A bound on the speed of sound from holography?

Posted by Aleksey Cherman

May 14, 2009

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This post is authored by Aleksey Cherman (on the left) and Abhinav Nellore (on the right). Aleksey is a graduate student in the nuclear theory group at the University of Maryland, College Park, working with Tom Cohen, and Abhi is a graduate student in Steve Gubser’s group at Princeton. Dmitry.

We all know that sound travels at about 343 m/s in air, and much faster than that in many solids. But just how much faster could sound travel if given the chance? Could there be a medium in which the speed of sound can approach the speed of light? Or might there be some more stringent fundamental bound on the speed of sound?

As it happens, the answer to the last question is no. There is a medium in which the speed of sound ( v_s^2 ) can approach the speed of light : a gas of pions (each of mass $m_{\pi}$ ) at a finite isospin chemical potential $\mu_i$ . When $\mu_i > m_{\pi}$ , but is small compared to $4 \pi f_\pi$ , it is possible to calculate the speed of sound using using chiral perturbation theory, and it is not hard to show that it approaches the speed of light in the chiral limit of m_pi going to zero. (see Son and Stephanov.)

So clearly there’s no general speed limit on the speed of sound for all consistent field theories. But let’s lower our ambitions a bit. Might there still be some broad class of theories that doesn’t include the counterexample above where there IS an interesting speed limit for the speed of sound?

As it turns out, the answer to that question is yes! But conditionally: we have to lower our ambitions still further. Working with Tom Cohen, we have been able to show is that in an a certain class of strongly coupled systems, v_s^2 must approach 1/3 c^2 from BELOW at high temperatures. That is, at least in this class of theories, and at least at high temperatures, there is indeed an interesting speed limit for sound. We do not yet know whether the speed limit applies away from the high temperature limit – that’s a subject for future work.

(We’re not the only ones who have worked on this: Paul Hohler and Misha Stephanov independently got the same results we did using different methods, and our papers appeared simultaneously. )

To calculate the speed of sound, we used the gauge/gravity duality. The gauge/gravity duality is a marvellous tool: it lets us calculate observables in some strongly coupled gauge theories just by doing classical calculations in theories with gravity. But as with most good things in life, there is a catch: there are no known gravity duals for any of the gauge theories currently used to describe nature. This means that the duality can’t be used to make quantitative predictions for nature: at best, one can hope to learn some interesting qualitative lessons about the behavior of strongly-coupled systems.

In our work, we considered 3+1 dimensional systems that have gravity duals with a single scalar field $\phi$ . These systems can be thought of as strongly-coupled large N conformal field theories deformed by the addition of a single relevant operator $\mathcal{O}_{\phi}$ , which is dual to the bulk scalar $\phi$ . These single-scalar gravity theories are the simplest non-conformal gravity duals. Different choices of potentials for the bulk scalar field correspond to different dual gauge theories.

In a 4D conformal field theory, the speed of sound is simply a constant: v_s^2=1/3 , as can be seen from the general fact that $v_s^2=dp/d\epsilon$ ( and $\epsilon$ are the pressure and energy density of a system), and the fact that the trace of the stress-energy tensor vanishes in a CFT, so that $\epsilon=3p$ . In a non-conformal field theory, the speed of sound depends on the temperature and other properties of the theory. The theories we decided to work with are relevant deformations of a CFT, so they should look like CFTs in the far UV. This means that at high temperatures, the v_s^2 should approach 1/3. What is not so obvious is whether v_s^2 approaches 1/3 from above or below: that is, is the 1/3 an upper bound?

To answer this question, we developed a high-temperature expansion for the geometry and the profile of the bulk scalar. At very high temperatures, the geometry should look like an AdS_5 Schwarzschild black hole, with a vanishing scalar field – that is, the system should become approximately conformal. Thus, our high-temperature expansion is an expansion around an AdS_5 Schwarzschild geometry. It turns out that the leading correction to the geometry is almost completely insensitive to the details of the scalar potential: one only needs to know the UV scaling dimension of $\Delta$ of $\mathcal{O}_{\phi}$ . Note that $2 < \Delta < 4$ , where the upper bound is due to the restriction to relevant operators, and we use the lower bound to avoid a subtlety involving the BF bound for stability of scalars in AdS_5 spaces (this does not affect our conclusions).

Once we found a way to calculate the high-temperature geometry, calculating the speed of sound was easy. In systems at zero chemical potential, the speed of sound can also be written as $v_s^2=d \log T/d \log s$ , where is the entropy. The entropy and temperature can be read off from the geometry, and we find that in the high temperature limit

$v_s^2=1/3 – C(\Delta) (\pi L T)^{2(\Delta-4)}$

where

$C(\Delta)=\frac{1}{18 \pi} (4-\Delta)(4-2\Delta) \tan \left(\pi \Delta/4\right)\frac{\Gamma(\Delta/4)^4}{\Gamma(\Delta/2-1)^2}$

Clearly, the correction away from 1/3 is always negative for $\Delta$ in the allowed range. So for all systems in the class that we considered, $1/\sqrt{3} c$ is the ultimate speed limit for sound, at least at high temperatures! It would be very interesting to see whether (and under what circumstances!) this speed limit still holds for lower temperatures.

Well, at this point you might say that this is all very well, but this was in the context of a pretty limited class of theories. That’s a fair point. As it happens, our result holds in systems with gravity duals with more than one scalar field as well, as will be discussed in a paper we are now finalizing.

But in fact, to our knowledge, $v_s^2 \le 1/3$ in all 4D theories with gravity duals, at least when the systems in question are energetically stable (i.e., in a state of lowest free energy). This is true both in fairly ad-hoc models like the ones we worked with in our paper, and in more sophisticated top-down models using various brane constructions.

So what are the next steps? First, it makes sense to look for counterexamples, to try to figure out the right domain of validity of the sound bound. Just how broad (or narrow!) is the class of theories to which it applies?

In a similar vein, it would be great to come up with some general argument that would show whether this kind of speed limit is a general property of holographic theories, or of some interesting subclass of them. Maybe such a speed limit can tell us something interesting about holography? If v_s^2 is indeed bounded by 1/3 in all theories with gravity duals, the next obvious thing to look at would be 1/N and finite coupling corrections to the speed of sound, to see how robust the results are away from the supergravity limit where N and the ‘t Hooft coupling are infinite…

For more on all of this, see the two papers below, and the references in them.

1. A. C., T. D. C., and A. N., http://arxiv.org/abs/0905.0903

2. P. M. Hohler and M. Stephanov, http://arxiv.org/abs/0905.0900

AdS/CFT, Hydrodynamics, Master Mason, Quantum field theory, String theory

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3 Comments »

Comment by Dmitry

2009-05-14 15:02:10

Dear Aleksey and Abhi,

thanks for the nice post! I would like to ask a somewhat arrogant question: why this bound seems surprising to you (and Stephanov)?

First of all, any relativistic matter has an equation of state $\epsilon=3p$ . I think there is no surprise that any plasma becomes ultrarelativistic at high temperatures (that is, $T\gg{}m_{\rm eff}$ ), where $m_{\rm eff}$ is the largest among effective masses of degrees of freedom of the theory.

Bare is of course zero, since you consider CFTs, but due to interactions between different degrees of freedom their effective masses can be non-zero.

If this is so, I guess, there is also no surprise why this bound is upper bound since at lower temperatures plasma should become non-relativistic. What do you think?

Cheers,
Dmitry.

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Comment by Abhinav Nellore Subscribed to comments via email

2009-05-14 22:03:43

Hey Dmitry,

Thanks for the question! It’s not true that all relativistic matter has \epsilon=3p. Consider the pion gas example treated by Son and Stephanov and mentioned in the blog post Aleksey and I wrote. This is clearly a relativistic system, but it nevertheless has v_s->the speed of light, and \epsilon does not equal 3p. CFTs have \epsilon=3p because the trace of the stress-energy tensor vanishes. So CFTs trivially have v_s^2=1/3. The system treated by Hohler, Stephanov, and my collaborators and me is conformal in the limit T->\infty. What’s not clear is whether 1/3 is approached from above or from below as T->\infty. What we show is that it is approached from below for a wide class of theories.

Abhi

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Comment by Dmitry

2009-05-20 14:17:40

Dear Abhi,

sorry for the tardy response, I am currently in US and my laptop does not want to work

EOS $\epsilon=p$ is what’s called stiff matter EOS, it only holds for relativistic scalar field and in the regime where kinetic energy completely dominates over potential energy. If the opposite limit is true, for example, EOS is $\epsilon=-p$ , the one of cosmological constant. Your counterexample (gas of pions, and pions are scalars) belongs to this class.

Now, if you deal with relativistic plasma with many different components (the N=4 plasma you deal with in AdS/CFT belongs to this class), EOS is typically $\epsilon=3p$ (check out for example Landau-Lifshitz, vol. II). So, the real question is which degrees of freedom give dominating contribution into overall energy-momentum tensor of N=4 plasma – if scalars are important, then you might see deviations from $\epsilon=3p$ , if vectors/fermions dominate, then the latter EOS is the most natural to expect. I guess what you ultimatly find is that scalars are not that important for the overall dynamics of plasma.

Cheers,
Dmitry.

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A bound on the speed of sound from holography?

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