Monday, March 03, 2008

Singularities and stringy geometry

Clifford Johnson wrote a nice report about David Morrison's lecture about the resolution of singularities in string theory and I want to add a few words, too. Most of them will be very general and philosophical, some of them will be slightly more technical.

Singularities as mantinels of our current knowledge

Singularities are objects or theoretical constructs in which something is infinite, uncontrollably high, dense, maximally squeezed, and so on. But the adjective "infinite" is usually a mere artifact of our incomplete knowledge about the particular physical situation or a class of questions. When we look more accurately, we see that the old quantities cannot really become infinite as we used to believe. Instead, there is a new layer of phenomena or knowledge - or "new physics", as physicists like to say - where new objects and theoretical constructs become finite. Our microscope or telescope can focus on the new realm and inform us about things that were impossible to seen in the previous picture of the world.

Let me tell you a few examples.

Millenia ago, people were scared by thundestorms, hurricanes, and rainbows. They were apparently created by an infinitely powerful god who was flying at an infinite altitude above the flat Earth. Of course, when the people gained a little bit of self-confidence and some skills, they realized that this particular god was finite and lived at an unimpressive altitude of a few miles above the surface. The atmospheric science was born and it has instantly regulated the singular notion of the climatological god. The replacement of an anthropomorphic and anthropogenic god in an infinite atmosphere by a finite and natural atmosphere was arguably one of the greatest developments in climate science and judging the current climate science by the fashionable global warming religion, much of the current progress goes in the negative direction once again. ;-)

The Earth used to be visualized as an infinite plane which led the thinkers to all kinds of paradoxes. These paradoxes were solved by a new picture of infinity. The Earth isn't really infinite, after all.

There are many examples of this kind but I want to get closer to modern theoretical physics in the conventional sense. There are many quantities in physics and they have certain units. We may imagine the value of each of them to be infinitely large or infinitely small. But something new and surprising may happen whenever you take it to the limit.

Eliminating units and solving infinite paradoxes

We may start with the distance but it is more historically and physically meaningful to start with the velocity. 103 years ago, you could have asked many questions about the phenomena that occur when velocities of objects become infinite. However, Albert Einstein developed his special theory of relativity. According to this framework, the velocity can't really grow infinite. In fact, it can't exceed the speed of light. Instead of a velocity v, it is more natural to talk about the dimensionless ratio v/c. The mysterious realm of an infinite velocity has been resolved. All paradoxes based on infinite speeds have evaporated.

In this particular case, v/c can't be greater than one and the sharply defined number one has a special meaning. In other contexts, we will see different things going on.

When 19th century physicists studied the atoms, they could have asked all kinds of questions what happens when an electron approaches the nucleus too closely. Quantum mechanics taught us that there is a whole new realm of physical phenomena that become relevant at very short distances.

Quantum mechanics

Because of the uncertainty principle, electrons can't really collapse to the nucleus, at least not for too long periods of time. The angular momentum can't go to zero: its spectrum is discrete. Note that just like 1/v can't go to zero, the angular momentum can't either but the detailed explanation is different. In the case of 1/v, it is always greater than 1/c and its allowed values are continuous. In the case of the angular momentum J, it must be a multiple of hbar/2. Instead of the angular momentum, I could also talk about the action. When the action - the integrated Lagrangian - associated with a physical process becomes too small, comparable to hbar, new physics of quantum mechanics starts to dominate.

Again, quantum mechanics has introduced a new layer of knowledge. When any physical quantity approaches a number comparable to Planck's constant, we must start to be very careful. New insights protect us from paradoxes and even in the absence of paradoxes in the old classical theory, quantum mechanics can tell us some new, unexpected answers to our questions.

Special relativity allows us to relate meters and seconds in our system of units: the speed of light is the conversion factor. Quantum mechanics allows us to identify energy with frequency - there are many equivalent ways to say what it is doing - and it reduces the independent units to one. Particle physicists would probably choose one GeV and its powers as the unit for anything. ;-)

In quantum field theory, one would naturally talk about the substructure of matter and effective field theories at ever shorter distance scales, as organized by the techniques of the renormalization group. They explain what is happening inside the atom, inside the nucleus, inside the proton - objects that used to be viewed as infinitely small. But instead, let us look at a related example involving quantum gravity.

In the quantum field theory setup, we ended up with the unique unit of one GeV and its powers.

Quantum gravity

But one independent unit is still one too many. When you set Newton's constant equal to one, or something equivalent, all quantities may be expressed as dimensionless numbers. Once you do it, you shouldn't be shocked if there is new physics whenever a quantity that used to be much greater than one approaches the values of order one or smaller - or vice versa, when an old quantity that used to be much smaller than one approaches or exceeds one.

If we set c=hbar=G=1, we deal with units of quantum gravity. The distance is naturally expressed in multiples of the Planck length. So we shouldn't be surprised that new physical phenomena occur when distances shrink down to one Planck length or times shrink to one Planck time.

This result of a dimensional analysis is pretty universal - regardless whether you understand string theory or not - and the only really plausible yet simple way to change this dimensional analysis is to change the dimension: the dimension of space. With large or warped extra dimensions (or something that must be effectively equivalent to them), new fundamental units of length may occur - higher-dimensional Planck lengths or string lengths of various types. They may be substantially different from the conventional Planck length as calculated by Max Planck.

The new physics near the Planck length is not only expected, with the logic sketched above, but it is badly needed. We know that things can become extreme. If you allow mass to collapse, the density of matter could a priori converge towards infinity, much like when you approach the Big Bang. As you get closer to the infinite value, it is becoming increasingly clear that something is missing in our knowledge.

Singularities in hidden dimensions

The singularity inside the black hole is actually a subtle type of a monster that is still not completely understood. In string theory, we can solve problems that are - from a purely geometric viewpoint - in the same universality class but that offer us better tools to nail the question down. Why? We can study vacua with extra dimensions - such as our Universe - and imagine that certain geometric features of the extra dimensions shrink to zero size just like in the case of the black hole. However, this singularity may still be extended along all three conventional large dimensions of space. And one can actually achieve this arrangement without any energy density.

Our stories about special relativity and quantum mechanics above represented two templates what may happen when some characteristic distances on the shape of extra dimensions shrink to very small values. Some people from the loop quantum gravity sect and related denominations assume that the quantity called "length" in quantum gravity must have the same properties as the "angular momentum" in quantum mechanics, namely that it must have a discrete spectrum.

This idea is extremely far from being necessary. In fact, one can see that it is naive, wrong, and reveals a profound lack of imagination, education, and talent of the proponents of such ideas. In some sense, the analogy of "length / Planck_length" with "c/v" in special relativity is much more correct in many cases: the spectrum of "length" of a geodesic inside the extra-dimensional shape remains continuous but something repels it from the value zero. A discrete spectrum would lead to contradictions with special relativity and with the existence of scalar fields.

But it is indeed true in certain situations that the length - or a volume - cannot shrink below a certain limit once you take the new physics of quantum gravity into account. In other cases, the value of zero can be achieved but it becomes completely harmless. There is a new, dual language in which a geometry with very short distances or volumes of spheres can be reinterpreted as a smooth theory expanded around a pretty normal point in the configuration space. Such a new description - such a duality - is another scenario how the paradoxes of the infinitely shrunk geometries can be resolved in many cases.

Constraints of geometry and mathematics

What is important and what the fifth-class physicists probably can never understand - because they are missing a lot of required neural cells to get this point - is that there exist hugely stringent mathematical constraints on the candidate new physical laws that supersede the normal laws of geometry and classical physics in the extreme conditions. Other physicists not only understand that these constraints exist but some of them can, in fact, fully take them into account in their active research.

When you listen to Dave Morrison's lectures, you may get very privy to this kind of reasoning. You may get familiar with examples that were known to mathematicians before the correct answers were actually given by string theory. These mathematicians who studied the complex manifolds realize very well that string theory is clearly the right framework to address all these issues and to nail them down.

For example, there exist "moduli spaces" of manifolds that solve Einstein's equations of general relativity: the whole family is Ricci-flat. You can also show that this "moduli space" of possible solutions of general relativity also contains geometries similar to the "conifold" - manifolds that include a singularity. So the existence of a singular manifold is clearly compatible with the approximate laws of physics - general relativity - and it is thus necessary to have a consistent picture what happens when your world approaches this shape.

String theory has shown that the moduli space is actually complex: an inherently stringy degree of freedom, the stringy generalization of a Wilson line (the integral of a stringy B-field) - an integral of a 2-form potential over a 2-cycle of the geometry - adds the "imaginary part" to the previously well-known "real part" - the inverse area of the 2-cycle. With this "complexification", one can describe the space of possible shapes of the manifold and possible topology changes beautifully. You can circumvent the singular point if you wish, just like you can make a trip around the point z=0 in the complex plane.

Again, there exists more than one answer to the question what happens to spacetime when it shape approaches the singular shape of a "conifold". One answer is based on type IIA string theory and another one on type IIB string theory. But it seems very clear by now that all legitimate and consistent answers are part of string theory: string theory should really be defined as the set of all possible complete sets of physical equations that describe geometry in the heavily quantum regime. And be sure that all the consistent solutions so far have been very tightly interconnected.

One thing is obvious. Whoever claims that he is a theoretical physicist and he or she can answer these fundamental questions without learning string theory is simply a crackpot.

Let me try to present a more detailed summary why it is so. The space of small compact geometries that solve Einstein's equations is arguably essential for everyone who wants to understand general relativity in the regime of very short distances. These spaces can be shown to include singular points - such a point corresponds to a whole geometry of space that has singular points in them itself and you shouldn't confuse these two singularities in the moduli space and in the real spacetime.

At any rate, the singularity in both contexts is a potential source of infinities, paradoxes, and unpredictability: you should ask what happens when the space approaches this particular shape which is what it can apparently do according to Einstein's rules of the game. Possible answers are extremely constrained. Why?

End of the world

For example, when I discussed things like the flat Earth at the beginning, I should have mentioned the "end of the world" where ships fall down the hill. An end of the world is an extremely dangerous theoretical construct because it is not easy - or possible - to design physical laws describing what happens when you get there. Many readers surely remember how they were solving this problem - what happens when you come to the end of the world - and it was tough. There are solutions but they are constrained a lot. For example, there can be a perfectly reflective mirror at the end, the so-called orientifold plane.

Or you can order an angel to remove the souls from the end of the world but such an angel theory - even if you accepted theories "glued from pieces" in such a way - will still not explain what the people who approach the end of the world actually see. It is tough. Another example: An end point of a string can exist (a part of an open string) but it must have either Neumann or Dirichlet boundary conditions; the boundary terms in the variation of the action would otherwise fail to vanish which would ruin the equations of motion in the bulk, too. It is not true that anything goes.

Similar huge constraints exist not only for the real space but also for various moduli spaces, more abstract versions of spaces that appear in physics. A moduli space is the space of possible values of all massless (or light) scalar fields that appear in a theory (or parameters that describe its solutions). If a quantity or a scalar field - such as the area of a 2-cycle - can only have values between 0 and infinity, you should always know what happens when the size goes to zero. Will it bounce? If it does, then the value really goes from -infinity to +infinity and it is another case of the mirror. But a mirror requires the values of -x and +x to be equivalent. Are they?

In the case of the shrinking conifold, this is actually not the right solution. The correct solution involves the "complexification" of the moduli space sketched above. But once you deal with complex numbers, you are extremely constrained, too. Recall that when you know an analytical function in a small region of the complex plane, you know it globally. If your theory includes complex numbers and holomorphic functions, believe me that you are constrained a lot.

The people who will tell you "all quantities are discrete and all problems are solved" are as naive as the primitive humans who say that an omnipotent climatological god is responsible for all weather phenomena. These people are just dense and they have nothing to say about these hard and exciting questions. In real science, different questions often have different answers. The questions how a particular singularity is "resolved" are surely examples.

The people who will argue that everything goes and a universal answer such as the "discrete spectrum of geometric quantities" is perhaps good enough to explain everything about quantum gravity - but who will actually tell you nothing about the spectrum of allowed values of the relevant quantities etc. - are just religious bigots who are not interested in the actual scientific answers to tough questions but who prefer to repeat clearly wrong and naive, non-quantitative dogmas for long enough time so that some intellectually challenged listeners will start to take them seriously. Lee Smolin is indisputably a textbook example of these manipulative pseudoscientists.

These questions are tough but they can be answered and many of them have already been answered. We know that one can "circumvent" various singularities in a complexified moduli space, jump to another branch of the moduli space describing a different topology, and identify the topology change with a condensate of physical objects living in the previous topology - or at least a condensate of objects that continuously transform into something that becomes a well-known physical objects when the geometry is de-singularized and "large".

Geometry and physics - background and particles or other objects - get mixed with each other. They rely on each other, non-trivially interact, can transform to each other, and can be equivalent to each other. The people who still want to study quantum geometry and physics within this geometry in isolation have completely misunderstood everything important that has been discovered about quantum gravity.

Finite diversity of resolutions

There are many kinds of possible explanations and dual descriptions what happens in these extreme conditions. For example, mirror symmetry implies that physical phenomena in two universes whose extra dimensions have two extremely different shapes are nevertheless completely equivalent if the manifolds are related by "mirror symmetry".

This is a concept that is obviously deep according to any good mathematician who is interested in complex geometry - but more generally, it is also crucial for any physicist who is interested in compact solutions to Einstein's equations. The full physics of string theory is clearly the most complete framework to answer and clarify properties of mirror symmetry at the most fundamental level. Because two very different shapes can lead to exactly equivalent physics, string theory shows us that there is something more fundamental about their "cosmological code" than the conventional concepts of geometry.

At the same moment, this insight - whose universal explanation remains mysterious despite the available technical proofs in many classes of formulations - is so captivating only because we can actually show that the objects related by mirror symmetry are continuously connected with large manifolds where the standard rules of geometry and old-fashioned physics do work. If we were not able to show that we are talking about two a priori different representations of a geometry, something that we have known from our previous approximate theories of reality, their equivalence would be unspectacular. We can show the equivalence of many pairs of objects that are non-geometric.

So if someone provides you with a model but he cannot show that it is equivalent to general relativity in the regime where it should be - long enough distances (and most typically, one can actually show that it cannot be equivalent to it) - then this model has no relevance for the questions about the fate of singularities in general relativity.

It has been overwhelmingly clear for quite some time that string theory is absolutely crucial to properly answer questions about the extreme geometrical environments and that the answers to questions in the well-known, highly supersymmetric contexts obtained from string theory are correct. Whoever tells you in 2008 that he is going to solve some well-defined puzzles of quantum gravity in contradiction to string theory is a retarded charlatan and I've been simply overwhelmed enough by the constant flow of this trash that it has helped me to escape the officially active research.

Sorry but I can't afford to tolerate this scum in the comments.

While it is true that theoretical physicists have not classified all possible answers to similar questions, all qualified theoretical physicists as well as mathematicians focusing on geometry are able to tell you whether a particular story about the fate of a singularity is true or not, when it is fully formulated, and whether it is plausible or not, when it is partially formulated. At least in all cases we have seen so far, it is the case. Everything boils down to pretty rigorous mathematical questions and be sure that answers to mathematical questions are usually less vague than postmodern babble of spin-foaming pseudoscientists.

The known stories - collections of new physical concepts and phenomena relevant for physics under extreme circumstances - pretty much cover all of the imagination of the best theoretical physicists and mathematicians in the field. In fact, many of the cute answers provided by mathematics and physics were surprising and people only learned them when the equations forced them to discover them.

But the diversity of the solutions doesn't allow you to say that everything goes. The number of qualitatively different possible outcomes of a shrinking conifold is finite and each of the outcomes is completely well-defined. Good physicists in the field know these things, the bad ones are confused and generate loads of "possible alternative answers" that every qualified person is able to rule out within a few minutes.

I think it is an absolutely paramount requirement for science to eliminate answers that can be shown to be wrong and, at the sociological level, to eliminate the people who have shown that they are only able to produce wrong answers. It is extremely bad for science if the journalists, media, and other non-scientific channels are helping to flood science with politically convenient loop quantum gravities, spin foams, dynamical triangulations, their proponents, and gigatons of similar garbage in order to dilute the concentration of valuable physics and valuable physicists (and the resources available to this group) to arbitrarily low values.

The point where the concentration of serious physics in the ocean of nonsense converges to zero is another example of a singularity and I hope that this one will also get resolved soon - by imposing a cutoff that won't be breached too easily.

And that's the memo.


  1. You know, Lubos, not all scientists feel the need to demonstrate their contempt for and ignorance about religion and G_D in order to justify their entirely unrelated scientific accomplishments.

    In fact, it is with the desire to 'see behind the curtain' if you will, that is responsible for most scientific accomplishment. Only religion (with, of course, at least one noteable modern exception) has guided that impulse in any culture, because those who deny the existence of something higher have no need to explore.

    Consider those who don't waste their time mocking others, but spend it in tireless pursuit of the Truth, wherever they may find it.

  2. Dear Ytba, there are many ways in which I respect religion but religion as a method to determine how the Universe works at the fundamental level - e.g. at the Planck scale ;-) - is not something I am going to respect, sorry, and this comparison of certain scientists' methods to religion is surely not meant as a compliment.

  3. Dear lumo

    "...religion as a method to determine how the Universe works at the fundamental level - e.g. at the Planck scale ;-) - is not something I am going to respect..." -- lumo

    Of course. Determining how the universe works is not (nor should it be) the purpose of religion, which is the edification of human character according to the moral dictates of the Creator. Of course, if one can come to a deeper understanding of the philosophical underpinings of science by invoking valid religious principles, something the great scientists have never been afraid to do, how can those who haven't yet reached their level critize them, let alone disrespect them?

    Anyway, what I was mainly objecting to was the setting up of the "straw man" of "religion dictating science" just to mock it, which is what is being done in the section titled "Let me tell you a few examples."

    There the "straw man" of a tiny subset of a subset of religious ideologies is set up (after some significant misrepresentations) in order to discredit them all when that specially selected one is shown to be false. That might work if they were all dependent on the truth of a part of that part, but they are not. Such biased and shoddy "logic" is something I can't respect.

    Also, since the point of the article could have been made without any self-serving cheap shots like those, and others throughout the piece, (unless that WAS the point of the article?) they actually detract from rather than enhance the whole.