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I often hear people say that general relativity predicted its own demise because of the singularities it predicts. If I'm not mistaken, this is also a problem in QFT. I wonder why singularities garner so much hate. To me, it seems like negative or complex numbers: we used to hate these things but now they are more generally accepted. Why can't it be that nature just has singularities? We might not get it, we might not like it, but it may be so.

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    $\begingroup$ Most view singularities as flaws in their mathematical apparatus. No scientist is going to be comfortable with such. $\endgroup$ – Lewis Miller Aug 3 '16 at 2:54
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    $\begingroup$ I am not scared of ghosts, either. Just like singularities, they don't exist. $\endgroup$ – CuriousOne Aug 3 '16 at 3:34
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    $\begingroup$ Maybe it's just because we cannot deal with singularities/infinities. We cannot see them, we cannot manipulate them &c. We just don't encounter such "anomalies" anywhere in the "real" world, hence our models of how things work in the "real" world just don't apply there. That makes them hard to deal with mentally, because it is non-trivial to find the rules in something that's so much inaccessible. Because of that, there are often more than one possible answer to questions we have, and it's hard or impossible to say which one is correct. That is uncomforting to many. $\endgroup$ – JimmyB Aug 3 '16 at 11:49
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    $\begingroup$ @CuriousOne I'm not sure if I belive in singularities or not but the only thing that scares me is Keyser Soze. $\endgroup$ – David Richerby Aug 3 '16 at 13:36
  • $\begingroup$ I ain't afraid of no ghost either. Ghosts are so useful though. Have you met Faddeev-Popov the friendly ghost? $\endgroup$ – AHusain Aug 4 '16 at 21:40
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"To me it seems like negative or complex numbers. We used to hate these things but now they are more generally accepted. "

Indeed. And in a general context, the infinite answer that some equations return might not be a problem. We have all kinds of rigorous notions of infinite quantities in set theory; see, for example, aleph and beth numbers and infinite ordinals) and topology: for example compactification of sets through the addition of a point at infinity. In the context of the Riemann sphere, the $z_\infty$ returned by a meromorphic function at a pole is altogether reasonable - it's a point on the sphere utterly like any other.

Crucially, though, in physics, most numerical results of theories either correspond to measurements, potential measurements or at least influence potential measurements in such a way that singularities imply infinite readings for potential measurements. Most physicists would agree that an infinite voltage is not a realistic measurement from a voltmeter.

Physics is not mathematics: it is mathematics together with the need to make descriptions wrought in mathematical language correspond to real world, experimental observation. Since no one has ever witnessed an infinite reading (unless you do something really daft and decide to define a voltage scale that is the tangent of an SI voltage reading in volts, for example), we reason inductively (not mathematically inductively) that no reasonable measurement will ever be infinite.

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    $\begingroup$ I would add that we can also usually dismiss particular theories that predict infinities even if we don't directly measure them, since infinities tend to propagate and we don't observe the effects. If there were an infinite voltage source somewhere in the universe, it would produce infinitely energetic electrons which would radiate an infinite luminosity which would vaporize the planet in an infinitesimal time. I think a lot of people don't appreciate that you can't hide infinity by tucking it away in a quiet corner of the universe. $\endgroup$ – user10851 Aug 3 '16 at 6:03
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    $\begingroup$ @JanDvorak: If the infinity happened in the past then surely it could not possibly be infinite, only very, very large. If it was truly infinite it would never be exhausted and would continue to happen now and into the future. $\endgroup$ – slebetman Aug 3 '16 at 10:22
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    $\begingroup$ @JanDvorak but then they would have to have been formed after the inflation period. The homogeneity of the CMB is a good indication that there was nothing too crazy in a very big chunk of the early universe. $\endgroup$ – Davidmh Aug 3 '16 at 10:35
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    $\begingroup$ But this brings up an interesting point, as you yourself said, if you measure something ridiculous like "tangent in the voltage" you could end up with an infinite measure. How do we know that our intuitions of "mass" and "energy" aren't the same way related to the most natural/fundamental quantities to measure in the universe? Perhaps singularities have infinite amount of each, but if one measures say $InverseAckermann(e^{ENERGY/e^{MASS}})$ that happens to be well behaved everywhere. $\endgroup$ – frogeyedpeas Aug 3 '16 at 17:18
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    $\begingroup$ @frogeyedpeas, that would be an example of a coordinate singularity . $\endgroup$ – Solomon Slow Aug 3 '16 at 17:38
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Those infinities can mean the breakdown of the theory. Blackbody radiation predictions using classical theory says that a blackbody will radiate an infinite amount of energy. It meant the theory breaks down, and it was only able to be explained right with quantum theory.

For General Relativity (GR) it is the same situation. At true singularities (not coordinate singularities which transfer away under a change of coordinates, like the black hole horizons, which are not the same problem, though since they tend to hide singularities maybe also means something useful for quantizing gravity) some of the scalar curvature values will become infinite, and since scalar this can not be fixed with a coordinate transformation. We need a quantum gravity theory to explain the black hole singularities. Or something.

But in GR it is not simply that, it also means that the spacetime where that singularity exists is incomplete - at the singularity the geodesics simply end. If you are traveling on one, and manage to still be something as you near the singularity, in very short and finite proper time your worldline will end. Like they tend to do in your palm, but this would be real, if it existed. Well, that's not so bad because nobody is stupid enough to go there.

But it has other physical implications on that spacetime. It also says that the spacetime is not predictable, that causality is violated there. And if geodesics a-causally end at the singularity, anything could also come out of the singularity, and extend over that spacetime. It would just not be causally safe. A dragon could come out or your 10,000 years old ancestors riding a rocket ship. That spacetime becomes acausal.

It is causally disconnected from the spacetime outside the black hole due to the horizon, so none of that makes any difference outside. But if there was a black hole without a horizon, then all of the connected spacetime would be acausal. Such a singularity is called a naked singularity. There is a censorship principle that says that there are no naked singularities. It's a conjecture, not proven mathematically, once in a while somebody finds a possible but pretty strange and typically considered not physically plausible solution of the GR equations, with a naked singularity.

I still agree with @CuriousMind, real singularities probably don't exist. But it does mean that the currently best known theories cannot be all right, that there needs to be a solution to the singularity problems, probably with a still unknown quantum gravity theory.

Hope you now don't fret too much if you might not be living in a causal spacetime. But if it drives you to find the solution, good for physics.

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  • $\begingroup$ The thing is, GR singularities aren't really singularities, are they? The wordlines don't end - as the spacetime gets more and more curved, time also spreads more and more, so nothing can ever reach the "singularity". And when you add quantum evaporation, the wordline will eventually be "liberated" as the spacetime flattens over time. They seem more like the result of a simplification - ignoring how the mass/energy gets to the singularity itself. The mass/energy collapse can proceed forever in GR, but it will never result in a "real" singularity. $\endgroup$ – Luaan Aug 3 '16 at 9:12
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    $\begingroup$ @Luaan: the worldlines do indeed end. An observer falling into a Schwarzschild black hole for instance, reaches the singularity at finite proper time, in a pure GR system. That's why GR people are uncomfortable about them. $\endgroup$ – tfb Aug 3 '16 at 11:00
  • $\begingroup$ @tfb In a static universe, you are right. But it still kind of nods more toward the "the universe isn't static" to me rather than "GR is wrong, because it matematically allows a singularity to be described". Don't get me wrong, I get that there's a lot of issues that crop up when reconciling GR with QM, but it doesn't seem to me that the problem is present in pure GR. For example, the wordline in a pulsating universe might be "liberated" to a new Big Bang (granted, GR isn't used for describing such early universe, but...). But ok, that still makes the model uncomfortable to an extent. $\endgroup$ – Luaan Aug 3 '16 at 11:29
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    $\begingroup$ @Luaan Singularities (in the sense of geodesic incompleteness) form in rather general cases in GR unfortunately. There's certainly no requirement for the universe to be static. They really are a problem for GR (although less of one if it could be shown that they were always censored, of course). $\endgroup$ – tfb Aug 3 '16 at 12:48
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To me it seems like negative or complex numbers.

Not at all. Negative or complex numbers are useful tools that, while maybe not having a direct equivalence in physical terms (you cannot have "-3 atoms" in a volume and no ruler will give you complex numbers as a readout), they are just a mechanism which collapse down to "realistic" numbers at the end, if you are so inclined to actually perform a real world calculation.

Note that not every infinity is a singularity. Some infinities have their uses; be it in infinitesimals (=> differentials, integrals), real/imaginary/transcedental etc. numbers (no matter whether they physically exist in the real world), or thinking about what happens if you keep going and going and going... those are just mathematical tools, based on the general concept of infinity, to get the job done.

Not so a singularity. 1/0 can never, by any means, be undone by further calculations - it's over. Undefined. Wall of bricks across the street. Yes, you can get close by shrinking some Epsilon, but that's not what people are talking about. They are interested in what is actually physically there. Never being able to reach that spot which is clearly encapsulated by an Epsilon-volume is not fun.

This is all very much in layman's terms. But it might show why people view singularities differently than those other phenomena.

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    $\begingroup$ My thermometer reads negative measurements for like a quarter of the year. $\endgroup$ – industry7 Aug 3 '16 at 17:04
  • $\begingroup$ In K? ... pretty sure you knew exactly what I meant. ;) $\endgroup$ – AnoE Aug 3 '16 at 18:46
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    $\begingroup$ As an aside, it's a trivial exercise to modify a ruler to give complex readouts. You don't multiply lengths to get a length, and without such a multiplication, there is no meaningful distinction between the real line or the imaginary axis or any other line through the origin of the complex plane. $\endgroup$ – user5174 Aug 3 '16 at 20:09
  • $\begingroup$ @AnoE I'm actually not exactly sure what you meant. I think that what you meant was that negative numbers don't have a physical meaning in cases where the measurement of a physical phenomenon is defined such that negative numbers don't make sense. And btw, scientists have actually measured temperatures below zero K: “The inverted Boltzmann distribution is the hallmark of negative absolute temperature; and this is what we have achieved,” says Ulrich Schneider. $\endgroup$ – industry7 Aug 25 '16 at 16:09
  • $\begingroup$ My comments regarding negative/complex were meant to pick up the train of thought of the OP: like negative or complex numbers: we used to hate these things. $\endgroup$ – AnoE Aug 29 '16 at 8:44
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Science is based on observation, and every event horizon shows that observation is restricted by some screens, and that we will never be able to observe what is found behind the screen (except if we are travelling ourselves behind the event horizon, with awkward consequences).

The lack of tolerance of singularities by science is also due to a lack of correct understanding of the principle of relativity: The twin paradox shows that observation and reality are two things, observation does not correspond always to reality: The twin having made a round trip near speed of light is observed having returned after 20 years, but in reality he aged only by 5 years. It is exactly the same principle which makes us observe infinities near event horizons. The twin paradox is a case of special relativity, and the event horizon is a case of the Schwarzschild metric.

If we measure and observe infinite values, that does not mean that there i s infinity, and general relativity does not "break down" when some event horizon prohibits the observation of certain regions of the universe.

Spacetime is relative, and our spacetime is our manifold of observation. If our observation is not able to follow certain geodesics that does not mean that this geodesic is not following the laws of GR (where it is not excluded that the laws of GR are completed by some other laws, quantum theoretical or not).

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    $\begingroup$ The twin paradox shows that there is no objective time. We have two different observations that are equally valid - but both correspond to reality. Saying that the observation does not correspond with reality is like saying "He measured 120 ml, but I measured 0.12 l - clearly, the observation doesn't correspond with reality." $\endgroup$ – Luaan Aug 3 '16 at 8:45
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    $\begingroup$ Sigh. "Science is based on observation and..." ...and until we have more acquaintance with seemingly impassable things we now imagine as "event horizons", we can have only mild, weak predictions how they really behave and what hides inside them. Period. $\endgroup$ – kubanczyk Aug 3 '16 at 9:05
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    $\begingroup$ So? How does that make one of the observations not correspond to reality? The universe doesn't have objective time. Both observations correspond exactly to reality. Does the image produced by a gravity lens not "correspond with reality" just because the light went on a wordline that wasn't "straight"? It may not correspond with your model of reality, if you assume there is such a thing as objective time, but that's a problem of your model, not the observation nor reality. $\endgroup$ – Luaan Aug 3 '16 at 9:06
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    $\begingroup$ @kubanczyk: Yes, GR does not permit predictions about the inside of the screen. But the fact that predictions are not possible is just a natural limit of GR and does not mean that GR "breaks down" in the sense that it must be abolished and replaced by something else. $\endgroup$ – Moonraker Aug 3 '16 at 9:12
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    $\begingroup$ Note that the event horizon is not the singularity... $\endgroup$ – AnoE Aug 3 '16 at 11:20
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In some sense infinities are an integral part of renormalisable field theories. There measurable quantities can be computed naively and turn out to be infinite. This can however not be right because no measuring apparatus goes up to infinity. Put another way, particles propagate, they must have a finite mass. Finiteness of measurements is a (somewhat tautological) experimental fact. Field theories are therefore 'fixed' by absorbing the infinities in quantities that are not observable.

In a Renormalisation Group (RG) language, at every (spatial) scale there is a different effective theory. The theory at the observation scale contains (by definition) finite parameters. When RG flow equations are solved and smaller scales are probed, it turns out that some parameters grow without bound. Therefore, if space is continuous, then the truly microscopic parameters are infinite. If you insist on defining a mass for the electron at the smallest possible scale, it must be infinite. We can't really measure it, but we can measure how it grows as the observation scale $s$, is lowered, $$m(s) = \frac{m_0}{s^{\eta}} \, .$$ $\eta$ is a positive exponent. $m_0$ is finite and measurable. $m(0)=\infty$ would be the microscopic mass.

I think that we don't run away from infinities. It is just that by definition they are not measurable. Finite numbers can be compared to each other. They are therefore measurable. Infinities on the other side are all the same. Sometimes they can be converted to finite numbers with constructions such as the RG or the Riemann sphere and related to something physical.

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It is entirely consistent with the mathematics of general relativity that the universe does not exist at or beyond a 20,000 kilometer sphere surrounding the Earth. Since there is no universe at that boundary, GR can make no predictions about what happens there, and thus allows any arbitrary thing to happen there.

Obviously this is unsatisfying, but it also has a trivial fix: the mathematics of GR can be continuously extended to and beyond 20,000 sphere, so we simply insist that it does so. We have no reason to believe otherwise, anyways.

There is (likely) a boundary infinitely far future and another infinitely far and past that would provide an obstacle to this continuation process, but that's okay because it's infinitely far in the future or the past, so isn't really relevant to anything.

The problem with the type of singularity that people make a big deal of in the context of GR is that it is forms an obstacle that can be reached in finite time.

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    $\begingroup$ If the universe does not exist 20,000 km from the Earth, that excludes the Sun. Are you claiming that the Sun does not exist? $\endgroup$ – sammy gerbil Aug 3 '16 at 20:57

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