The Undefinable

If I can teach one lesson to all of you, it’s this: be precise. In physics, we try to state what we mean as precisely as we can. If we can’t state something precisely, that’s a clue: maybe what we’re trying to state doesn’t actually make sense.

Someone recently reached out to me with a question about black holes. He was confused about how they were described, about what would happen when you fall in to one versus what we could see from outside. Part of his confusion boiled down to a question: “is the center really an infinitely small point?”

I remembered a commenter a while back who had something interesting to say about this. Trying to remind myself of the details, I dug up this question on Physics Stack Exchange. user4552 has a detailed, well-referenced answer, with subtleties of General Relativity that go significantly beyond what I learned in grad school.

According to user4552, the reason this question is confusing is that the usual setup of general relativity cannot answer it. In general relativity, singularities like the singularity in the middle of a black hole aren’t treated as points, or collections of points: they’re not part of space-time at all. So you can’t count their dimensions, you can’t see whether they’re “really” infinitely small points, or surfaces, or lines…

This might surprise people (like me) who have experience with simpler equations for these things, like the Schwarzchild metric. The Schwarzchild metric describes space-time around a black hole, and in the usual coordinates it sure looks like the singularity is at a single point where r=0, just like the point where r=0 is a single point in polar coordinates in flat space. The thing is, though, that’s just one sort of coordinates. You can re-write a metric in many different sorts of coordinates, and the singularity in the center of a black hole might look very different in those coordinates. In general relativity, you need to stick to things you can say independent of coordinates.

Ok, you might say, so the usual mathematics can’t answer the question. Can we use more unusual mathematics? If our definition of dimensions doesn’t tell us whether the singularity is a point, maybe we just need a new definition!

According to user4552, people have tried this…and it only sort of works. There are several different ways you could define the dimension of a singularity. They all seem reasonable in one way or another. But they give different answers! Some say they’re points, some say they’re three-dimensional. And crucially, there’s no obvious reason why one definition is “right”. The question we started with, “is the center really an infinitely small point?”, looked like a perfectly reasonable question, but it actually wasn’t: the question wasn’t precise enough.

This is the real problem. The problem isn’t that our question was undefined, after all, we can always add new definitions. The problem was that our question didn’t specify well enough the definitions we needed. That is why the question doesn’t have an answer.

Once you understand the difference, you see these kinds of questions everywhere. If you’re baffled by how mass could have come out of the Big Bang, or how black holes could radiate particles in Hawking radiation, maybe you’ve heard a physicist say that energy isn’t always conserved. Energy conservation is a consequence of symmetry, specifically, symmetry in time. If your space-time itself isn’t symmetric (the expanding universe making the past different from the future, a collapsing star making a black hole), then you shouldn’t expect energy to be conserved.

I sometimes hear people object to this. They ask, is it really true that energy isn’t conserved when space-time isn’t symmetric? Shouldn’t we just say that space-time itself contains energy?

And well yes, you can say that, if you want. It isn’t part of the usual definition, but you can make a new definition, one that gives energy to space-time. In fact, you can make more than one new definition…and like the situation with the singularity, these definitions don’t always agree! Once again, you asked a question you thought was sensible, but it wasn’t precise enough to have a definite answer.

Keep your eye out for these kinds of questions. If scientists seem to avoid answering the question you want, and keep answering a different question instead…it might be their question is the only one with a precise answer. You can define a method to answer your question, sure…but it won’t be the only way. You need to ask precise enough questions to get good answers.

11 thoughts on “The Undefinable

  1. Q

    Unless the question is being phrased in an operational way, such as “If I perform experiment X, what outcome will I observe?”, the user is inadvertently asking not a question about the real world but a question of the model’s internal representation of the world. The operational question receives a prediction to an outcome of a hypothetical experiment. The answer to the question about the model lives purely inside the model. The translation between the model and reality might be intuitive, such as with Newtonian mechanics, but when it comes to quantum mechanics or general relativity it’s often not. Even if the user poses the question in a sufficiently precise way, the answer might be meaningless to them.

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  2. ohwilleke

    One of my favorite maxims is that when choosing between definitions it is important to know the purpose for which it will be used. A definition isn’t good or bad in the abstract. It is good or bad when it makes it possible to get the answer the real (deeper or more practical question) that caused you to need it.

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  3. Dimitris Papadimitriou

    There is indeed a widespread confusion about the singularities in GR. When relativists talk about the ” center of the black hole” they mean the coordinate center, not the spatial center. They also know very well the difference between spacelike and timelike singularities ( after all, the ” strong” version of the Cosmic censorship hypothesis has to do mainly with the latter).
    Another source of confusion is that people sometimes forget that GR is about Spacetime geometry, not Riemannian and the difference is crucial!

    As an example, the Big Bang singularity cannot be localised.
    As you said in your post, the singularity itself is not part of the manifold, but you can always define hupersurfaces arbitrarily close to ” it” , and they’re spacelike, just like the hupersurfaces r= constant inside the Schwartzschild horizon. Even if you change coordinates, the spacelike character of the Schwartzschild singularity does not change!
    Remember that the concept of the ” trapped surfaces” that Penrose used back in the 1960s is invariant, and the r= const. surfaces are a special case of this.
    Even if you change coordinates, the fact that the spacetime geometry inside the hole is dynamical ( for the Schwartzschild between 2Gm> r> 0, and for the subextremal rotating black holes between the event and the Cauchy horizons) does not change!
    That was already well known in the GR community for decades, ( recently I saw a paper from T. Jacobson from the late 90s that mentioned this).
    Even for timelike singularities, the r= 0 is not necessarily the spatial center! ( A typical example is the Kerr spacetime, where the r= 0 is a disk surface, not a point. On the other hand, the Reissner – Nordstrom black hole has indeed a pointlike singularity!
    But these timelike singularities are considered unphysical, even in the classical theory, because of the well known instabilities ( blue shift/ mass inflation) that occur in the vicinity of the inner horizon, so only spacelike ( or null) singularities are considered seriously- at least in the classical theory- that’s roughly the strong cosmic censorship.

    Going back to the Schwartzschild black hole, it is obvious even from a Penrose diagram ( one that depicts a collapsing object to a BH), that the r= 0 center of symmetry coincides with the spatial center of the collapsing star only before the singularity forms.
    After that the r=0 is akin to a spacelike surface ( a horizontal line in the 2d diagram); essentially a singular future ” boundary” for the spacetime( informally).
    This is not magical at all! Intuitively, one could say that spacetime geometry is stretched and squeezed similarly to what happens to everything that falls in.
    Essentially it is a consequence of the fact that the growth of the tidal forces inside the horizon is time dependent!

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    1. 4gravitons Post author

      Thanks. One thing I wanted to clarify: the stack exchange answer I linked in the post seems to say that the intuition you get from the Penrose diagram for a Schwarzschild black hole is not actually sufficiently invariant to provide a clear answer to whether the singularity is a surface or a point, because there are multiple ways to formalize that intuition that disagree. Do you think the post is wrong about that, or am I misunderstanding what it’s saying?

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      1. Dimitris Papadimitriou

        I read the 1st answer from stack exchange that you mentioned. It was a careful answer and I , mostly, agree. There are some remarks I have to make, though:
        a) About the Kerr singularity: there is not really a controversy about it, there are so many studies already.. For a recent good paper about it : arXiv:1912.06020 ( Chrusciel et al).

        b) About the Penrose diagram ( of a spherically symmetric collapsing object):
        Every point in the diagram represents a sphere. The r=0 center of symmetry ( that coincides with the spatial center of the collapsing star) is the line in the left side, that goes upwards in the time direction, as usual, until the formation of the singularity. An observer that falls in later, long after the collapse of the star, cannot reach the spatial center anymore, because this center is already in her/ his past. Whatever manoeuvre that observer tries to do, wherever he/ she tries to go, there is only a finite amount of time before the end, so the singularity, in that sense, is everywhere inside the hole, and acts like a future ” boundary”, so that any timelike curve (or null ) is doomed.
        This is what the diagram depicts.
        As the ” Area- radius” r coordinate , that is timelike inside the horizon, goes to zero, the tidal forces that are, roughly, inversely proportional to the r^3 , grow unbounded. Everything ( including the geometry inside!) is stretched in the horizontal ( in the diagram, that represents the ” proper” – let’s say- radial direction, in contrast to the “temporal” coordinate radius r ) direction and squeezed in the perpendicular ones.
        As r=0 is approached, the infalling object is torn apart completely. At the final stages its remains become causally isolated from each other, contrary to the common belief that the infalling objects are squeezed to a single point!

        So, although the singularity itself is not a part of the manifold, what really matters is that , close to it, every r= constant hypersurface is spacelike, and the interior of the hole does not resemble an empty sphere with a ultradense pointlike thing at the center as the common myth goes..
        Instead, it is a hupertube that ” closes” in the past, when the singularity forms ( this is the spatial center) and extends spatially in the ( spacelike inside) time dimension of the external asymptotic observers.
        I know that its not easy to describe this with just words , here..
        For a more intuitive picture, the 3d Eddington/ Finkelstein diagram is more suitable, i think.
        The tube closes at the bottom, like the edge of a spindle, when the singularity forms.
        Then it continues upwards and grows bigger as the black hole ages. In the diagram there is also another spatial dimension that represented by the angle phi, while the other angle ( theta) is omitted.
        There is also a maximal hypersurface at r= (3÷2)Gm that can be used for the calculation of the – maximal- spatial volume inside the black hole ( that is slice independent / invariant).
        As expected, this maximal volume grows hugely as the external time goes by, even if one counts the Hawking evaporation, as the estimations show ( as a reference: Christodoulou / Rovelli, arXiv:1411.2854 – there are more references at the end of this paper).
        As a general remark, the important thing has to do with the interior spacetime geometry/ causal structure of the black hole, not the singularities themselves, as physical entities.
        After all, there are, probably, no divergent quantities in the physical world.
        On the other hand, the attitude that this is is only a topic of academic interest misses the point, I think.
        Even when a deeper theory will replace singularities with something else ( some extension through a QG region, or something totally different) , there has to be a ,more or less, ” smooth” connection between what the classical theory predicts and the QG extensions;/ corrections otherwise new problems will appear.

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        1. 4gravitons Post author

          Thanks! From what you’re saying, I think one nice way to think about this is that there is a concrete observational question that “is the singularity a point?” might be trying to ask: namely, “if objects approach the singularity, do they come together?”. And if that’s the concrete question you’re asking, then there’s a concrete answer: no, objects approaching the singularity are separated, becoming causally disconnected the closer to r=0 they get.

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          1. Dimitris Papadimitriou

            This is an intuitive way to think about it – for black hole singularities, where the Weyl curvature diverges.
            For approximately isotropic and homogeneous universes, like ours, the initial singularity cannot be ” localized” either, although the Weyl curvature was, presumably, low in the beginning ( only the Ricci curvature was infinite then). Things are even more trickier if our universe is spatially infinite..

            It’s difficult to find an all encompassing, rigorous definition ( of the kind that satisfies a philosopher of physics ,e.g.)for all kinds of singularities in general.
            Although they’re not part of the manifold, their specific characteristics are related with the geometry and topology of the” host” Spacetime.

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  4. tomdickens

    Thanks for this interesting post and the link to the Physics Stack Exchange answer!

    This reminds me of the question of the volume enclosed by a black hole’s event horizon. This is another question without a unique answer, depending upon the coordinates chosen for the spacelike surfaces in the BH space time. There are several interesting papers on arXiv that discuss this, for example “How big is a black hole?” by Marios Christodoulou and Carlo Rovelli (arXiv:1411.2854). They show that their definition of the interior volume grows with time since the BH collapsed, eventually becoming extremely large.

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    1. tomdickens

      Sorry, I just noticed that Dimitris have already mentioned the BH volume problem! (I tried posting it the other day but somehow the post failed.)

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  5. Dimitris Papadimitriou

    “Is the Wavefunction ” real”, or is it just a mathematical tool for calculating probabilities in Quantum mechanics?”
    This is an obvious controversial question that we forgot to mention!
    Some people consider it not well defined, while others answer it with much confidence..
    Many years ( decades) of debate and still no consensus exists, even for what is the definition of
    ” real” in the question.

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