Q: What are singularities? Do they exist in nature?

Physicist: Singularities are just artifacts that fall out of math.  They show up a lot in theory, and (probably) never in nature.  The “singularities” most people have heard of are black hole singularities.

In practice, when you’re calculating something in physics and you find a singularity in your calculation (this happens all the time), which usually looks like “1/x”, then that means that there’s a mistake somewhere, or you’re looking at something that never happens, or there are physical laws or effects that haven’t been taken into account.

The simplest mathematical singularity: 1/x.  The closer you get to zero, the more 1/x blows up, and at x=0 the function is undefined.

For example, when you drain water from a tub or sink the water will spiral into the drain.  A back of the envelope calculation (based on known principles) shows that the speed, s, that the water is moving is s=\frac{c}{r}, where r is the distance to the center of the spin, and c is a constant that has to do with how fast the water was turning before you pulled the plug.  Notice that this equation implies that as water gets closer and closer to the drain, it will move faster and faster, and that right over the drain it will be moving infinitely fast.  So how does the universe find an out?  This will look familiar:

One of the slick tricks that water uses to avoid spinning infinitely fast at the center of a vortex: not being there.

Even when you’re deep underwater there are outs; turbulence, cavitation, that sort of thing.

A slightly more obscure example is the energy of a charged particle’s electric field.  If you have an electron just sitting around, the energy, E, of its field outside of a distance, R, from the electron is E =\frac{e^2}{8\pi\epsilon_0}\frac{1}{R}.

The electric field being considered is the light blue area. The most intense part of the field is close to the electron, so as R gets smaller and smaller, the total energy gets bigger and bigger.

Most of that is just equation porn.  The important bit is the “\frac{1}{R}“.  Once again there’s a mathematical singularity in an equation describing a physical thing.  But, once again, the universe (being sneaky) finds a way out.  This is a hair less intuitive than the whirlpool thing, but in quantum mechanics an electron is described as being “smeared out” (in an uncertainty principle kind of way).  It doesn’t exist in any one place, so the idea of getting infinitely close doesn’t really make sense.

The whirlpool thing and the electron thing (and hundreds of other examples) are examples of singularities that show up in the math, but can be explained away through experiment and observation, and shown to not be singularities in the physical world.

In general relativity, the shape of spacetime near a spherical mass is given by:

c^2 d\tau^2 = \left(1-\frac{r_s}{r}\right)c^2dt^2 - \left(1-\frac{r_s}{r}\right)^{-1}dr^2 - r^2\left(d\theta^2 + \sin^2{(\theta)}d\phi^2\right)

Now, unless you’re already a physicist, none of that should make any sense (there are reasons why it took Einstein 11 years to publish general relativity).  But notice that, as ever, there’s a singularity at r=0.  This is the vaunted “Singularity” inside of black holes that we hear so much about.

Something like a star or a planet doesn’t have a singularity, because this equation becomes invalid at their surface (this equation is about the empty space around a mass).  But, for a black hole the gravity is so intense, and spacetime is messed up so much that it looks as though there’s nothing to stop matter from becoming infinitely dense.  However, unlike the singularity over the drain in your sink, the singularity in a black hole can’t be observed.  Which is frustrating.

I suspect that what we call the singularity in black holes either doesn’t exist (there is some law/effect we don’t know about) or, if cosmic censorship is true, the nature of that singularity both doesn’t matter and can’t be known, since it can never interact with the rest of the universe.  There are some theories (guesses) that would fix the whole “black hole singularity problem” (like spacetime can only get so stretched, or some form of “quantum fuzziness”), but in all likelihood this is just one of those questions that may never be completely resolved.

The whirlpool photo is from here.

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16 Responses to Q: What are singularities? Do they exist in nature?

  1. angel says:

    Great article, I’m not physicist but always enjoy the manner that attack the problems, in particular the form so fluid that the article was written, I try to incorporate that style when do my document, unfortunatly I only have read this style in english and have some problems to incorporate in spanish xd.
    Anyway thanks for the great article

  2. qubit says:

    Interesting and very informative article. I think the energy equation is incorrect. It’s only epsilon not epsilon square.

  3. The Physicist The Physicist says:

    You are completely correct. Fixed!

  4. Russ says:

    I came here looking for an answer about black holes that involves several singularities I suppose. Initially I thought that black holes have a singularity where all the matter occupies the same point in space. Given that premise, I was confused about the spin of a black hole. If a black hole inherits its spin from the star it was formed from and spins faster due to conservation of angular momentum, then wouldn’t the singularity have an infinatly fast spin? Since there are black holes with “not infinite” spin, then does that imply that the matter didn’t collapse to a single point, or is it, like in the article, something else we don’t understand going on? And finally, I’d we could observer the spin of a star before it went supernovae and the spin of the resulting black hole, what would you suppose the difference would tell us about the interior if anything?

  5. Mickey says:

    I have a couple questions about the supposed singularity in a black hole and the curvature of space time:

    1) What is curved space-time? Time is curving with regards to a spatial dimension? If so does a particle traveling to a center of a blackhole with a finite speed actually reach a singularity or just move forward in time?

    2) Wouldn’t a distribution of mass over curved 4 dimensional space make it impossible to have infinite density if a most of the mass were at different time coordinates?

    3) If Hawking radiation is correct doesn’t that seem to suggest intuitively that there is no singularity in a black hole?

  6. If Singularity does exist in nature it will eventually break the law of conservation of Mass-Energy, conservation of charge etc… of the Universe, because nothing can escape from Singularity. and moreover how can a zero volume can occupy Infinite density, does it make sense at all…..

  7. If Singularity does exist in nature it will eventually break the law of conservation of Mass-Energy, conservation of charge etc… of our Universe, because nothing can escape from Singularity. And moreover how can a zero volume can occupy Infinite density, does it make sense at all ? Thus all the Matter entering into the singularity will end up there. Is it necessary that a ” Black-Hole” should have a Singularity ? or Is a Singularity is just a mathematical frame-work and might not exist in realty. There is no, proper definition of Singularity. Can’t we construct a Black-Hole model which doesn’t require any sort of Singularity…….?

  8. Tyler Parker says:

    When you spin a quarter, right before it stops the quarter’s axis of symmetry’s rotation relative to a central axis is unbounded. I had a professor who verified this with a couple grad students using a vacuum chamber, laser, reflective disk and really smooth surface.
    This is the closest i’ve thing I heard of to an actual physical finite-time singularity.


  9. jake the snake says:

    If space is infinitely divisible then there is no smallest size. The volume cannot reach zero. Just like you can’t reach infinity. The black hole will just shrink forever and never attain zero volume. If it always has a volume there can be no singularity.

  10. jason says:

    It sounds like your saying that a singularity is like water down the drain. Now depending on which hemisphere your in water spins right or left. However if everything that goes into a singularity if it stays in this plane of reality . It would mean that so much is compressed into one point making pretty impossible. However like a drain . Do singularity’s empty into another plane of existence ? Or is it so heavy that it able to just keep packing material in one space?

  11. Todd says:

    Probably the most familiar example is walking from one spot to another. Theoretically, the intervening distance can be divided to infinity…first taking half the distance, then half that, then half that and so on. But in day to day experience we traverse the distance without incident.

  12. Wilfredo J. YAMBAO says:

    Very interesting article. No, I’m not a physicist but your explanation cleared now my understanding: singularities don’t exist in reality

  13. BLeRg Zylacone says:

    If a point singularity is shrinking into less space, couldn’t it be said that the matter in it is in motion flowing towards its’ center? If particles of matter are all flowing towards the center of the object containing the singularity, won’t those particles eventually collide? Does this collapse in volume accelerate matter? If the particles of matter do accelerate, could they exceed the speed of light before colliding?

  14. AlanD says:

    There are various ways to say what the author says, I guess, but I’m a mathematician and want to say it this way: The General Relativity equation quoted, with a 1/r term, is NOT even a mathematical equation at r=0. That is, the equation has no mathematical meaning at all at r=0 because division by 0 is not and can not be defined mathematically in any way. The problem is LOGICAL, not physical. The equation can be a valid math and physics equation for r as near to 0 as you want, but is nothing at all except a nonsense statement at r=0. Maybe it’s good poetry; I don’t know, but it has nothing to do with mathematics. Therefore, if the expression r=0 can even truly have physical meaning to a physicist in this situation, then he or she MUST find some other equation to describe that meaning. The universe does not have to strain or sneak around here; the physicist simply has to find a different equation which has actual mathematical meaning. Of course I do not know what that equation needs to be, but that’s not my problem!

  15. Ian Barber says:

    Yay! This finally explains away needing a moving singularity for a moving black hole.

  16. Ian Barber says:

    Does planck size apply in a black hole? If yes then does that rule out singularities?

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