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Emergent gravity: positive and negative insights

A conference at MIT was dedicated to emergent gravity and I find the approaches of a vast majority of the participants (except for Subir Sachdev, who was talking about AdS/CMP) to this question unreasonable and hopeless; see the program and Backreaction for a report. As Moshe Rozali wrote, it is strange that no string theorists were invited, especially because many actual experts on emergent gravity were present in the same MIT building during the conference.

It is possible, and in fact likely, that the metric tensor - a set of fields introduced to physics by Einstein that define the spacetime, its dynamical geometry, and that underlie the gravitational force (because it results from the spacetime curvature) - is not a truly fundamental entity.

See also: Emergent space and emergent time (2004)
But it is very important to know some "details" how the gravitational force is actually supposed to emerge. There are two basic classes of approaches to the mechanisms of emergence:
  • Preconceived, simple ideas that "should work" but they have never worked when their details were investigated (but people are often not ready to give up)
  • Unexpected ideas that people didn't consider a few centuries or decades ago but that naturally emerged in some equations of physics and they actually do work (but people are often afraid to study them too much and to go too far from the old tracks)
Quite clearly, your humble correspondent denounces the first group while the second, mostly string-theory-based group is thought to be very important.

Very generally, different researchers and their teams may try different approaches. But their research should never stand on increasingly shaky and unlikely pillars. Every scientist may have a slightly different interpretation of the existing evidence. But his assumptions shouldn't become crazily unlikely: he should still rationally evaluate the available evidence. I will dedicate six more paragraphs to this sociological question, before I return to emergent gravity.

Some people say that different approaches to similar questions should be "supported" by the society. What they mean is that every scientist should try to assume whatever he wants, even if it looks extremely unlikely (and maybe even increasingly unlikely) given the available evidence, and bureaucrats or politicians are those who should ultimately regulate the number of people who study various approaches (via funding), probably depending on their P.R. departments.

I completely disagree with this kind of a "scientific method". If we talk about the real science, every researcher should individually try to obtain the right answers about the reality, instead of looking for biased answers, hoping (or claiming to hope) that his or her bias (or the bias of his or her team or community) will be compensated by others. Even when it comes to the very big questions - and, in fact, especially in the case of the big questions - scientists should try to evaluate the existing evidence as well as they can to decide which research approach they should choose. To do so, they must know at least something about the "competing" approaches. They should never leave the big questions to the bureaucrats or politicians because the big questions are a part of science, too (in fact, the most important one).

For example, it is foolish for a scientist to dedicate a whole life to an idea (with unspectacular consequences) if the probability that the idea is correct is much smaller than 10^{-10}, the inverse population of Earth.

Why? You can see that the contributions of such a scientist to science are likely to be negligible. How can you see it? Multiply his contributions by the population of Earth. This product now includes all the scientific work of mankind but it is still unlikely that at least one idea investigated by at least one person is correct. So even the expected value of the results obtained by the whole mankind would be pretty small: they would probably not find the correct theory in their lifetime. What one person is doing is 10^{10} times smaller so the value is really tiny.

It means that if scientists study pretty much the same problem - e.g. quantum gravity - they must be interested in all relevant arguments and evidence, instead of segregating themselves into different "fields". More concretely, loop quantum gravity is not a "different field" than string theory. Loop quantum gravity is the same field as string theory, namely quantum gravity - the only difference is that it is done by people who can't properly evaluate the scientific evidence and they end up with wrong answers. Unfortunately, they don't seem to care about real physics and real evidence whenever it shows that they're doing something incorrectly.

In the very same way, it is not true that string theorists should "isolate themselves" from the attempted alternatives. They should know what the alternative approaches are saying and they should have a qualified opinion about it. If they think that the alternative approaches are likely to be true or that they say something important, they should incorporate it in their work (or to completely change their approaches). If they think that the alternative approaches are simply incorrect, as I do, they should behave according to this conclusion, instead of being "nice" and support a wrong answer (and huge amounts of wasted man-hours and dollars paid to this hopeless approach). There is nothing "nice" about supporting bullshit.

Fine. Let me stop with sociology and return to the mechanisms of emergent gravity.

How space emerges in string theory

The oldest example began in 1919. Let me assume that the reader knows what the Kaluza-Klein theory is, at least at the level of Chapter 4 of The Elegant Universe.

Kaluza-Klein paradigm

In Kaluza-Klein theory, one can relate higher-dimensional theories to lower-dimensional theories by the trick (or mechanism) called compactification. There are two types of a description of a physical system in this context: the higher-dimensional one and the lower-dimensional one.

In the higher-dimensional description, one assumes that there exists a rather simple theory - for example, a field theory with a small number of fields - that is defined in a higher-dimensional space. In Kaluza's original example, a five-dimensional spacetime was equipped with pure Einstein's gravity.

The lower-dimensional example is "simpler" in one sense only, namely that the spacetime dimensionality is smaller. However, it has many more degrees of freedom. For example, in the original Kaluza-Klein theory, you need to Fourier-transform the five-dimensional fields over the fifth dimension. The individual Fourier modes generate a tower of infinitely many mostly massive four-dimensional fields. Their precise interactions are constrained - the infinitely many masses and interaction constants are linked to each other - because they remember their five-dimensional origin. So you can view this physical system as a four-dimensional theory but it is a special type of a four-dimensional theory, with infinitely many fields and infinitely many constraints on their properties.

Kaluza's original idea was to unify electromagnetism and gravity, the only two forces that the anti-quantum physicists of his era (such as Einstein himself) found worth studying. Qualitatively speaking, the idea worked - after the corrections by Klein who really realized and appreciated that the extra dimension was compact. However, using quantitative laws, the simplest version of the idea could have been ruled out rather soon. Note that the Fourier expansion is something that the ancient Greeks didn't understand well: it was only invented two centuries ago.

In string theory, the idea was generalized and a generalized Kaluza-Klein theory is compatible with several classes of realistic models of the real world within string theory. What are the generalizations? Well, first of all, the compact dimensions don't have to be a circle. They can be multi-dimensional shapes, such as the Calabi-Yau manifolds. The Fourier expansion has to be replaced by the expansion into the eigenmodes of some more general operators. More complicated shapes, such as conifolds, orbifolds, non-Kähler manifolds, warped throats, non-geometric compactifications, or many others have been studied and fascinating, coherent insights have been found in most cases.

But string theory has generalized the concept in many other ways that couldn't have been expected at the beginning. For example, the integer label distinguishing the different Fourier modes can have many interpretations. In the Kaluza-Klein theory, it has to be the momentum along the compact (fifth) dimension. In string theory, it can often be represented as a winding number or a wrapping number of strings or branes in a dual description.

Because of this unexpected property, new dimensions of space may emerge out of completely new phenomena. If there is a very short non-contractible circle in your compactification, strings can be winding around it many times. The winding number (how many times they're wrapped around the circle) becomes effectively continuous and can be reinterpreted as the momentum in a new dimension that becomes a new, large, effectively non-compact circle.

This phenomenon is called T-duality (and it can be rigorously proven in perturbative string theory). Compactifications on very short and very long circles are equivalent to one another once extended closed strings are allowed to wrap these circles. And they must be allowed to do so because an unwound string is always able to pair-create two opposite wound strings (as long as any interactions are allowed at all). T-duality can be applied to several independent circles in the stringy geometry: mirror symmetry is a remarkable example of a triple T-duality that relates two beautiful, but a priori unrelated six-dimensional Calabi-Yau shapes.

In principle, you could imagine that the whole three-dimensional space around us (let me omit time now; this limitation was discussed in the 2004 article) can arise from a conspiracy of many particle species in 0 dimensions. In this sense, the whole space is emergent. However, there are many ways how infinitely many objects in 0 dimensions can interact with each other. The knowledge of their higher-dimensional origin, or something equally constraining, is clearly necessary to find realistic quantum mechanical models in 0 spatial dimensions.

Just by saying that physics in 3 spatial dimensional is equivalent to a system in 0 dimensions, you haven't solved much. But indeed, we know several ways how physics of a field theory (or something that is approximated well by a field theory at long distances) in many dimensions can naturally be rewritten in terms of a theory in 0 dimensions. The BFSS matrix model is an example. However, we needed to know a lot about the higher-dimensional physics to find such a special model. Almost every "obvious" modification of the BFSS matrix model ends up with a quantum mechanical model without any higher-dimensional interpretation.


Holography is another, and perhaps even more famous example of emergent dimensions. In the case of Kaluza-Klein theory and its generalizations, we didn't have to rely on gravity. T-duality works even before gravity gets turned on. However, holography - the equivalence of a gravitational theory in D+1 dimensions and a typically non-gravitational theory in D dimensions - does depend on gravity very strongly. Why? Holography is related to the entropy bounds: the maximum entropy one can squeeze into a given volume is achieved by a black hole and the resulting maximal entropy only scales as the surface of the region, and not the volume as you might expect, in Planck units. And black holes need gravity to exist.

The anti de Sitter space is the most specific and successful realization of holography in action, because of Maldacena's AdS/CFT correspondence. The "bulk" of the anti de Sitter space is emergent here: we can describe its physics in terms of a theory that only lives on the boundary of the space (at infinity, in this case).

You might think that it is just another example of the Kaluza-Klein paradigm: we are making some kind of Fourier transform over the extra (radial, holographic) dimension. You might think that the boundary theory will have "infinitely many" fields of the Kaluza-Klein type. But you would be essentially wrong. The boundary theory is extremely simple - in the case of AdS5/CFT4 duality, it is a cousin of QCD: an ordinary gauge theory with a finite number of "elementary matter fields".

Now, the infinite tower of fields that we knew from the Kaluza-Klein case hasn't quite disappeared. You can still create infinitely many fields but they are composite fields. For example, excited strings in the bulk are created by various composite operators in the QCD, analogous to Tr(ZZZAZZZZB), the BMN operators. However, the very momentum along the holographic dimension is not encoded in this way. The reason is that the boundary theory is non-gravitational and it also includes off-shell, local Green's functions: you can study correlators of operators represented by Feynman diagrams where the external particles don't satisfy the expected energy-momentum-mass dispersion relations. However, the equivalent theory in the higher-dimensional bulk is gravitational and only knows about the on-shell scattering amplitudes. It is meaningless to ask about the local, off-shell Green's functions of the gravitational theory in the bulk: this fact is indirectly related to the holography itself.

Other mechanisms

Kaluza-Klein theory and holography are just two examples how dimensions may emerge - and transform into something else - in string theory. I should also be talking about topology transitions, quantum foam in topological string theory, deconstruction, and many other interesting mechanisms where dimensions of space emerge from something else. Gravitons themselves are emergent in perturbative string theory because they are closed strings in a particular vibration mode: see Why are there gravitons in string theory.

However, if I started to talk in this way, I would never stop because all of string theory may be viewed as a generalization of gravity that is emerging from something else. For example, all closed string modes may be thought of as components of a gigantic stringy "gravitational multiplet" and all other branes are generalizations of a string (and they can be dual to strings in various dual descriptions). There is no way to strictly separate "geometry" from "non-geometry" in string theory. This statement shouldn't be surprising because it is really equivalent to the fact that string theory unifies gravity with other forces (and matter).

But some people don't realize that these statements are equivalent because they assign similar propositions with emotional labels. Unification is good, so it should exist, but a separation of geometry from non-geometry is also good, they think. Well, it's no good because such a separation would be the opposite thing than unification and unification is good, indeed. ;-)

In string theory, all concepts we know are kind of linked to each other and all of them may be viewed as generalizations of geometry (a quantum, stringy geometry). Moreover, the combinatorial graph indicating how different fields, objects, and concepts in string theory are linked to geometry (and to each other) is not a tree graph. It has loops, too.

For example, the dualities (equivalences between various stringy vacua) link all the theories (and their objects) into a complex multi-loop network. Each loop implies a non-trivial prediction - a consistency check similar to the transitivity conditions for the transition maps on a manifold. And all these consistency checks have worked well, so far: they are almost as powerful as experimental tests in proving that this is a theoretical structure that a theoretical physicist simply has to care about. It is a beautiful, robust structure. And this whole structure may be interpreted as a generalization of Einstein's general relativity in which the right tools to generalize have been fully exploited.

So instead of talking about all ways how space (and time?) can emerge and transform in string theory, which is what all the conceptual efforts in string theory research (thousands of papers) are focusing on, even if they don't say so explicitly, let me return to the bad type of the research of emergent gravity.

Discrete and condensed-matter gravity

Some people are very impressed by the unexpected ways how macroscopic pieces of material can exhibit new types of behavior in condensed matter physics - superconductivity, superfluidity, Fermi liquid, highly correlated fermions, metals, fractional quantum Hall effect, and so on. I have surely missed some of the best examples.

I am also impressed except that Nature doesn't guarantee that similar ideas will work at many places. Some places need completely different ideas and we can often see what they are.

Other people talk about discrete physics, imagining that a finite volume of space is always made out of a finite number of easily visualized, "discrete" elementary building blocks. The space is a spin network and the spacetime is a spin foam (or a causal dynamical triangulation), they say. For some philosophical reasons, they find such a philosophical picture pleasing. But in physics, something's being philosophical pleasing is an entirely different criterion from something's being physically correct.

Assuming that something is only composed out of discrete blocks is a huge assumption - much like the assumption that a random, measurable, a priori real quantity will be integer-valued. It is very unlikely that it will be integer-valued. In the very same way, it is very unlikely that all of physics may be encoded in "discrete" quantities (even though, in some cases, both situations can occur - but there is usually a well-known argument, not just wishful thinking).

More concretely, these discrete descriptions of space suffer from a couple of very general problems. They almost always break the Lorentz invariance which is always a huge problem because the Lorentz invariance is one of the key experimentally verified principles underlying modern science (special relativity is crucial in particle physics). They violate the Lorentz invariance because the vacuum is not really "empty". It contains a new kind of luminiferous (or gravitiferous) aether. Consequently, one expects that a privileged reference frame is picked.

Moreover, the aether seems to have a huge entropy density - probably the Planckian entropy density if the building blocks have Planckian dimensions. Such a huge entropy carried by the vacuum would completely destroy thermodynamics (for example, it would cause a huge friction because such a "vacuum" resembles a highly viscous liquid) as well as interference in all interference experiments (because the microstates of the vacuum are distinguishable and they can't interfere with each other). It would also spoil the Lorentz invariance by itself because the entropy density is a time component of a four-vector (am entropy current) and its nonzero (huge) value is non-invariant under the Lorentz transformations, too.

For the typical composite theories of space, one can show that these problems are real and huge. The Lorentz violation in these theories is not "small" in any sense. Among many other problems, this bug also makes perpetuum mobile possible. ;-) The only method how to achieve a tolerable situation is to have a system whose vacuum can be shown to be fully equivalent to a traditional "empty vacuum" in a Lorentz-invariant field theory. If you can't see a reason why such an equivalent description should exist, it almost certainly doesn't exist and your composite theory is ruled out because it disagrees with some extremely basic features of our world such as the "emptiness" of the vacuum (e.g. the absence of friction in the vacuum).

There are other huge constraints that make similar composite models of a graviton impossible. The Weinberg-Witten theorem is a textbook example of these no-go theorems. These two extremely famous physicists have shown that composite massless particles with spin exceeding one cannot exist; in fact, even theories with an elementary particle with spin above one cannot be renormalizable local quantum field theories.

This result is a "negative" one in the sense that it kills someone's hopes. But I am among those who view negative results to be as important for science as positive results. Whenever we understand how Nature doesn't work, we also understand something about the way how it does work. These "negative" results usually lead to a reduction of research activity in a certain direction which is why many people don't like them. They steal "jobs" from the people. But this is not an objective, unbiased criterion. I think it is great if you can save man-hours in this way! If you get rid of the bias, positive and negative true insights are equally valuable.

Needless to say, in the real world, not quite all such man-hours are saved. Many people continue to investigate theories that directly contradict some of the known no-go theorems. The perpetrators usually don't understand the no-go theorems well. Most typically, they either ignore the theorem completely or they invent some bogus explanations why their theories shouldn't be subject to these no-go theorems (recall e.g. Garrett Lisi's bizarre "explanations" why he doesn't care about the Weinberg-Witten theorem).

If these people were sane, they would realize that their explanation why they "can" circumvent the no-go theorem could be used by the author of every single composite graviton theory (for example, everyone could say that at the end, they want to add a positive cosmological constant, which is a subtlety that Weinberg and Witten didn't take into account, they emphasize).

Consequently, it would follow that the Weinberg-Witten theorem is completely vacuous and the authors themselves, Weinberg and Witten, would have to be completely deluded. For some reason, the "composite gravity" geniuses don't see any problem with such an inevitable conclusion of their thinking. Now, the thing that irritates me is not that someone is ready to believe that Witten and Weinberg are deluded and that they write vacuous theorems. Feel free to believe so. But you are still wrong because the theorem is actually very powerful.

For example, the cosmological constant is completely irrelevant and you can see that it can't change the conclusion (that the composite graviton theories are ruled out). The vacuum energy only creates errors of relative magnitude 10^{-120} or so, altering the conclusions from the flat space. But the composite character of gravitons induces much greater errors - of order one - so they can't cancel. These problems - the cosmological constant and the compositeness - have nothing to do with each other.

I think that whoever tries to study similar "composite" theories should carefully read the Weinberg-Witten paper and learn their methodology. He or she should try to apply and modify similar reasoning for his or her context, too. And I assure you that it is not hard to modify the Weinberg-Witten arguments so that they can rule out every research direction in which the "gravitons" or the "metric tensor" are composite fields in the bulk of spacetime (that is however non-emergent).

String theory guides us to the right highway

String theory is able to circumvent the theorem in very unexpected ways, for example in the AdS/CFT correspondence. The reason why gravitons can be emergent or associated with composite operators is that it is not only the gravitons that are emergent: it is a whole dimension of spacetime (or more) that emerges, too. Such a broader process of emergence is harder to imagine a priori which is why people haven't tried it before the phenomenon was understood by string theorists (who tried to understand why some consistency checks concerning black hole entropy worked better than they expected).

So it is fair to say that most of the "very minimal" and "philosophically pleasing" pictures how gravity, geometry, and other things may emerge can be falsified. On the other hand, string theory exactly tells what kind of non-minimal extensions of these ideas you should pursue in order to have a chance that your theory won't be instantly ruled out.

There are lots of examples of this phenomenon - that string theory is able to "guide you" in finding many non-trivial and non-minimal mathematical structures and theoretical frameworks that share some features with the "cool and simple ideas" that you might invent after a few minutes but, unlike the cheap and fast ideas, these non-minimal ideas work. We can check that they do achieve what they claim.

For example, you may imagine that Cartan, Killing, Dynkin, and others wouldn't have been able to classify Lie groups and to find the exceptional ones. Nevertheless, string theory could have been discovered even without this insight. After a few years, they would also find the heterotic string theories and realized that one of their versions has 248 copies of a massless gauge field. They would be inevitably led to the E_8 group (and there are other constructions that are natural in string theory that lead you to E_8). Soon afterwords, they would discover the other exceptional groups as the subgroups of E_8.

There are many other, more complex examples of this kind that are behind the popular phrase that "string theory is smarter than us". For example, supersymmetry, a new symmetry that, morally speaking, circumvents the Coleman-Mandula no-go theorem (implying that symmetries that non-trivially mix the spacetime symmetries with the internal ones are impossible), was found in the context of string theory, too. It was sufficient for Pierre Ramond to try to incorporate fermions into the old bosonic string theory. He found both the superstring as well as the (worldsheet) supersymmetry.

Just like the experiments have often been necessary for the theorists to realize certain things that they could have discovered by pure thought (by they didn't), string theory is able to do the very same thing. It often helps us to search for a "needle in a haystack" even though we often find the farmer's daughter instead - i.e. we find a non-trivial, non-minimal construction similar to other constructions that have been looked at, except that the people were not able to combine all the right ingredients to make it work.

Even though Nature and string theory are demonstrably smarter than us, and smarter than Einstein and others, some people still try to be smarter than Nature and string theory. The only problem is that their attempts never work. The airplanes don't land. Virtually no interesting constructions - potentially relevant for Nature's mechanisms at the fundamental level - have been discovered in this way and most of the people who continue to investigate e.g. the emergent gravity - while the spacetime in their framework is not emergent by itself - are stuck with some excessively simple theories that have already been falsified.

Meanwhile, string theorists keep on finding fascinating new things, by making their careful "experiments" with string theory. These new discoveries have existed in the abstract Platonic world of cool mathematical ideas and they have only been "discovered". Certain people may feel happier if they "invent" completely new things except that this approach, while useful in engineering, hasn't led anywhere in theoretical physics. Nature can't be "invented". Its secrets already exist out there and we are only allowed to "discover" them. To do so, we must listen to Nature - through the experiments or string-theoretical calculations - in the right way to hear what She is trying to tell us.

You may try to be smarter than Nature but you will eventually fail because She rocks.

And that's the memo.

P.S.: A particularly silly viewpoint on gravity will emerge in 2010 when Erik Verlinde will claim that gravity is an entropy-driven force.

That can't be the case. Such a "LeSage theory of gravity" would imply that the gravitational phenomena are inevitably irreversible and that they don't apply to individual particles because they would destroy the interference patterns.

However, gravity works even in microscopic experiments, neutron interferometry has shown that the whole pattern is moved exactly as you would expect from a "freely falling" wave function. And gravitating bodies don't lose energy and don't create any macroscopic entropy. Erik Verlinde's picture is born as a dead baby.

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