation singularity. The singularity is reached before the null surface and is then a spacelike surface similar to the Schwarzschild horizon. Whether for the eternal black hole this is overcome by being on a frame where the null rays constituting this are sufficiently red shifted is not clear. The problem of Hawking final burst upon reaching r_- still remains.

    The extension of a spatial surface either from this world through the black hole into the idealized entangled world, or into the timelike interior is a diffeomorphic freedom, like a gauge choice, that physics needs to fix the problem, but then discards or "mods out" of the path integral. The physical aspects of this I think are with some transition between entanglements between states in the outside world and with entanglements between exterior and interior states. Spacetime singularities say, "Here there be quantum dragons."��

  2. Of course there are other essential singularities in "physics".

    I think you mean there are no esential singularities in the standard model of particle physics or statistical mechanics.

    That's true.

    But consider an electron. Its self-energy density is infinite at its exact center. That singularity cannot be observed in the standard model: its "renormalized" away.

    But the standard model of particle physics is wrong. It excludes gravity, which is not renormalizable. The stress-energy tensor's scalar reduction is infinite at the center,
    so is its gravitational potential. The resolution, of course is the same one as in black holes: a working theory of quantum gravity.

    1. You're assuming that the mathematical model is reality which is an open question. We kinda know it isn't, don't we? If eg everything is smeared at a Plank length then real numbers are an approximation and we actually need a different type of number to describe reality which may not have any singularity problem. That's a difficult to observe of course and yes, real numbers have been extremely useful.

  3. Sabine, your drop have an unrealistic cartoon-like form.

  4. Sabine,

    You noted that:


    I’m fine with the idea that such a transition would appear smooth, but I do have a trio of questions for anyone about how this would look to the infalling observer.

    Firstly, as best I’ve been able to understand it, an infalling observer should see an image of the black hole surface rise up around her to form an ever-deepening well.

    That’s because light emanating from very close to the event horizon will behave like a weakly tossed ball, mostly falling back onto the horizon nearby before it can escape. Only light that is almost perfectly vertical will resist falling back, and even then only at a tremendous red-shift energy cost. The angle over which light can reach escape velocity grows narrower as the light source gets closer to the absolute horizon. Rather intriguingly, black hole information isolation thus begins before the object crosses the horizon, rather than behaving like an abrupt shut-off exactly at the horizon.

    Since these ballistic light trajectories apply in both directions, the infalling observer will see an increasingly magnified image of the local event horizon, one that blocks most of her view of the outside universe. By the time she reaches the actual horizon, only an infinitesimally narrow cone of view pointing straight up will remain open.

    Now to be honest, I’m not even sure this narrowing effect is all that critical to my main questions. But at the very least it shows how an infalling object can grow asymptotically close to full black-hole isolation from the rest of the universe without actually falling past the event horizon. And in any case, I want to frame my questions as closely as possible to what she would actually see as she falls in. So, three questions:

    Question 1: As she peers upwards through her ever-dwindling cone of observation, will she see time speed up for events in the outside universe?

    Question 2: At the instant before crossing the absolute event horizon, just as her observation cone becomes infinitely thin, will time for outside events appear to speed up infinitely? If so…

    Question 3: If the infalling observer sees events in the outside universe occurring at infinite speed, doesn’t that also mean that regardless of the size of the black hole into which she is falling, it will necessarily evaporate via Hawking radiation before she can actually cross its horizon?

    A “yes” to all three of these questions would mean that singularities cannot form via infalling matter, since all infalling objects would by their own observations find themselves unable to cross the event horizon. Thus far from being a smooth transition, the absolute event horizon would behave like an impenetrable barrier, one that isolates infalling matter profoundly, but never completely, from the rest of the universe.

    Incidentally, the other popular line of reasoning (from Thorne I think) for how matter ends up inside of black holes is that when a large star collapses, matter inside the star is pinched off from the rest of the universe by the formation of the observed event horizon, like a water drop from a dripping faucet.

    The oddly simple argument (also from Thorne, ironically) against this singularity collapse mode is that the absolute event horizon — the only one that counts, since is the only one isolated from all inertial frames — forms first at the center of the star, and then moves outwards at incredible speed. If this moving event horizon is in fact impenetrable for the reasons I just described, it will push unabsorbed matter before it. That incidentally is my pet theory for how Type II supernova core rebound really works.

    Finally, in terms of arguments about whether or not black holes destroy information, this analysis would fall on the “no information is ever lost” side.

    1. Does this imply that you need an infinite time to cross a horizon, which implies that in the entire history of the universe no horizon has yet been crossed, nothing has yet fallen into a black hole?

    2. No, you cross the horizon in finite proper time.

    3. But the question is, how long does it take before I, a stationary outside observer, see you crossing the horizon if you fall into a black hole?

      IIRC, Susskind once stated that black hole evaporation looked from the outside as if the material that is still outside the horizon boils off.

    4. I am not a physicist, but please allow me to raise the Gedankenexperiment of falling into a black hole.
      Thinking that when approaching a black hole, due to the gravitational force, time slows down, what an imaginary entity would experience is that everything that is around it moves away from it.
      As Lawrence Crowell also says in another comment "falling into a black hole is not so much crushed, but rather pulled apart."
      If the time it takes to get to something tends to infinity, the distance to that something also tends to infinity.
      When you reach an area in which you experience a time that tends to be infinitely slow, the horizon itself would also tend to be infinitely far away, just like any other entity.
      All this happens for our imaginary entity in finite time and I am not sure at what moment what surrounds it stops making physical sense or if it stops being part of this universe. But I don't see how you can reach a singularity or how once a first unit of plank space reaches a density that generates a horizon, something that surrounds it can fall beyond this horizon.
      That there are supermassive objects is unquestionable but that their structure is the one proposed by the mathematics of general relativity seems to me as likely as that subatomic particles
      have no size and have infinite density.
      As you say in the video, nobody really knows what is inside a black hole, perhaps we should take with more skepticism all the theories that explain it to us in such detail.
      Thanks for indulge me with my naivety.

    5. Sabine, since you did not object to any of them, may I assume your answer to each of my three questions is yes?

      hector: Proper time is of course just local clock time, the time displayed by a clock that is close enough to the observer so that the temporal ambiguities of special relativity can be made negligible. Stated that way, proper time is just wristwatch time, your own personal clock.

      That is why I phrased my questions only in terms of the time and frame of references of the infalling observer. Switching observer perspectives, for example, is guaranteed to give nonsensical answers.

      Thus she sees the event horizon ahead of her and can calculate with precision when she will hit it. This is why folks nothing changes as she crosses the event horizon. That is simply true! So how can she not cross it a microsecond or so later?

      The resolution is oddly simple: The event horizon will no longer be there.

      That is, no matter how fast she is moving towards it, the event horizon will evaporate before she reaches it. In the case of a supermassive black hole she will have the unique experience of watching the black hole shrink at an astronomical rate, always maintaining a slight lead on her inward motion.

      If she has an especially durable spaceship, by the time it reaches what should have been the center of the hole, it will have disappeared. She will instead find herself hurtling outward from where the hole was at very close to the speed of light. There’s another way of saying that: She has become part of the Hawking radiation released by the final explosion of the black hole. That is, after falling in on one side of the hole, then by her clock she very quickly emerges from the other side of the hole, the antipode of her infall point.

      This “other side” emergence idea was first identified and named antipodal by Norma Sánchez in 1986 [1]. Gerard ’t Hooft rediscovered (or recalled?) Norma’s idea three decades later [1], and elaborated upon it in terms of momentum waves moving across the spherical event horizon [2]. If you combine the ’t Hooft elaboration of Norma’s idea with the restriction that nothing can ever actually cross the event horizon, it becomes equivalent to forcing all particles into momentum states within a spherical surface, much like how electrons are quantum delocalized in mirrors. Like mirror electrons, such particles can be simultaneously very energetic yet “cold” to outside observers, due that energy being tightly sequestered in momentum space. My own term for this (I’ve used it here) is the dark mirror model.

      The dark mirror model trivially allows the kinds of electrical conductivity and magnetic behaviors that have been postulated for black holes in recent years, and thus makes polar jets a lot easier. But unlike traditional super-simple black hole models, the states of matter in dark-mirror black holes are anything but simple. Such states might for example include band states and even forms of momentum-space interparticle bonding. With the exception of electromagnetism (think polar jets again), this complexity would just be highly sequestered.

      Notice that in the dark mirror model, Hawking radiation barely qualifies as quantum. Black hole radiation is instead almost entirely relativistic, not much more than infalling particles emerging very belatedly from their Sánchez-’t Hooft antipodes. However, the buzzsaw effect of spending a few trillion eons within a ferociously compressive spherical momentum space will make the problem of calculating exact emergence points a bit, um, problematic.

      [1] Norma Sánchez, Semiclassical Quantum gravity in Two and Four Dimensions.
      Gravitation in Astrophysics, Springer, 371–381 (1987).
      Preview pages:

      [2] Gerard ’t Hooft, Black hole unitarity and antipodal entanglement. Foundations of Physics, Springer, 46, 1185–1198 (2016).

    6. If there is a physicist that I respect, it is Gerard ’t Hooft and I think he is the first to take everything he proposes with a certain skepticism.
      We take structures and effects like Hawking radiation for granted. We consider that from the remains of a supernova we obtain a black hole and from there we develop models and theories.
      But do we have any idea how a horizon emerges? How happens in the place of greatest pressure in this supernova that a region of space is enveloped by a horizon and is separated from the rest of the universe? How is it possible, if, as we are proposing, nothing falls within a horizon before the hole evaporates, that mass is added to it? Supermassive objects exist and their mass is concentrated within a volume that resembles that of a theoretical black hole horizon, but is it configured as a singularity with its horizon, or are they crowded microholes? Or all the particles have become bosons and can occupy the same infinitesimal point? Or once a horizon is produced, it no longer makes sense to talk about mass and particles and what we observe is a tear in the fabric of space-time that cannot be re-sewn?
      The only toys we can play with are quantum mechanics and general relativity, but if black holes teach us something, it is that our toys are useless for this game.

    7. Addendum: Just to address a few questions folks might have about my earlier comments:


      I mentioned that if her spaceship survives, the infalling observer will sail through the region of the black hole at full speed, watching it evaporate literally in front of her, and emerge out the other side as part of the Hawking radiation debris from the explosion of the final micro blackhole. But that kind of sail-through can’t be the entire story for Hawking radiation, since otherwise how could the hole evaporate in front of her?

      Do you recall the deep-well effect I mentioned earlier, in which curved trajectories of light seem to cause the local surface of the black hole to curve up and envelope the infalling observer? It results in an every-narrowing cone of communication back to the external world as she falls deeper.

      Also, have you ever heard the explanation of Hawking radiation that compares the event horizon to a membrane that is thicker for large black holes and thinner for small ones? Particles quantum-tunnel across this membrane, with the probability of escape increasing as the membrane gets thinner. By the time you get to proton-sized black holes (mountain-ish mass range) the tunneling becomes so intense that the whole shebang just goes boom.

      Think of the well as being that membrane, with the cone of access as the only path over which tunneling still leads to the outside universe. For a large black hole this well gets very deep very quickly, and the odds of tunneling grow very, very small… but never quite zero.

      Such cones are the tunneling paths even for supermassive black holes, and that is how they evaporate. While the rate of evaporation is incredibly slow for outside observers, for the infalling ship the process occurs at blinding speed.


      And that may be another question that comes to mind for some readers: If the hole is evaporating due to quantum events that have nothing to do with infalling ship, how why should the hole evaporation rate keep pace exactly with the speed of infalling ship, as viewed by the infalling observer?

      The answer is that it’s the other way around: The black hole always slows down the infalling observer time just enough to keep itself eternally ahead of the ship. To an outside observer the ship would appear to remain at the event horizon for an unimaginably long period of time. But none of that matters to the infalling observer, who only sees her own smooth motion and the black hole seemingly evaporating to match her motion, regardless of how the external universe might want to interpret the same event.


      I also noticed that Lawrence Crowell aptly pointed out the incredible lateral ripping forces involved in falling onto an event horizon. Those are why anything surviving a black hole infall is, to say the least, problematic. But this is also where momentum space comes into play. Even in everyday physics, the best way to smear an electron out is not to tear it apart into still smaller “particle”-like pieces, but to transfer its bundle of conserved quantum numbers into momentum space, where they can then easily spread out into immensely thin sheets by xyz space standards. If you don’t believe me, look in a mirror! The surface electrons in that Fermi sea within the very thin sheet of metal behind the glass are the ones that bounce your image back at you, and they exist mostly in momentum space. Metals really are quite a bit more six-dimensional than we give them credit for. The nuclei and core electrons stay mostly in xyz, but it is the conduction electrons smearing themselves out in momentum space that are largely responsible for making the metal strong, conductive, ductile, and reflective.

    8. Addendum 2: Observational implications of the dark mirror interpretation

      This literally was not on my agenda for the rest of this week, but some part of my brain refuses to ignore dangling strings. So, if only to get on to other things…


      The most significant implication of the dark mirror model is that there is no such thing as Hawking radiation, and that its non-existence should be verifiable.

      Dark mirrors replace Hawking radiation with what I’ll call Sanchez-Whiting-'t Hooft or antipodal particle emission. This is accompanied by a purely electromagnetic form of radiative cooling whose rate depends inversely on the size of the body.

      Hawking radiation fails because it assumes erasure of all quantum states except mass-energy, charge, and spin. That in turn allows blank-slate quantum emissions where for example matter and antimatter particle emissions are equally likely.

      In the dark mirror interpretation the pro-matter quantum numbers get churned about but never erased. Any escaping quantum number bundles must still micro-collapse into ordinary matter particles, and that’s just ordinary tunneling, not the freewheeling equations of Hawking radiation. It also significantly changes what happens to black holes over astronomical time spans.

      Gerard ’t Hooft suggested that the antipodal interpretation permits a time-symmetric equilibrium,[1] which in turn suggests there is a threshold mass at which infall and antipodal outflow become equal and time-symmetric. Below this ’t Hooft threshold a dark mirror body can explode violently by emitting more particles than it takes in. This is the case I described earlier for a “smooth”, antipodal-emergence relativistic journey through an evaporating black hole.

      Larger bodies will no longer be capable of evaporating, since antipodal particle emission is neither as forgiving nor as forgetful as Hawking radiation. More bluntly, for larger black holes infalling objects just go splat, and are quickly and irreversibly converted into an excruciatingly compressed spherical state.


      However, there is also that second component of pure electromagnetic cooling. An infall pit of the type I described earlier is about as literal of an example of a black box with a small hole as one can imagine, so it will exhibit black-body radiative cooling. However, since the size of this opening shrinks as the hole grows larger, stellar black holes should reach thermal equilibrium much faster than supermassive ones. One area in which the concept of diverse internal thermal states might be interesting to explore is in the transition that supermassive black holes undergo from quasar-like young states to quiescent modern galactic centers.

      An interesting question is this: What is the final “cold” state of a dark mirror object after all of its internal heat has been dissipated?

      My own speculation, based quite unfairly on ’t Hooft’s momentum wave analysis,[1] is that it will condense into multiple homogenous, 2D momentum space bands (Fermi seas) with collocated spherical symmetry in xyz. The bands would at least begin as electrons, protons, and neutrons, though an end state of electrons and quarks is quite plausible. If the latter occurs, I suspect the quark bands would remain cross-correlated to deal with color confinement. (Hmm… why do I keep thinking about stock market prediction models?)

      If dark mirror objects — black holes — do indeed cool down to form homogenous electrically charged bands, there is a good chance the transition will affect the electromagnetic fields around black holes in odd and distinctive ways.[2]


      [1] 't Hooft, G. The Quantum Black Hole as a Hydrogen Atom: Microstates Without Strings Attached, arXiv preprint arXiv:1605.05119 (2016).

      [2] Balazs, N. L. Wave Propagation in Even and Odd Dimensional Spaces.
      Proceedings of the Physical Society. Section A, IOP Publishing, 68, 521, (1955).

  5. Hi Sabine,

    Just commenting on a couple of spelling issues. As to whether singularities are real in some sense depends on what one defines as real, something you talked about several times in your previous posts. I tend to agree with the common view that a singularity in a theory indicates a limit of applicability of the theory, nothing more.
    - "x squared" is preferred to "x square"
    - "your remainders" should be "your remains"

  6. Phase singularities are real physical things that one can observe in an experiment. They appear as essential singularities in optical field when they form complex zeros. We call them optical vortices.

  7. That Newtonian gravity blows up as the distance between two bodies goes to zero is why Roger Boscovitch, a Romanian physicist (1711-1787) postulated the notion of repulsive interatomic forces so matter is stable. I learnt this from a book by Paul Davies. I find this remarkable and it's a pity it's not more widely known.

    If we think of infinity as a kind of singularity then by saying singularities don't exist in physics we are also saying that infinities don't exist.

    I'd agree that black hole singularities signal that our usual theories break down there. But the search for quantum gravity is sooo difficult when we can't pribe the Planck distance. After all, it was the difference between the experimental evidence of blackbody radiation and its theoretical description that helped kick off quantum theory.

    It's a remarkable coincidence that in both cases we have a black body of some description that is key to moving physics forward.

  8. Does Pauli's exclusion principle form a real singularity for measuring? Can the separation of quanta by antipodal identification be considered as singularity? So, are the elementary particles singularities in spacetime topology (math, physics)?


  9. - In the same manner as infinities , singularities may be real in mathematics but they do no exist in nature. An example is the singularity at the center of a black hole. It simply indicates that the theory is not complete in those specific circumstances and something is not completely understood. In that particular case it is a hint, according to me, that something lies beyond our gravity and spatial environment This particular singularity may be difficult to resolve as it will require utmost imagination and modelling. M.

    1. Btw - I tend to think that the b-hole singularity indicates a phase transition to that other stuff where we came from.

  10. What people do not appreciate is that matter falling into a black hole is not so much crushed, but rather pulled apart. As I indicate below there is a reduction in volumeI think people have this idea there is some fixed volume of a ball V = 4πr^3/3 interior to a black hole. This is utterly wrong. The event horizon is not a boundary between the exterior universe and some volume of a ball inside. It is a gate from this world into another.

    The Weyl curvature C_{trtr} = 2GM/r^3c^2 describes the separation of two test masses with initial separation Δ_r along the radius as

    dU^r/ds = 2 C_{trtr}Δ_r(U^t)^2 ≃ 2GMD/rc^2,

    where U^t ≃ r and dU^r/ds = d^2Δ_r/ds^2. As one of these test masses approaches r = 0 you can see their separation approaches infinity. This means the region inside a black hole has “infinite” volume. Indeed, if you look at the conformal diagram for the Schwarzschild metric you see the upper triangle for the black hole interior has unbounded separation at r = 0. It also means the singularity is not a point, but rather a 3-dimensional surface where the Weyl curvature diverges, at least classically.

    The region inside a black hole from this geodesic separation equation is then growing madly. The volume grows with respect to Rindler time according to this separation. This is utterly bizarre, for this means the phase space volume here is exploding as well. This has connections to an exterior metric to Hilbert space. This is different than the interior metric for a black hole according to the Fubini-Study metric, and it describes a distance metric according to phase differences, what we call phasors (not Star Trek phasers) in basic physics. This describes the quantum complexity of a black hole. Interestingly, the dynamics of this in a finite element setting describes the Mandelbrot set, and with the Planck length as some cut-off to isolating a qubit fractal geometry describes the quantum gravitation an observer can detect. The chaotic dynamics and limit to computation is a running renormalization group flow for the limits of any possible observation.

    As matter approaches this inner singularity its volume increases. The Weyl curvature conserves the area bounding any region, which for this argument we assume contains some “blob” of matter. While the area is conserved it is stretched into a thin region. We are all familiar with how products on store shelves are tall, and annoyingly fall off easily, and inefficiently use encasing material as a result. This is a marketing thing. So as this area is conserved and deformed the volume inside decreases. This means the material inside is compressed in one sense of the word. The Weyl tensors for the angular components depend on r as C_{tθtθ} ~ C_{tφtφ} ~ 1/r and with U_t ≃ r as r → 0 this means the geodesic separation slows to zero and the distance between two test masses arrayed perpendicular to the radial direction approach each other as Δ_{θ,φ} ~ r. The volume as a cubic dependency does then mean the volume collapses as r → 0.

    We may then inquire as to what this means. For QCD quark-gluon bag it means hadrons are pulled apart into a quark-gluon plasma before reaching r = 0. If we invoke string theory, ignoring issues of positive vacuum energy etc, strings then get ripped apart into a vast number of fragments. The physics near the Planck scale is reached.

    1. Lawrence Crowell7:52 AM, December 13, 2020



      Not known to be Physics;


    2. I have my suspicions that string theory and supersymmetry are really quantum gravitation physics and not physics in positive vacuum energy or Λ ≥ 0 spacetime. For Λ < 0 thing are different. The local region near two colliding black holes is AdS_4 and there this physics may work. For Kerr or Reisner-Nordstrom metric in the interior region the vacuum is also negative.

    3. @Lawrence Crowell:

      I don't understand what you mean by 'the two test masses are arrayed perpendicular to the radial direction'.

      Generally, I think of test masses as ideal point masses. But that can't be the case here as they are arrayed 'perpendicularly'. Are they ideal rods instead?

      Given your description, a black hole sounds like a tardis. Bigger in the inside than it is on the outside.

      A 'gateway into another world' doesn't conjure up the right picture, at least for me, because given your description how the metric expands like crazy inside the space with everything being pulled apart it doesn't sound like anything like our worlds can exist there - in which case its not much of a world. If anything, it sounds to me like a machine for testing Planck scale physics as everything scales up! But where we have no control over what happens and nor can we ever find out what actually does happen. Two unfortunate liabilities for such a machine for present day physics ...;-).

      Of course we have Hawking radiation but as this is thermal that doesn't tell us anything about what happens inside the interior.

  11. I wonder if it would be legit to apply conformal geometry to the BH interior, with a flip-- let the singularity be a boundary at infinity, and then view the infalling matter as emanating from a point source. You could have a picture something like "Angels and Demons". From the exterior of the BH you would see something with finite dimensions. But if you were part of the space within the BH it would appear unbounded, a bit like an expanding universe, in fact. I wonder if you could get a re-scaling along the lines of CCC?


    The prediction of the technological singularity has been around for a while, but is it anything different from hoverboards and jetpacks and The Jetsons and The Way the Future Was? I now think that it is. As we all know, computers can do more and more. They can beat any human at chess, but were programmed by humans. They can now beat computers at go, but were not programmed by humans, but rather via machine learning. Of course, once computers can do something, they can do it orders of magnitude faster than humans. What about artificial intelligence? It's around now in primitive forms. Ideas differ on to what extent a general artificial intelligence is possible. If we assume that it is, then as soon as computers can be programmed to develop artificial intelligence, they will do so quickly, and whatever they develop could in turn program another computer and so on, resulting in a vicious circle.

    1. Phillip Helbig12:40 PM, December 13, 2020

      I have built a general AI which is more intelligent than 90% of humanity, a number of professional Physicists, a UK Astronomer Royal, a Physics Nobel Laureate and a Fields Medal winner:

      While (Universe Not= La-la Land)
      If there is no evidence for X
      Then X is not necessarily True;

    2. I also used to think of technological singularity as la-la land. But not anymore. But I'd disagree that it's something completely new. All of this can be traced back to the industrial revolution when processes first began to be industrialised on a mass scale. People began to then notice the social ills that it brought in it's wake. The infamous 'satanic chimneys' and children being exploited in factories. Thos is why enviromental laws and child labour laws were brought in. Likewise I suspect with AI. All this is subsumed under an unfashionable word today: Capitalism whose central concept is that of the commodity and whose driving force is profiteering, envy and greed. Its even co-opted our political system. Once we would have said that European democracies were liberal democracies. Now they are neo-liberal, where the neo means politics is conceptualised through markets. It's often referred to pejoratively as market fundamentalism.

      The dystopia that the technological singularity imagines is that of the human intellect being commodified. And this brings lots of dangets in it's wake; for example, I read a short note posted at the Oxford Internet Centre warning of the dangers of propaganda bots.

    3. "I have built a general AI"

      A quick analysis shows that it is actually a stupid chatbot which posts drivel in response to any post of mine. It doesn't even understand the idea of at least pretending to be on-topic, which spammers mastered decades ago.

    4. Phillip Helbig6:17 AM, December 14, 2020

      I make the on-topic point that for some people - you, Martin Rees, Crazy Luke, Geraint Lewis - the technological singularity has already happened.

      I'm going to name my AI "Helbig 2.0".
      Why don't you ask my AI if the universe is fine-tuned, Helbig 1.0?
      Computer Says No Evidence

    5. @Phillip Helbig:

      It's worse. It's not an AI. Nor even the imitation of an AI. It's a sock-puppet and I think I can see the wiring ...

    6. Still sore about Gentzen. Some people prefer being right to the truth. All Helbig 1.0 has to do to avoid deprecation in favour of Helbig 2.0 is provide evidence of universal fine-tuning.

      Strange that so many physicists claim universal fine-tuning - the UK Astronomer Royal, Brian Schmidt, Helbig 1.0, Luke Barnes, Geraint Lewis, etc. - but can provide zero evidence.

      I think there must be a further division within Homo Sapiens. It's difficult to consider these people as the same species as me.


    And "nothing" is not responsible for curvation of spacetime.(?)

  14. As an uninformed layperson, I would define 'singularity' as that point beyond the human mind's ability to calculate or contemplate

  15. My understanding is that many physicists believe that matter cannot be compressed beyond a certain density, say the Planck density, which is 3.36 x 10^96 kg/m^3. If this is true, then the Big Singularity would have had a radius that was fairly close to the classical electron radius (around 10^-15 meters), if one uses the presently accepted value of the mass of the Universe, 1.5 x 10^53 kg, in the calculation. A few people have accidentally stumbled upon this curious coincidence. Yet, I am skeptical, though there are too many such neat coincidences in our Universe.

    1. Density isn't observer-independent. The statement that it's bounded therefore isn't well-defined.


      Flippiefanus is correct:


      Optical vortices are a huge field these days, tracing back to work by Sir Michael Berry (famous for the Berry phase) in the early 1980's. Similar vortex wave singularities have also been found in electron beams, Bose-Einstein condensates, and I believe other media.

      Optical vortices are not the only actual singularities that have already been discovered in nature. Berry has long had a special interest in such things and he's the go-to guy there. This paper has examples of some very early discoveries:

      Berry, M V, 2000, ‘Making waves in physics. ‘Three wave singularities from the miraculous 1830s’, Nature 403 21 – 6 January 2000.

      (You might find a downloadable copy on his publications page.)

    3. Kris,

      The problem with these examples, as I said in my video, is that they appear in models we know break down at a certain distance scale. No one doubts that singularities appear in approximations. But are they are real thing or an artifact of the approximation?

    4. @Kris:

      I had a look at Berrys article, Making Waves in Physics. It was published in Nature in 2000, it's expository and historical and very well written - as one would expect for a Nature article.

      If I'm not mistaken these singularities are like the singularities of Vladamir Arnold's singularity theory where they can be resolved geometrically by adding dimensions. This isn't the kind of physical singularity we get when density goes to infinity signifying a breakdown of the theory itself.

    5. The singularity in a Kerr or RN metric is actually repulsive. Orbits around them are closed timelike curves. These are identified with some sort of monodromy, but I have not seen this worked out and for myself ponder this some. In this setting what is really important is the monodromy; the singularity is just an artifact of our mathematical language.

    6. Couldn't agree more Lawrence. I've identified the repulsive force as equivalent to a soliton or 'Mexican Hat' profile, reversing within the torus to emerge at the cusp. We then get what we actually find; all accreted matter is re-ionized and ejected in the 'quasar' jets. Indeed we find slightly MORE matter in the jet columns! Possibly due to pair production at the shear planes as Martin Rees found some decades ago?

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  18. Mozibur,

    You're right that Berry's singularities are not the type where density goes to infinity. However, as Sabine noted, "certain types of singularities do not even require any quantity to become infinite."

    The objects Berry is describing have a feature called "topological protection." That guarantees the existence of singular points in their mathematical descriptions which are physically necessary. This is what gives some types of optical vortices their amazing stabilities. Those involve singularities in either the wave phases or polarizations. The trick is that where phase or polarization is singular, the wave amplitude is also going to zero.

    Similar mathematical considerations led Dirac to speculate the existence of magnetic monopoles, which may yet turn out to be physically real.

    1. Is "topological protection" sufficient to produce the production of hawking radiation as deminstated by some types of optical solitons or is there some other factors involved?

    2. The 'topological protection' is there so that the so-called singularity doesn't vanish. The situation is modelled by vector bundles.

      There's an easy physical analogue which shows what happens: a Mobius strip has a twist and no matter how you stretch and pull the band that twist remains there. This is the 'topological protection'. The same is true for additional twists. After making the appropriate identifications, in mathematical physics, these are exactly the instantons, monopoles and the like. But as you can see there is nothing singular about them : it's an artifact of a partial description.

    3. By the way, do you work with optical vortices experimentally? I hadn't heard of them before and just looked them up on wikipedia and they seem fascinating - and the pictures don't look bad either. There were some pictures in Berry's article too, as I recall.

  19. Hi Sabine,

    Michael Berry's article cites William Whewell's discovery of a singular point (or "catastrophe") in the pattern of tides between the coasts of England and Holland. Is that a real thing or an artifact of the approximation?

    You could compare Whewell's analysis to the problem of identifying the highest point on a mountain peak. Assuming it has a continuous surface, topology would tell you some such point must actually exist. To find that in practice of course would involve a scale-dependent measurement, like measuring the length of the English coastline. At progressively smaller scales, you come to asking which atom is the highest. What about the electrons jumping on and off? And where are those, exactly?

    However, unlike the length of England's coastline, which blows up as you go to finer and finer scales, the location of the mountain peak converges. The same would hold for Whewell's singularity. So I think it's fair to call it a real thing -- at least as a real, well-defined geometric feature of something physically real.

    1. All these theories are effective theories, meaning they break down on small scales. On small scales there's just atoms (or subatomic particles) and nothing is singular, so all the singularities in effective theories are mathematical artifacts.

      As I said, the only instance we currently know of that could be a true singularity is the black hole singularity (or other singularities in GR). That's because we have no theory that could replace GR and make the singularity go away.

      Of course there's some people who insist on denying reductionism. I hope you don't want to go there because I'm somewhat tired of having to repeat that that's an unscientific argument. It's a deus ex machina attempt. To make it work, you basically have to say "and now a miracle happens" and somehow water is not longer made of atoms or something. Don't get me started.

  20. Actually, it has occured to me that the simplest example of a singularity is that most ubiquitous thing in physics, a point. As Euclid defined it, it is extensionless. Physically, that doesn't make sense either spatially, as a position; temporally, as a moment; or in terms of matter, as a point particle like an electron is supposed to be.

    Zeno was the first to realise that singular positions of space or time don't make sense physically. A point lost to many. His criticism is still calid today.

    To my mind, its one reason why I like string theory because it does posit extension; but of course it has extension in only one direction. Adding extensions in other directions gives branes.

    1. A point per se isn't a singularity. It's an example for something that has size zero, which is unphysical. I talked about last week. That in and by itself doesn't mean every point is a singularity. In fact that statement is entirely meaningless.

    2. Mozibur7:24 AM, December 17, 2020

      "Physically, that doesn't make sense"
      Yes, but the idea of a point is useful in models. Have you heard of Geometry or Calculus?

      "His criticism is still calid today."
      Zeno's paradoxes are avoided using Calculus. Have you heard of it?

      "its one reason why I like string theory"

      There is no physical evidence to support String Theory and 10^500 possible theories is a tad vague.

      "Adding extensions in other directions gives branes."

      I see no evidence of a brane in your comment.

  21. Hi Sabine,

    It's true Michael Berry's singularities are only within waves. Are you arguing that waves are not real on a fundamental level? If we should regard only elementary particles as fundamental constituents of the universe, well, what are those? Many of their properties and interactions can be modeled with soliton waves -- including their singular nature. Also, as Feynman has pointed out, our only working model of an electron is as a point particle.

    1. The only waves that are currently fundamental are gravitational waves. That doesn't mean other waves are not real. It just means they're not fundamental. They derive from something else. They "can be reduced to" something else, as the philosophers would have it.

      As I already said above a point is not a singularity. A point has zero size. You can interpret this as something that's infinitely small. That in and by itself is not a singularity.

      I am somewhat perplexed that people are so confused about this. (Especially after they watched my video explaining what a singularity is.)

  22. Dr. Hossenfelder,

    As always a very interesting read of all of the comments. The various theories and modeling that go into explaining a black hole and its singularity provide for great thought. The most interesting comment to me was that the event horizon was a "gateway." I believe that this is the most important point that we can think about. The Event Horizon is for simplicity sake the "point" where the acceleration of gravity equals the speed of light. At this "point" doesn't the physics we accept say that time and space come to a stop and any mass goes infinite? So, once you move through this gateway trying to explain or understand what happens should not be considered as being associated with the physics we know on our side of the gateway. If mass is infinite and space ends how can any model or theory talk about having anything of any shape or mass anywhere past the event horizon? How about throwing this one out there; one of the comments talked about the event horizon having some type of thickness. Well, this thickness equals the phase necessary to reach C^2 acceleration at which point everything exists in an energy form. What happens next I do not have a clue, maybe the energy recycles back into the vacuum energy to maintain field density as the universe expands.

    I guess the point that I am trying to make is that looking at black holes and singularities, if singularities really exist, needs to go beyond the physics we known. The event horizon gateway is very unique and we should be looking at some form of unique physics to go with it.

    1. Hi Steve,

      I believe you are confusing the escape velocity with acceleration. Also, the event horizon is not a point. It's a surface. In the simplest case it's a sphere. And no, neither space nor time stop there. There is nothing in particular happening at an event horizon. It is merely the boundary of a region from which you cannot escape. There is no singularity at the event horizon. The black hole singularity is inside the event horizon.


    They do not.

  24. @ Terry Bollinger




    No to these questions. Because she is freely falling you have not only the gravitational time dilation but also the relativistic doppler effect to take into account. The net effect is redshift, at the horizon z=1. So looking upwards she sees the outside universe speeding down.

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  26. It's good to drive prople think what are real things. In physic the only real thing is to compare observations to predictions, nothing more.

    Singularities are math. Is the number theory full of singularities? Mad idea. But the quantum measurable world can be very connected to the number theory. Applying range rules out direct measurability of pure math in phenomena but it's not excluded.

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  28. Yet another highly interesting function is

    f(x) = (sin(x) - x cos(x)) / (x - sin(x))
    = cos(x) (tan(x) - x) / (x - sin(x))

    (1) The zeroes are the Eigenvalues of the differential equation y''+ k y = 0 (e.g. F = m a)
    (2) It has a single (removable) singularity at x=0 with f(0)=2
    (3) For large x the function approaches f(x) = -cos(x)



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