I am a layman in physics and just read about black holes on the internet. I read that matter encounters geodesic incompleteness in the singularity in an uncharged black hole.

I heard an analogy of geodesic incompleteness as a straight line on a paper reach a hole on the paper, so it cannot continue. But in this analogy, isn't the straight line possible to continue into 3D (continue down the paper)? So, if matter reaches the singularity, is it possible too (to reach another dimension)?

I also heard that the matter is annihilated when reaching the singularity, does it mean it disappear from this world, and violates conservation of energy?

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    $\begingroup$ Couple of things to point out: 1) don't be adding dimensions to an analogy (I mean that both figuratively and literally). An analogy only works, and is only intended to work, in the exact circumstances in which it was posed. The line on the paper can't jump to 3D because the reason for having a line on paper is to exclude the 3rd D. Including it makes the analogy meaningless. 2) If matter is annihilated, it is usually transformed into energy, which preserves conservation of energy. However, in extremely curved space (black holes fit the bill), energy is not necessarily conserved $\endgroup$ – Jim Sep 1 '17 at 12:19
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    $\begingroup$ Yeah, I'm just saying don't extend analogies in general. They aren't intended to be accurate if extended and any conclusions from extensions of them are likely to be wrong. Also, in cases that energy isn't conserved, it no longer has to go anywhere. If it isn't conserved, the total amount simply changes. It's like asking where fire goes when you put it out. The total amount of fire isn't conserved; it doesn't go anywhere. You just put it out and no more fire. Similarly, if the total energy isn't conserved, energy doesn't come from or go anywhere, there's just a different amount $\endgroup$ – Jim Sep 1 '17 at 12:38
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    $\begingroup$ No, my fire analogy wasn't looking at the energy in fire; it was looking at the fire itself - the flame, not the constituents. Around us, energy is conserved; you can follow where it comes from and goes. The number of fires burning on Earth is not conserved; I can easily put a fire out without any explanation for where the missing fire is. Similarly, in the case where energy is not conserved (like in heavily curved spacetime), we could add or remove some without it going anywhere. If it went somewhere, it would be conserved and we are looking at the case where it is not conserved. $\endgroup$ – Jim Sep 1 '17 at 13:17
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    $\begingroup$ as for whether or not the law of conservation of energy exists in curved spacetime, that's another matter. You only adhere strongly to it because you've been told it always holds and have never seen a counterexample. But general relativity actually shows us circumstances where total energy conservation is not a physical law anymore. You are not going to encounter them on Earth, but they exist and you can find numerous questions on this site detailing the mathematics $\endgroup$ – Jim Sep 1 '17 at 13:20
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    $\begingroup$ For the record, I totally agree that it's just like magic. Which is further reason for you to dive into it and learn the whole thing; then you'll know magic. The thought that I know magic makes me smile every day $\endgroup$ – Jim Sep 1 '17 at 13:41

Strictly speaking geodesic incompleteness doesn't mean the worldline of the particle ends at the singularity, but rather that we can't predict what happens to it. The trajectory of a freely falling particle is given by an equation called the geodesic equation:

$$ \frac{d^2x^\alpha}{d\tau^2} = -\Gamma^\alpha_{\,\,\mu\nu}\frac{dx^\mu}{d\tau} \frac{dx^\nu}{d\tau} $$

It's a scary looking equation but you don't need to understand all the details to see what the problem is. What happens at the singularity in a black hole is that some of the parameters $\Gamma^\alpha_{\,\,\mu\nu}$ become infinitely large and we're left with an equation that has infinity on the right hand side. Since we can't do arithmetic with infinity (because it's not a number) we have no way to calculate the trajectory of the particle at the singularity.

Incidentally much the same happens when we try to work backwards in time towards the Big Bang, and that's why it's commonly said that time started at the Big Bang. See my answer to How can something happen when time does not exist? for more on this.

Anyhow, the upshot is that GR cannot tell us what happens to matter falling into a black hole when it hits the singularity. However most of us believe that general relativity ceases to be a good description of the physics when we get close to the singularity and some form of quantum gravity theory will take over. The trouble is that we currently have no theory of quantum gravity.

  • $\begingroup$ If geodesic incompleteness occurs just because we cannot calculate it, then quotes like“here ends spacetime ” or “the matter is annihilated” is misleading, as we just cannot know where they are, but not disappear from this world, isn't? $\endgroup$ – lyk Sep 1 '17 at 11:32
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    $\begingroup$ @lyk: yes statements of that sort are misleading, but the trouble is that any popular science level statement about GR is likely to be misleading to some degree. Whether it really matters is debatable. $\endgroup$ – John Rennie Sep 1 '17 at 12:09
  • $\begingroup$ Also, does the trajectory in your answer means trejectory of time?I am also a little convinced of that continue to calculating the trejectory of time is a bit useless as even through we can calculate the trejectory of particle in singularity, the trejectory after wards will be same as the metric will still be infinity even it continue to increase.Am I wrong? $\endgroup$ – lyk Sep 1 '17 at 13:40
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    $\begingroup$ @lyk: it isn't a limitation of the maths. Most of us believe infinities don't actually happen in nature, and when we get infinity in our calculations it means we're using the wrong theory. In this case it probably means we should be using some form of quantum gravity theory. $\endgroup$ – John Rennie Sep 1 '17 at 15:13
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    $\begingroup$ I don't think this is right. Although it is possible to have geodesic incompleteness in a case where the spacetime could be extended in order to allow the geodesic to continue, normally we are not interested in such things, and they are not normally considered to fall within a careful definition of a singularity. There is a nice, very readable paper on this, "What is a singularity in general relativity?," Ann Phys 48 (1968) 526. then quotes like“here ends spacetime ” [...] is misleading No, not misleading at all IMO. This is a very accurate characterization. $\endgroup$ – user4552 Sep 1 '17 at 18:05

The answer is we don't know. I think it is uncontroversial to say that the prediction of singularities by GR is a sign that the theory is failing: we don't expect there actually to be singularities. But we don't have a theory which works (which does not predict singularities in other words) where GR predicts singularities -- such a theory would need to unify QM and GR -- so currently the best we can say is that we don't know.

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    $\begingroup$ A true scientist willing and happily admits what they don't know. It's what we don't know that drives us to learn! $\endgroup$ – corsiKa Sep 1 '17 at 16:14
  • $\begingroup$ "such a theory would need to unify QM and GR" - String theory does that right, doesn't that work? $\endgroup$ – Paul Sep 1 '17 at 20:00
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    $\begingroup$ @Paul String theory isn't a theory: it's more like a framework for theories, one of which might be a precursor to a working theory. So no, it doesn't. $\endgroup$ – tfb Sep 1 '17 at 20:09
  • $\begingroup$ But does it mean that black holes (in which GR describes them) don't exist? $\endgroup$ – user168013 Sep 1 '17 at 20:22
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    $\begingroup$ @rus9384 That's a complicated question. I think it's clear that singularities are physically implausible, but also clear that objects which are essentially indistinguishable to distant observers from the black holes which GR predicts do exist. $\endgroup$ – tfb Sep 1 '17 at 20:39

General relativity does not tell us much. In the Schwarzschild metric the singularity is a spatial surface. I use a diagram for the truncated Schwarzschild metric with hawking radiation. The $r~=~0$ singularity is continuous with the exterior region when the black hole quantum evaporates or explodes.

Fava and others have proposed the singularity of a black hole as a condensate of tachyons. Tachyons are these oddities which are thought in some ways as particles which move faster than light. However, it is more that they are a case of a completely unstable vacuum. The invariant interval is given by $$ (mc^2)^2~=~E^2~-~(pc)^2. $$ I will set $c~=~1$. For standard relativistic motion we have $m^2~\ge~0$, where it is zero for massless particles. For the tachyon $m^2~<~0$.

We now consider the Klein-Gordon equation from the invariant interval with $E~\rightarrow~i\hbar\partial/\partial t$ and $\vec p~=~-i\hbar\nabla$ so the invariant interval is the field equation $$ \frac{\partial^2\phi}{\partial t^2}~-~\nabla^2\phi~=~m^2\phi. $$ We then have that $m^2$ is a sort of potential and for the tachyon it is negative. This means the vacuum is completely unstable. This may then be a model for the singularity of a Schwarzschild black hole. It is completely unstable and explodes in the exterior region as Hawking radiation.

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  • $\begingroup$ This sounds like crank material. Has this been published anywhere? $\endgroup$ – user4552 Sep 1 '17 at 18:06
  • $\begingroup$ Vafa proposed tachyon condensates and there is Sen, Ashoke "Tachyon condensation on the brane antibrane system". JHEP. 8 (8) (1998). $\endgroup$ – Lawrence B. Crowell Sep 1 '17 at 19:06
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    $\begingroup$ The Sen paper, arxiv.org/abs/hep-th/9805170 , doesn't say anything about black holes or singularities. $\endgroup$ – user4552 Sep 2 '17 at 0:13

Nobody knows what's real, and you can do worse than thinking about an analogy involving sheets of paper, so let's just fix that analogy.

as a straight line on a paper reach a hole on the paper [...] But in this analogy, isn't the straight line possible to continue into 3D (continue down the paper)?

Don't think of it as "punch a hole into a sheet of paper". Your image should be more of a very deep, in fact infinitely deep well made by the sheet of paper. It goes ever on and on (in the 3D space into which the paper is embedded).

In this analogy, your line does not have to end, it also does not have to re-appear on the other side of the well (which is infinitely deep), and it will not leave the paper either, and nothing in particular will happen to it.

On a side note, this analogy extends to give an idea how wormholes could be imagined - instead of an infinitely deep well, curve it around and attach it to another bit of your paper...

Again, I have to stress that these are all just imagery or illustrations to guide your mind into having anything to think about. It would be very weird if the complex real world would function as simply as a sheet of paper.

By the way, I like another analogy more than the paper: a reasonably big lake of water with fish with it and a hole in the middle through which the water can flow out freely. Very near the hole, the water flows faster than the fish can swim (="speed of light") with all the consequences you can imagine. Obviously, nobody knows what happens to the fish when it is finally sucked in. :)


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