A few days ago, Roger Penrose and two less famous co-recipients gave the 2020 Nobel Prize lecture.

The Nobel Prize is a relatively famous award, arguably the most famous one in the world, and Penrose is one of the most famous living mathematical physicists. Still, the video above has 16,000 views now; this version of Despacito has over 7 billion views. The ratio is about half a million which places the public interest in science into a perspective.

And indeed, I haven't listened to the Nobel Prize lecture, either, arguably because I think that there would be nothing really new and interesting to learn (which may be a wrong expectation!). The Penrose-Hawking singularity work was done basically half a century ago and the fate of singularities according to the state-of-the-art theoretical physics has profoundly changed in several subsequent minirevolutions, most of which are unknown to Penrose himself.

A singularity is a point in space, either the real \(\RR^3\) space or the \(\RR^4\) spacetime or a moduli space or basically any other space, where some quantities are ill-defined, usually infinite in a limit. The previous sentence contains the word "infinite" and just like this word, the word "singularity" induces some religious feelings inside a layman (including a physicist-beginner).

Why? You know who is also infinite? God. He is great. The human stupidity and the Universe are infinite but I am not sure about the latter, Einstein properly quipped. We are so small relatively to the giant Infinity. We can't get there. We can't beat it. We must be humble. Infinity may be inconsistent in mathematics etc.

All these feelings are great and they represent the overlap of mathematics, science, theology, and philosophy. I haven't avoided them, either. At the same moment, mathematics and science ultimately becomes rational about everything including infinities and singularities. The laws of mathematics and physics and the tools to think about all the topics "beat" infinities and singularities – much like they tame God as long as He is meaningful or worth a discussion.

So you may discuss functions such as \(y=1/x\) which have a singularity, like one at \(x=0\) in this case. Singularities become a dull thing. They may also be classified. Something breaks down at these points but such a breakdown isn't the end of the world. It's just the end of quantitative discussions about the behavior at the given point – which is no catastrophe, partially because "almost all points" are away from this singularity (and the word "almost" may be given a rigorous meaning in measure theory). Similarly, manifolds may involve points where the geometry is not smooth. Those are singularities as well, some curvature invariants are infinite at these points, too.

One may ask whether singularities are physically possible. Again, rich people like these discussions because they make them feel metaphysically deep and theological. For example, Ray Kurzweil's ideas about the technological singularity represent a pile of clearly invalid gibberish that makes some billionaires feel like Jesus Christ, the carriers of some truly qualitative paradigm shift (that cannot arrive, partly because there always exist new processes that slow down any exponentially growing process in the real world).

Most laymen still imagine the black hole to be an infinitely dense object. The black hole is all about the singularity, they think. This view is completely wrong. The singularity is just a nearly untestable (and therefore uninteresting) episode at the "very end of the life" of the infalling observer. Instead, the truly shocking and defining traits of the black hole already start at the event horizon. The event horizon is what defines a black hole (the event horizon makes it a hole and it makes it black, too) – and the curvature and the matter density near the event horizon is perfectly finite. And the event horizon, the boundary between the interior and the exterior, plus the relationship between the interior and the exterior is what is responsible for the bulk of the information loss puzzle; the singularity is nearly irrelevant.

Nevertheless, Penrose and Hawking have used the singularity as an ally to prove that black holes unavoidably arise in rather generic astrophysical situations. The curvature invariants exceed any "classically reasonable" threshold which means that in a classical approximation which is still accurate enough, a singularity unavoidably arises. With some extra assumptions (that eliminate the naked singularities, in the sense of Penrose's Cosmic Censorship Conjecture), it is guaranteed that these unavoidable singularities are covered by the unavoidable event horizon, too.

Three days ago, Suvrat Raju, a brilliant physicist, my former TA, and a Marxist, posted a 156-page-long review of the black hole information puzzle. It's called a review but make no mistake about it, the title may be considered misleading because a "review" may be expected to contain "utterly uncontroversial knowledge". Lots of the stuff in that review is controversial enough (among researchers) although I agree with almost everything; many conclusions are extracted from very recent papers (the key ideas are partly conclusions, partly assumptions of the recent papers; the logic isn't "fully proven" yet). One of the conclusions that Suvrat considers settled is that "the black hole interior is completely dispensable", as I often emphasized. It's just a dead end street. Equivalently, a copy of all the physics is always encoded in the degrees of freedom that may be fully tied to the exterior.

Because the infinity is ill-defined (it can't appear on a voltmeter or any other "quantitative apparatus" used by an experimental physicist), amateur (and bad) physicists tend to jump to a premature conclusion. Infinities and singularities are totally evil, inconsistent, and they have to be banned. Everything must be smoothened out around the putative singularities. The failure to do so is politically incorrect.

Well, this is a wrong opinion based on a sloppy reasoning and exaggerations. Singularities appear in legitimate physics and they may be exact. For example, some heat capacity etc. may be infinite near a phase transition. This particular singularity may be blamed on an approximate, limiting theory – thermodynamics or statistical physics. If you send the number of molecules \(N\to\infty\), the functions describing the behavior of matter become really "pure" and they exhibit a singular behavior near the phase transitions. If you realize that you have a finite number \(N\) of atoms and if you study some quantities describing these atoms accurately, you will find out that the quantities become smooth in the very vicinity of the would-be singularity, too.

However, singularities and infinities appear even in quantum mechanics, a theory that properly explains why the atoms exist in the first place, a theory that is expected to make "everything smooth" and "ban singularities". Well, if you think about this "summary of quantum mechanics and its impact on physics", this expectation is very dumb for an obvious reason. Yes, it is true that quantum mechanics introduces atoms and it generally says that matter cannot be infinitely fine (and the electron can't sit infinitely close to a proton), and that is also why you cannot trust the formulae for the heat capacity etc. in an arbitrarily vicinity of the phase transitions. But does quantum mechanics make everything smooth? Not at all. The very term "quantum mechanics" was chosen because the energy (of an atom, for example) jumps discontinuously. And a discontinuous jump means an "infinite energy change per unit time", i.e. a singularity!

In fact, singularities are more omnipresent in quantum mechanics than they are in classical physics. You need to work with operators such as \(1/(H-E)\) where \(H\) is the Hamiltonian (operator). Those are important for the evolution and they display a pole when \(H=E\). Poles in the scattering amplitudes (written as functions of energy and/or momenta) are not inconsistencies. They are signs of the existence of stable states (initial, final, or intermediate).

The important lesson is that only the final predictions for "the numbers shown by an experimental apparatus" must be finite but any intermediate step in the calculation may involve infinite or singular quantities. It's just fine! So quantum field theory, a combination of the special theory of relativity and quantum mechanics, produces infinities in the loop diagrams. They may be removed by the renormalization, a "black magic" that nevertheless works just fine and allows far more precise predictions than a calculation where these loop diagrams would be completely banned (because their infinities may be said to be politically incorrect).

In the process of the renormalization, even terms contributing to the probabilities may be infinite. But as long as the "outcome" is well-defined and possible to be realized, its probability must be a finite number between zero and one. And a viable theory may include infinite terms contributing to the probability – which must perfectly cancel at the end, however. Again, the very notion that infinities may cancel may be called (and has been called) dangerous, a black magic, politically incorrect. But one may organize the calculations so that the calculation is damn real, controlled, and the result after the cancellation agrees with the experiments (almost) perfectly.

So results for the truly doable experiments must be finite; everything else that is used during the calculation is allowed to be infinite and any efforts to ban such infinite intermediate constructs or results are totally misguided symptoms of a knee-jerk reaction, the pathological precautionary principle, and sloppy reasoning. This is also true about the spacetime geometry in general. May the spacetime geometry include singularities or does it have to be smooth?

A certain widespread culture from popular books – which affected me as well when I was a kid – says that the spacetime must be smooth everywhere. Singularities must be banned and if they're present in any description by classical general relativity, it shows that the classical general relativity is incomplete. A more complete theory surely replaces the singular regions by similar but smooth ones. Well, not really. Sometimes, the singular classical geometry is indeed replaced by a smooth one (after corrections are incorporated). Sometimes, the corrected geometry only becomes smooth in some variables (string units vs Planck units). Sometimes, the corrected geometry remains singular and qualitatively equivalent to the classical one.

First of all, the singularities that arise in the Schwarzschild and similar black holes are the "final answer" in the sense that the life of an observer must really end there. There cannot be a consistent theory where the observer gets slightly deformed, the geometry behaves a bit differently than in the classical GR due to some corrections, and the observer survives and keeps on living "somewhere". Well, for curvature invariants smaller than the Planck scale, the corrections are still small and the classical GR may still be trusted which really means that the curvature invariants may get at least close to the Planck scale. But a Planck-like curvature is really so huge that not only carbon-based life is dead for a long time there; you may argue that the notion of a smooth spacetime has broken there, too. The corrections can't do much about it. On top of that, you may analyze the possible "loopholes" by which the spacetime could continue beyond the singularity and you will find out that none of them is really consistent with the basic rules such as causality. So yes, the black hole singularities are "real". And they don't mean any real inconsistency of the theory.

Don't get me wrong. I am not saying that the classical GR is exactly true. It surely isn't. I am just saying that the particular trait of classical GR, the fact that it may have singular points in the spacetime, isn't a trait that must be completely banned or rejected by a more complete theory. In fact, string theory is a perfectly consistent quantum theory of gravity. Nevertheless, it works just fine at spacetimes which aren't smooth everywhere. Also, Penrose has assumed that the classical GR must be complete enough so that naked singularities are banned at the level of classical GR, and that's why a deeper theory (which would be needed to predicted what a naked singularity emits) isn't too needed. This Cosmic Censorship Conjecture was pure faith. There have been some partial positive evidence in favor of weakened forms of the conjecture; any reasonably general form of the conjecture is believed to be wrong by now (counterexamples exist) although some interesting enough weakened versions are supported by new evidence, e.g. by the equivalence with the more tangible Weak Gravity Conjecture of ours.

The most innocent type of a singularity that is allowed in string theory is an orbifold singularity. An orbifold is a quotient of a space, like \(T^4/\ZZ_2\) where \(\ZZ_2\) flips the sign of the four coordinates of the torus. The 16 fixed points of this \(\ZZ_2\) map on the torus are singular; the space around them isn't equivalent to a piece of \(\RR^4\). Does string theory allow strings to propagate on the background of an orbifold spacetime geometry? Or does string theory confirm that even such singularities must be cancelled in the sense of the cancel culture warriors?

The answer is unambiguous. String theory perfectly allows orbifold backgrounds of this kind. In fact, the consistent treatment of orbifolds is one of the amazing features of string theory where the "stringiness" is most directly relevant. Aside from reducing the number of degrees of freedom by demanding that the fields are invariant under that \(\ZZ_2\) group that defines the orbifold, string theory does something else that the point-like particle theories don't. It also increases the number of degrees of freedom. By a factor that may be said to exactly cancel the reduction! String theory does so by adding the twisted sectors. New closed strings – whose points' position in the spacetime is periodic on the closed string, but only modulo the \(\ZZ_2\) transformation – have to be incorporated into the spectrum.

The stringy treatment of the orbifolds is a simple yet amazing piece of the string theory mathematics which already shows how simply clever string theory is. The particular \(T^4/\ZZ_2\) orbifold happens to be a singular point in the moduli space of the K3 surfaces. Most K3 surfaces are smooth but there are singular points in the space of shapes of these K3 surfaces, the "moduli space", where the K3 develops a singularity. \(T^4/\ZZ_2\) and \(T^4 / \ZZ_3\) are special points of K3 surfaces' moduli spaces. (Note that the singularities are present both in a K3 surface itself; as well as in the moduli space of possible shapes of the K3 surfaces).

Less trivial but equally consistent are orbifolds of curved manifolds; and conifolds. In fact, string theory vindicates the existence of any "spatial geometry with a singularity" that has some beauty and good reasons to expect that the spacetime where this geometry is stationary in time solves Einstein's or similar equations governing the evolution in time. Singularities that require time dependence are much less understood, partially because supersymmetry is almost unavoidably broken and many useful cancellations no longer hold. But I think that it is reasonable to say that string theory allows lots of time-dependent classically singular spacetimes, too.

All these constructs are physically consistent because the total probability amplitude for a process occurring within such a spacetime is ultimately finite. In the case of string theory, you don't even need the ultraviolet divergences as the intermediate results (terms whose infinite parts cancel). All terms are ultraviolet-finite. But they may still be terms describing the propagation of strings on a singular spacetime geometry. Well, the probability amplitudes for strings on top of an orbifold are just sums of e.g. \(2^2=4\) terms labeled by the boundary conditions in the two directions of the world sheet, \(\sigma\) and \(\tau\). All these four terms are completely analogous to terms on a non-singular smooth torus. The fact that the total spacetime geometry must be interpreted as a quotient (which therefore has an infinite curvature invariant at the fixed point) is not harmful at all. The terms adding up to the probability amplitudes are as finite and smooth as they were in the smooth space.

I must say that just like the religious feelings are suppressed by rationality in the case of "infinities and singularities in physics", the morally equivalent transformation has happened in mathematics, too. Are there infinite sets whose size is in between the continuum and the countable set (of integers)? This is the continuum hypothesis. The answer was impossible to reach for a "finite human being". But the 20th century mathematics has adopted a self-confident, less religious attitude. If some questions are really (and in principle) inaccessible to a human being, they are meaningless! This principle has the obvious importance in physics where every idea is ultimately a tool to explain the truly doable experiments. If ideas are in principle disconnected from observations, they're an uninteresting non-physics talk, not "big words that we must worship"! But it's important in mathematics, too. The question is whether you can prove the Continuum Hypothesis. And it was proven that you can neither prove it nor disprove it if you only have the other, more common axioms of set theory. Mathematicians say that "the Continuum Hypothesis is independent of the ZF or GB axioms [of modern set theory]". In this sense, many opinions about the "world of the infinite sets" are a matter of conventions. You can choose axiomatic systems that say "Yes" and systems that say "No" and those must be considered "equally good" because both may be consistent.

In the past, I've had goose bumps when I thought about lots of such things. Most of these goose bumps have gone away. Sometimes, I am a little bit sorry of this loss. The goose bumps were cool. It was great to be afraid of the infinity and the singularities. On the other hand, I only had the goose bumps because I really wanted to know the truth. It was largely unavoidable for me to learn about this truth; and understand (and perhaps, in some cases, help to discover) the answer to many such questions. The goose bumps have been replaced by the rational, cold understanding which may be boring. But in some cases, it is prettier than the unrefined fear and "infinity worshiping" that I started with. Various special, singular points may appear in the real space and moduli spaces. The symbol \(\infty\) may be a label placed on some special points of all these spaces (in most cases, more specific labels that say \(\infty\) but add some more detailed information about the singularity type are used instead). But the "big picture" of the theory including the "sometimes singular" spaces used by the theory is still mathematically and physically consistent. The mathematical consistency boils down to the absence of provable contradictions; the physical consistency involves the unique finite predictions for the experiments that are so "really" measurable that their results really have to be finite.

We may say that mathematicians and physicists have become the "masters of the infinities and singularities". In the past, the infinities and singularities were infinitely far, on the boundary of all the real or conceivable pictures, and therefore eternally inaccessible. (And the black hole singularity had to come and still has to come at the end of the observer's life.) However, by the late 20th century, mathematics and physics has become a full of diagrams of "theories and gadgets that really work fine" where the singularities and infinities may be spotted right in the middle of calculations and right at the center of many diagrams. We have tamed a lot of infinities and singularities and turned them into our employees, slaves, pets, and parrots in the cages, too.

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