## Wednesday, September 21, 2005

### Viscosity and Andreas

There are many interesting people now at Harvard, and also many interesting visitors. For example, our physics buddy Andreas Karch - who is now in Seattle - is visiting us, too. He finds "AdS/QCD" to be the among the most interesting topics these days to work on. I want to write some general comments about it later.

One of the things that he clarified for me was the origin of the AdS/CFT derivation of the lower bound for the viscosity that we discussed for example here. In a huge class of theories, the viscosity must be greater than "1/4.pi" times the entropy density, counted in fundamental units. It seems to be true in general. In a subclass of the theories, one may construct a gravitational dual of the theory. How do you derive the bound from the gravitational dual? Let's start in the field theory. The viscosity may be calculated, in the field theory, from some two-point function of the stress-energy tensor. This quantity is directly translated to the gravitational picture.

The relevant calculation involves a graviton propagator in the background of a large (greater than the AdS curvature radius) AdS Schwarzschild black hole; such a black hole is generically dual to the lowest-viscosity environment. It's a rather tedious calculation but you may get the result including the numerical pre-factor. A simpler calculation - one that Andreas found less comprehensible - involves the quasinormal modes. But Andrei Starinets comments:

Viscosities (shear and bulk) of thermal theories with gravity duals can be computed basically in three ways (all inter-related, of course):
• via Green-Kubo formulae (graviton's absorption on the gravity side)
• by computing the (retarded) correlator of stress-energy tensor in (Lorentzian signature) AdS/CFT
• by computing the lowest quasinormal frequency of the relevant gravitational background. This frequency is precisely the hydrodynamic pole in the above mentioned correlator. Computing quasinormal frequencies is sometimes technically easier than computing full correlators, but it is absolutely rigorous. If you like, please look through our recent paper with Pasha Kovtun, hep-th/0506184, where we sort of summarize this.

1. Dear Lumos,

In a previous post you wrote:

"The ratio of shear viscosity to the volume density of entropy seems to be always greater than a fixed constant "1/4.pi" (times hbar over Boltzmann's constant). The inequality is saturated for a large class of strongly coupled interacting quantum field theories - corresponding to a kind of ideal fluids - and one can explain it by the fact that they are the holographic dual of a gas of black holes in some kind of anti de Sitter space."

In the new post you write: "The relevant calculation involves a graviton propagator in the background of a large (greater than the AdS curvature radius) AdS Schwarzschild black hole; such a black hole is generically dual to the lowest-viscosity environment. It's a rather tedious calculation but you may get the result including the numerical pre-factor. A simpler but less rigorous calculation involves the quasinormal modes."

Are you saying that in field theories you can have viscosity present due to the field in space? If that is what you saying then there will be a drag force on a particle moving through space which is proportional to the viscosity.

If the spacetime fabric is a 'gas of black holes' in 5-D, these appear as radiation on the 4-D hologram if I understand what is suggested. The spacetime fabric as seen in 4-D is radiation without viscosity. However, I'm probably on a different wavelength and shouldn't comment.

Best wishes,
Nigel

2. Hi Lubos,

Found the New Scientist article that started your previous post on this subject:

"Exotic black holes spawn new universal law
16:24 23 March 2005
NewScientist.com news service
Jenny Hogan

"Black holes may define the perfect fluid, suggests a study of black holes that only exist in a theoretical 10-dimensional space. The finding may have spawned a new universal law in physics, which puts constraints on the way fluids behave in the real world.

"Dam Thanh Son from the University of Washington, US, and his colleagues used string theory to model a 10-dimensional black hole as a liquid. String theory tries to explain fundamental properties of the universe by predicting that seven dimensions exist on top of the known three spatial dimensions. While the concept is currently unproven as a cosmological model, the tools of string theory can sometimes provide answers to real quantum problems.

"That means that while the 'black holes' modelled by Son are not astrophysical black holes, but mathematical objects that exist in string theory, their findings may have relevance to the real world.

"The fluid has two properties that relate to the black hole's surface area: viscosity, which describes how thick the liquid is, and entropy density, which is a measure of the internal disorder. Son's team found that the ratio of these two properties is a constant which can be expressed as a mixture of fundamental constants from the quantum world."

My initial reaction above was inspired by the reference you made to a "holographic dual of a gas of black holes" in space.

However, I don't care about the entropy of a black hole. All I'm interested in is whether black holes have relevance to the spacetime fabric. The holographic conjecture seemed to me to imply that for dealing with the space time fabric, it is admissible to treat the gauge bosons which cause gravity as radiation which in 5-D spacetime is equivalent to black holes (ie, mass), so the radiation is associated with an equivalent mass of black holes.

Best wishes,
Nigel

3. To be clear, this is my point again:

Spin and extra dimensions

Does the difference between the known differing spin of bosons and fermions derive from the freedom to spin in an extra dimension?

If indeed black holes (mass) in 5-D spacetime are equivalent to radiation on the 4-D hologram, does this equate fermions in 5-D with bosons on the 4-D hologram? Is spin altered by freedom in extra dimensions?

In electromagnetic theory, a boson like a photon contains equal amounts of positive and negative electric field energy travelling at light speed with integer spin and ‘no rest mass’ (it is never at rest anyway).

A fermion like an electron has just one type of electric field energy (negative in the case of an electron), and has half integer spin with a rest mass (it can be at rest, while still spinning around some internal axis).

I think this is the kind of deep question that should be addressed. Is the distinction, between normal fermions and bosons, that spin can include an extra dimension?

Nigel