Perfect fluids, string theory, and black holes

The omnipresence of very low-viscosity fluids in the observable world is one of the amazing victories of string theory. The value of the minimum viscosity seems to follow a universal formula that can be derived from quantum gravity - i.e. from string theory.

The story of this insight is quite remarkable. But let us begin with two recent scientific announcements to see that this is a good time to revisit this topic.

Recent reincarnations of perfect liquids

Four weeks ago, the ALICE experiment at the LHC announced its first results from the lead-lead ion collisions. It was very fast, see e.g.

ALICE announces the first results (Symmetry Magazine)

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Charged-particle multiplicity density at... (preprint)

The LHC confirmed the previous findings that the Universe in the initial superhot nuclear state acted like a liquid (Google News), a nearly perfect fluid. This insight was previously known from RHIC - as well as from string theory. The viscosity was very low. So the environment was nothing like a gas; I will discuss the viscosity of gases later.

Six days ago, Duke researchers measured the viscosity of lithium-6 atoms at temperatures of 10^{-9} Kelvins or so:

Fahrenheit -459: Neutron Stars and String Theory in a Lab (Duke press release)

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Universal Quantum Viscosity in a Unitary Fermi Gas (Science)

John Thomas and co-authors were interested in the scaling of the viscosity of this material. They found a power law. They also calculated the ratio of the viscosity to entropy density and were able to see that it was just by a factor of 4-5 higher than the universal lower bound dictated by string theory even though the conventional estimates of the viscosity produce results that are higher by orders of magnitude.

Back to 2004: string theory and viscosity

These results are just steps in a much more impressive sequence of advances. The main statement that can be derived from string theory is that many strongly-coupled (i.e. hardly calculable) systems are able to approach the minimum possible viscosity and the minimum possible viscosity (valid for any dimensions and very diverse systems) is given by

eta / s = hbar / (4 pi k_B) = 6.08 x 10^-13 Kelvin second

Here, "eta" is the viscosity and "s" is the volume density of entropy. The numerator "hbar" is the reduced Planck's constant, "4" is "2+2", "pi" is approximately "3.14159265358979" (yes, I have memorized the first 100 digits but that's off-topic haha), and "k_B" is Boltzmann's constant. Even in the usual units, the ratio doesn't depend on the speed of light which may make it relevant for non-relativistic systems as well.




The calculated viscosity-to-entropy ratio is usually much higher if you use sloppy "conventional" methods that are optimized for mundane physical systems and that don't fully appreciate that the real systems are really strongly-coupled.

For a couple of years, the derivation of the bound was known from string theory. If you're at least somewhat interested in the topic, I recommend you to read a renowned 9-page 2004 paper by Kovtun, Son, and Starinets:

Viscosity in Strongly Interacting Quantum Field Theories from Black Hole Physics (PDF)

The paper defines all the concepts you need and derives the lower bound from the properties of a black hole in a higher-dimensional anti de Sitter space. The systems whose properties are captured by the stringy black hole picture are some of the dirtiest examples of "low-energy experimental physics" while the calculated bound resulted from some of the most formal investigations of "quantum gravity according to string theory".

This union is noteworthy. String theory began to eat the theoretical research in physical disciplines that were thought of as impossibly remote. Macroscopic phenomena of heavy ion physics, superconductors, Fermi liquids, non-Fermi liquids, and even hydrodynamics became candidates for a description in terms of black holes.

The possibility to explain complex phenomena using simple models that makes correct quantitative predictions is counter-intuitive and it changes some expectations that people - including the physicists - used to have. We used to think that:

If the interaction strength or the density of elementary particles or other quantities describing the "complexity" of some physical system becomes very large, the physics of the system becomes very difficult and almost impossible to calculate or describe, even qualitatively.

The opposite limits of all parameter spaces are occupied by sea dragons.

For more than a century, people knew that the statement above can't be quite correct. For example, in the thermodynamical limit (very many atoms), statistical physics of many atoms hugely simplifies. The averaged values of temperature and entropy capture all the key aspects of the macroscopic behavior and many things may be predicted. The large number of gas molecules makes the qualitative physics easier, not harder. So it is sensible to think that some "extreme limits" could actually make the physics simpler.

However, the second superstring revolution in the mid 1990s has strengthened this reasoning tremendously. In fact, it seems that the previous displayed paragraph has been replaced by a very different expectation:

Whenever some quantities defining "complexity" such as the interaction strength, the density of atoms, or the number of colors are sent to extremely large, infinite values, physics doesn't become infinitely incomprehensible. On the contrary, it greatly simplifies. The remaining "truly difficult" problems can only survive for the "intermediate" values of those quantities such as the coupling constants, densities, or the number of colors.

That's a totally different philosophy. You may live with the following metaphor.

Before Columbus, the distant places of the ocean could look very mysterious and the same was true about different continents across the sea. They could have been occupied by totally unfamiliar creatures. There could even have been an end of the world which looked very different than any mountains you have ever seen (see the picture).

However, Columbus has shown that while it was still easier for the Europeans to walk on the European beaches and study them, they can do the same thing with the American and other beaches as well. And they turned out to be qualitatively similar. The opposite limits of the oceans became understandable and the dragons were chased out of them. People no longer believe in the dramatic ends of the world, either: the Earth is round, after all.

In the same way, the mystery only survives in the middle of the ocean. There could be Atlantis in the middle of the Atlantic Ocean but it's pretty difficult to swim in those places, at least for extended periods of time, so the existence of the island may remain mysterious (unless you can use the satellites as well to see that the island is a myth).

The character of the new descriptions may differ. Many of those descriptions that are relevant for the strong coupling of a theory may be literally identical to the original description. For example, if the interactions between fundamental F-strings in type IIB string theory is adjusted to an infinite value, you obtain the same IIB string theory with dual F-strings - originally D-strings - whose coupling goes to zero: the coupling constants "g" and "1/g" are equivalent.

In some cases, this S-duality relates two different theories. Type I string theory with coupling "g" is equivalent to SO(32) heterotic string theory with the coupling "1/g".

A large value of the coupling constant is not the only quantity that may make the physical phenomena mysterious, even at the qualitative level. You may also compactify some dimensions on manifolds that are extremely tiny. By doing so, you produce many new light particle species out of the wound strings, wrapped branes, and similar new light states.



However, as one can show e.g. by the Dualities vs Singularities (preprint) methods, all compactifications of M-theory or type IIA/IIB string theory on any extreme rectangular tori with arbitrary scaling relationships between the radii are equivalent - by U-dualities - to a limit that is qualitatively understood (M-theory or weakly coupled type IIA or IIB string theory on large radii, in the appropriate units).

Just like you can cover the Earth by maps so that the full surface is "understood" today, at least in the "zeroth approximation", the same is true for the moduli space of M-theory, at least for simple types of compactifications. The picture above is an example.

Here, U-duality is the equivalence between two (or more) theories - or two (or more) compactifications of string/M-theory - that map large/small radii and large/small values of the coupling constants on each other in non-trivial ways. The concept generalizes and unifies S-duality, that only exchanges the weak and strong coupling, as well as T-dualities, that exchange small and large radii of compactifications.

I may be even more general than that. In the past, people would think that the same physical building blocks may behave in infinitely many ways. Just because the parameter spaces seem to be infinite, they should be covered by infinitely many possible types of behavior. The mankind could only understand a tiny portion of those possibilities - phases in a phase diagram, if you wish. However, it turned out that it's not the case; the "very humble" belief wasn't right. The regimes of qualitatively different behavior may be pretty much fully understood and their number is essentially finite.

Path towards the viscosity insights in string theory

Because the Internet is filled with tons of junk written by aggressive crackpots who have no clue about science but who use Google Alerts to detect any article about string theory and supplement it with a flow of meaningless yet disgusting lies, and Peter Woit is just one major example of this subhuman trash, I feel the urge to emphasize that the insights about the viscosity are not some "adjacent discipline" to string theory that is only vaguely related and has nothing to do with the unification and the formal investigation of the structure of string/M-theory.

Quite on the contrary. As I have already explained in the article

Why unification sits at the core of string theory

and others that are linked over there, the insights about the low-viscosity fluids emerged from the most formal research of string theory that analyzed string theory as a theory of quantum gravity and the unification of all forces.

In early 1996, Strominger and Vafa macroscopically calculated the correct Bekenstein-Hawking entropy of a string-theoretical black hole - a result that was later generalized to huge classes of black holes, including multi-parameter near-extremal, non-extremal, rotating, as well as some black holes we can see in the telescopes.

But before many of those technical generalizations emerged, the insight has led to some more important, conceptual progress. In particular, Juan Maldacena was shocked by the fact that many of the black hole entropy calculations worked perfectly even though no one could really explain a reason. Finally he found an explanation that made the coincidences more comprehensible - his AdS/CFT correspondence.

At least some non-gravitational scale-invariant (conformal) physical systems are exactly equivalent to the vacua of quantum gravity (really string theory) in an anti de Sitter space that has an additional radial, holographic dimension.

This insight has led to approximately 10 thousand followup papers (that have verified the predictions in many contexts and generalized them to others) and is often considered by the top theorists as the most important breakthrough in theoretical physics of the last 15 years, to say the least.

The AdS/CFT correspondence may be viewed as a relationship analogous to the S-duality: one still explains the strong coupling behavior of a theory. For example, the relevant coupling of a gauge theory with "N" colors and the gauge coupling "g" is described by the 't Hooft coupling, "lambda = g^2.N". If "lambda" is small, perturbative diagrams converge well. If "lambda" is greater than one or so, the higher-loop diagrams get extra powers of "lambda" ("g" from the vertices and "N" from the colors that may run around the faces of the diagram). The complicated diagrams - and non-perturbative effects - begin to dominate and perturbative physics becomes a bad estimate of the qualitative behavior.

But exactly when this occurs, when "lambda" exceeds one or so, there is a new description - the higher-dimensional curved quantum gravity - that becomes weakly coupled because its compact dimensions become reasonably large (so that geometric intuition in them is OK and general relativity has small corrections) and the curvature becomes low, too. The transition between the two descriptions is gradual, of course. But there is a sort of trade-off over here: when one description becomes very hard, another one must become very easy.

Note that besides a large "coupling constant" and a tiny "compactification radius", a tiny "curvature radius" is another extreme circumstance that may make a theory "strongly coupled" and "difficult to calculate". In this sense, there exists no asymptotic, large region that is filled with mysterious dragons. If one of the descriptions becomes "severely inapplicable", another one may be found and it works.

However, Maldacena showed that the new description may often brutally differ from the old one. Its spacetime may have a different number of dimensions while it is affected by the gravitational force, while the original lower-dimensional description is not.

All this progress may be understood as a sequence of steps towards the "understanding and classification of all conceivable phenomena". And all of this progress requires us to understand string theory and the relationship between the concepts it unmasks. For those reasons, string theory became much more "a theory of everything" than it used to be.

We would think that a "theory of everything" could only describe the fundamental building blocks while the bound states containing infinitely many of them would remain incomprehensible and they would require calculations that had nothing to do with the "fundamental level" of string theory. This expectation turned out to be incorrect: the most canonical equations and concepts of string theory describe both the "maximally elementary" limits of physics as well as the "maximally composite" limits.

This progress was mostly unexpected. Gravity was thought of as being unified at the Planck scale, pretty much in the same way as the electromagnetic and weak interactions are unified at the electroweak scale. However, no one suspected that gravity would also become unified with complex manifestations of non-gravitational phenomena.

Also, no one would think that "quantum gravity" may ever become "experimentally accessible" because its typical distance scale, the Planck length, is so tiny. But the AdS/anything developments showed that quantum gravity is (and quantum properties of black holes are) experimentally testable although we test it (or them) in some "internal regions" of an AdS space only, not in the UV regions "near the boundary" where the conventional "elementary process" and the normal "unification with other forces" becomes manifest. But those two regions are connected - in the usual geometric sense.

Role of black holes

String theory predicts worlds whose basic "sketch" agrees with our world. At long distances, there are fields with the spins and interactions that we know from the Standard Model, General Relativity, and similar theories. However, string theory also predicts many more objects and processes - strings, branes, topological transitions of hidden dimensions, and many more.

However, all the objects and processes I mentioned above depend on the configuration. Whether there are light fermions or light strings depends on the "vacuum" you choose. However, there exists one force that is universal in all stringy vacua: it is the gravitational force.

The gravitational force is omnipresent because as Einstein first understood, gravity is the dynamics of the space itself. As long as spacetime is a part of a theory, things will gravitationally attract. See also Why gravitons exist in string theory.

Gravity as refined by general relativity makes many general predictions (such as the Big Bang cosmology) and the black holes are the main "localized objects" predicted by general relativity. You can't avoid them, especially because the Penrose-Hawking singularity theorems imply that pretty conventional configurations of matter in the Cosmos inevitably end up as black holes. They exist in all the vacua of string theory and their possible properties - shape and charges - only depend on the low-energy physics. If you're only interested in macroscopic properties of the black holes, all the other details about a chosen stringy vacuum are irrelevant. The large degeneracy of the "landscape" doesn't affect any of these issues.

The high-mass spectrum of string/M-theory - the particles much heavier than the Planck mass - are inevitably dominated by the black hole microstates (an observation known as the asymptotic darkness) and the macroscopic properties of the black holes boil down to long-distance physics once again. So even in this sense, the only "mysterious" yet qualitatively important numbers survive in the middle - they're the properties of objects whose masses are comparable to the Planck scale. The objects that are much lighter or much heavier than that are understood by long-distance physics.

The black holes have the maximum entropy among all bound objects of the same mass and conserved charges simply because they're the final stage of a gravitational collapse. The statement that the viscosity-to-entropy ratio is minimized by the black holes may also be phrased in the opposite way: the entropy-to-viscosity ratio is maximized by the black holes. Black holes simply do maximize the entropy. The eventually evaporate by the Hawking radiation which has an even higher entropy - but the Hawking radiation is not a bound (localized) object.

So the black holes are universally important in the physics of gravity. Their properties, including the quantum properties such as the high entropy, are therefore universally important in the physics of quantum gravity. And because we have said that large classes of non-gravitational field theories may be rewritten as quantum gravity in anti de Sitter-like spaces, black holes are universally important for all of them, too!

Myths about perfect fluids

Perfect fluids are often misunderstood by many people, including physicists who begin to study similar questions. First of all, it is often thought that the viscosity may be arbitrarily low. Gases and superfluids are often presented as examples.

However, they are actually not examples of the minimal-viscosity fluids at all. Superfluids in the real world are de facto composed out of two components, the normal one and the superfluid one. The normal one gives the superfluid its pretty high viscosity.

Moreover, the superfluids and perfect fluids differ - and in some sense, they're the opposite objects. A perfect fluid is an idealized environment whose stress energy tensor resembles what is known as the "dust" in cosmology and is fully described by the energy density, pressure, and velocity at each point. The off-diagonal elements of the stress-energy tensor - and the spatial-spatial ones are really what is called the "stress" (the word that appears in the stress-energy tensor) - are effectively absent. And it's the stress that ultimately defines the viscosity.

In other words, the "momentum component in a direction" doesn't flow in "another direction". This actually also means that there is no heat conduction at all: perfect fluids must be perfect heat insulators. That's exactly the opposite property than the superfluids whose thermal conductivity is actually infinite!

One must be very careful about words such as "perfect" and "super-" that may sound similar - in the colloquial language, they're almost the same thing - but they may mean exactly the opposite things in science. Physics doesn't have a universal "moral sense" so it can't tell you what is "good" and what is "bad". An important example: a hypothetical "warmer" climate is not "bad" in any sense. You must be very careful what inequalities and limits are actually included in a priori vague verbal labels such as "perfect" or "super-". There is no "intuitively ethical" way to answer any of these questions. Physics - and science - crucially depends on separating factual arguments from the emotions.

Another fact contradicting the myths is that gases don't have a particularly low viscosity because the viscosity may be written as the product of the mean free path and the average molecular speed; the mean free path is very long for gases, so the viscosity isn't really low. Moreover, the entropy density of gases is pretty low so the viscosity-to-entropy ratio actually comes out very large for the gases.

Summary

All these insights help us to understand the unity of Nature as well as all the mathematical tools she could have used when She was designing Herself. The most universal prediction of string theory, namely the existence of quantum gravity, plays an essential role in these advances. The unity of phenomena has crossed borders that were thought of as impenetrable intellectual barriers: the boundary between "weakly coupled simple" and "strongly coupled complicated" phenomena and objects.

The AdS/anything revolution has shown that theories in those two "opposite" camps are often equivalent as well. These days, string theory classifies not only the possible consistent theories describing the elementary fields including the gravitational one (in terms of the stringy vacua): it also classifies large classes of emergent phenomena.

The most universal localized objects predicted by any relativistic theory of gravity are black holes which is why black holes play such an important role in the description of many strongly coupled and complicated systems.

For some situations, the AdS/CFT correspondence is completely well-defined and in principle, it relates doable and arbitrarily accurate calculations on both sides. The duality between the N=4 gauge theory and type IIB string theory on AdS5 x S5.

In other cases, the gravitational side of the duality remains somewhat vague. It remains unclear whether an alternative definition of the gravitational side of the duality may be defined - differently than by the non-gravitational definition - at an arbitary accuracy. It is plausible that a very particular compactification of string theory is exactly equivalent to QCD; it is equally plausible that such a precise compactification cannot be found because it doesn't exist.

But even in the latter case, string theory has surely led us to think about the complicated setups correctly at the qualitative level: it told us what is qualitatively going on and it implies inequalities that don't depend on the exact knowledge of the system. We learn the right "zeroth order approximation" of the right description from string theory and string theory is crucial even for this step. Remarkably enough, sufficiently strongly coupled complicated systems love to approach this limit very closely.

Degrees of freedom that are numerous and interact strongly - and with many others - inevitably resemble black holes whose degrees of freedom have the same properties taken to the extreme limit.

And that's the memo.