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A Layman's Question About the Nature of Space

  1. Jun 11, 2010 #1
    While reading Lee Smolin's book "Three Roads to Quantum Gravity" in which he talks about the Bekenstein bound and a smallest fundamental unit of area, the following occured to me:

    Suppose there does exist such a smallest unit, call it A.

    Then there exists a smallest volume, V(a).

    A singularity (in the sense of a black-hole), by definition has infinite density.

    By the definition of density and the above lower bound on volume of a region V(a), the singularity must have infinite mass.

    What gives?


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  3. Jun 11, 2010 #2


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    But do such singularities exist in nature? I don't know any professional who thinks they do. So far they come up in the "classical" (vintage 1915) theory of gravity called General Relativity.

    The conventional expert view, for many years, has been that this simply shows that GR is wrong. GR doesn't continue to apply down there in the hole, it fails to compute sensible numbers. So one tries to develop an improved theory to replace it. The improved theory should not predict infinite density or infinite curvature---i.e. should not blow up. There are various candidate models.

    There is a good one-page essay about this at a German research institute's public service outreach website. The AEI has a website called "Einstein Online". Check it out. It explains what I was saying and answers your "What gives?" question. The most useful essay about the confusion people have over the "singularity" idea is titled "A Tale of Two Big Bangs".
    You will see it at the E-O menu.

    I have a link in my signature, for quick reference. But you can also simply google "einstein online cosmology" and you'll get it.
  4. Jun 11, 2010 #3
    Marcus, using the idea of a singularity and it's infinite density doesn't necessarily show that GR is wrong though, does it? It shows that it's predictions are only compatible with a continuous space. In other words, singularities would be possible if space were not discrete, right?

    Thanks for taking the time to answer my question.

  5. Jun 11, 2010 #4
    Pediciting singularities shows that it's hit the boundaries of its applicability. If GR predicts singularities for the systems it models - smooth manifolds (continuous space in your terminology), then something about the "gravity-is-modelled-by-GR-on-a-smooth-manifold" scenario is incomplete.
  6. Jun 11, 2010 #5
    In the new papers by Ted Jacobson he seems to talk about actual physical singularities though. Alluding to Penrose and Hawking. To quote Arxiv blog:

    "Astrophysicists are different. They have such extraordinary faith in their theories that they believe singularities actually exist inside black holes. The likes of Roger Penrose and Stephen Hawking have even proved that singularities are inevitable in gravitational collapse."
    -- http://www.technologyreview.com/blog/arxiv/ [Broken]

    What do they mean by that then?

    Last edited by a moderator: May 4, 2017
  7. Jun 11, 2010 #6


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    Adam what Sheaf says is right.
    In physics the normal thing to do when a math model stops giving you reasonable numbers at a certain point is you don't trust it around that point. You consider it inapplicable.

    This has happened with other theories in the past. People found they had "singularities"---which means points where they break down, blow up, stop computing. So that just means the theory has a "limited domain of applicability". You apply it only where it works and keep well away from places where it blows up.

    Eventually the flawed or incomplete theory (in those other cases) was replaced by an improved theory that got rid of the singularity. A famous example is the theory of thermal radiation (the heat glow of hot objects) which had infinities until around 1900. Those singularities were fixed by Max Planck who replaced the flawed theory by one that didn't have those failure-points---the new theory didn't blow up.

    There was a somewhat similar problem with the earliest models of the hydrogen atom that had the electron orbiting the nucleus. That problem was fixed too. The old theory had glitches and got replaced by a new one that didn't have the problem.

    In the case of cosmology, commercial media and science popularization books have spread bunk and misconception about singularities amongst the public. The recognized failures in the vintage-1915 (socalled "classical" GR) theory are just a sign the theory has a limited range of applicability.
    There are already several candidate replacement theories, in various stages of development, which get rid of those particular glitches.

    Try reading "A Tale of Two Big Bangs" at the Einstein-Online website. They give more detail than I want to.

    Eventually the newer nonsingular cosmology will get out into the popular book market.
  8. Jun 11, 2010 #7


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    Heh heh, what does who mean? The MIT magazine article is not professional and doesn't even give an author. I looked at the Jacobson Sotiriou papers cited and they say nothing about singularities actually existing in nature. Or about "gazing into infinity"!
    It is not true that Penrose Hawking proved singularities exist in nature. They proved they must occur in the classical (1915) GR model.

    I have heard Jacobson say outright (in professional company at a KITP workshop) that he does not believe there is a singularity at the pit of a black hole. We don't know, but he says his hunch is that time-evolution continues---no blow-up. What is interesting is to figure out what happens instead of a singularity. There was a 2-week workshop at Santa Barbara KITP about just this thing, and Jacobson was one of the star players. The video is still available online, or was the last time I checked.

    You have to realize you can't trust journalism like that MIT magazine thing. It is not a technical or scholarly paper. Since it doesnt say who the author was, nobody's reputation is on the line. It is garbage.

    Just to make it convenient for people to see. Here is what the "Technology Review" anonymous article said:

    "To any ordinary physicist, a singularity is an indication that a theory has broken down and some new theory is needed to describe what is going on. It is a matter of principle that singularities are mathematical objects, not physical ones and that any 'hole' they suggest exists not in the fabric of the Universe but in our understanding of it.

    Astrophysicists are different. They have such extraordinary faith in their theories that they believe singularities actually exist inside black holes. The likes of Roger Penrose and Stephen Hawking have even proved that singularities are inevitable in gravitational collapse.

    For them, removing the event horizon around a black hole raises the exciting prospect of revealing a singularity in all its naked glory. When that happens, we will be able to gaze at infinity.

    That seems bizarre.

    That is utter bull. It is not true that "Astrophysicists are different" from other scientists. If you look at the literature the overwhelming bulk of "quantum cosmology" papers are about fixing the theory, what happens instead of the singularity. It is not true that Jacobson and
    Sotiriou are talking about "naked glory" and "gazing at infinity".

    They are talking about a way that one might in principle destroy the horizon and get a look at what is really there instead of "infinity". At least I thought so when I read the two papers! Could be wrong, why don't you have a look?

    I posted the two Jacobson Sotiriou links and abstracts two days ago, in the quantum gravity biblio thread. If you want you can check them out.
    Last edited by a moderator: May 4, 2017
  9. Jun 11, 2010 #8
    Yes I actually read those papers before the article on the blog. That's not to say I understood them though. :). I follow what you post eagerly and even though most of it is way over my head, sometimes there's some extremely interesting stuff there. It beats reading about politics or sports any day. Anyway, they actually say the same thing the paper which is where they more or less just cut'n'pasted it from:

    "But are spacetime singularities physically relevant or are they simply mathematical peculiarities of special solutions to Einstein's theory? As a matter of fact, Penrose and Hawking have managed to show that singularities are not only relevant but actually inevitable in gravitational collapse."
    ---- http://arxiv.org/abs/1006.1763

    But yes, I understand that it's probably a case of the theory "breaking down", not an actual physical "singularity". Whatever that is... :)

    Black holes, I think, are the most intriguing and interesting things in the entire universe. I'm actually a little sad that answers probably won't come in my lifetime.
  10. Jun 11, 2010 #9


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    You are right about interpreting what they say in that paper---a submission to the FQXi essay contest. I was wrong in my interpretation of what they seem to be saying.

    Now I am puzzled. Maybe they mean something new by "singularity", maybe what they are talking about is what arises in nature instead of the infinite density, infinite curvature glitch.

    It could be that my concepts are obsolete in this case. Could singularity mean something new, that we don't yet know what it is?

    I should apologize for an apoplectic moment of strident overstatement. Your reading sounds modest and levelheaded, Sbrothy.

    But "infinite density/curvature" doesn't mean anything to me. At the relevant scale geometry would be quantum and we don't know what spacetime is in quantum regime, or so I imagine. The question is still, for me, what in reality replaces the classical singularity?
    And if they want to call it a singularity----a new kind of singularity, say a "quasi-singularity" or a "quantum spacetime singularity"---I can't object. I have great respect for Jacobson.

    I wish they'd use a different word because traditionally it means a breakdown of some manmade theory, or the place where the breakdown occurs---not something in nature.

    What he actually said at the KITP workshop, with Steven Shenker pushing him to hazard a hunch, is that he thinks "time evolution continues."
    At the pit of a black hole, where classically the time-evolution stops, where timelike geodesics terminate, there (said TJ) in a quantum treatment they wouldnt end. Time-evolution would continue on thru somehow. If pushed to say, that was his hunch.

    Maybe we are coming to where we will apply the term "quantum spacetime singularity" to that situation----where the classical singularity has been avoided or resolved---where in a conventional mathematicians sense there is a non-singularity :biggrin:

    I should get the link to that KITP discussion session that TJ co-led, in case anyone is interested.


    The discussion started out being led by Gary Horowitz and then TJ took over halfway through, around minute 30:
    The critical segment was minute 42-45. Steve Shenker pressed TJ to say what his hunch was. The discussion got lively. His hunch was time evolution continued thru the pit of a black hole---into something. The word "quasi-singularity" was used, I think by Shenker or maybe Horowitz.
    If you download the quicktime movie you can then drag the pointer to minute 42 and start there. You don't have to watch the first 42 minutes unless you want to.
    Last edited: Jun 11, 2010
  11. Jun 11, 2010 #10
    Thats a quantum viewpoint.

    There are lots of different types of singularities just like there are lots of different types of infinities. And I believe many mathematicians and physicsts would have trouble agreeing on what either means. In any case, both quantum mechanics and general relativity breakdown at big bang and black hole singularities. So nobody has any real idea whats going on at either singularity. Ansd remember it was Einstein who firmly believed that black holes did not even exist, and produced a paper explaining why....so everybody gets bedazzled.

    yes, another piece of the puzzle.

    If space and time really transform as GR says, I don't know why time evolution continues.
    My own amateur hunch is that space, time, energy and forces all have a single origin that has yet to be discovered. It's incomprehensible to me that the strong, weak electromagnetic forces would be reconciled, but gravity,for example, is something different....that's an ugly idea for me.
  12. Jun 12, 2010 #11
    I think that's definitely the case, because in the paper they say

    So I read it as a location where you will be able to view new (i.e. not described by GR) physics, rather than a location where you will have infinite curvature.
  13. Jun 12, 2010 #12


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    Thanks, sheaf. That sums it up nicely. So presumably no "infinite density" stuff (as suggested by Adam in post #1.)
  14. Jun 12, 2010 #13


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    I read it as "Are the solutions of the Einstein field equations with singularities relevant for describing what we observe?".
  15. Jun 14, 2010 #14
    Due to holographic principle all information of the Black Hole have to be encoded in its Even Horizon in a number equal its surface / 4 Planck length squared. Therefore the infinity density in a singularity is not possible.
    If the amount of the information (mass) increases the density decreases.
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