Short-lived modes from hydrodynamic dispersion relations
Department of Theoretical Physics, University of Geneva, 24 quai Ernest-Ansermet, 1214 Genève 4, Switzerland
We consider the dispersion relation of the shear-diffusion mode in relativistic hydrodynamics, which we generate to high order as a series in spatial momentum for a holographic model. We demonstrate that the hydrodynamic series can be summed in a way that extends through branch cuts present in the complex plane, resulting in the accurate description of multiple sheets. Each additional sheet corresponds to the dispersion relation of a different non-hydrodynamic mode. As an example we extract the frequencies of a pair of oscillatory non-hydrodynamic black hole quasinormal modes from the hydrodynamic series. The analytic structure of this model points to the possibility that the complete spectrum of gravitational quasinormal modes may be accessible from the hydrodynamic derivative expansion.
As an effective theory of conserved currents near equilibrium, the strength of hydrodynamics lies in its universal applicability. This is reflected in the fact that at any given order in the derivative expansion only finitely many terms can be written down, reducing any theory with the appropriate limit to a set of transport coefficients.
Recently the convergence properties of the hydrodynamic derivative expansion have been investigated. In the context of boost-invariant flows in various models, it has been shown that the late proper time expansion of the stress tensor has a zero radius of convergence , with many further interesting investigations in this context [2, 3, 4, 5, 6, 7, 8, 9, 10, 11]. The divergent nature of the series was argued to be connected to the factorial growth in the number of allowed terms at each order in the hydrodynamic expansion. These works demonstrated that the perturbative expansion can be understood as one piece of a more general transseries, containing non-perturbative contributions from short-lived non-hydrodynamic modes which decay exponentially in proper time. The contribution of these non-hydrodynamic modes were inferred from the perturbation series using the machinery of Borel-Padé. Another example is provided by the summation of the hydrodynamic expansion applied to a cosmological setting . See also the recent review for related results .
It is an interesting question in general, as to how much information about the underlying microscopic theory can be extracted from a hydrodynamic derivative expansion. In this work we adopt a different approach to this question, focussing on the large order behaviour of the dispersion relations of hydrodynamic modes. This allows us to directly access short-lived non-hydrodynamic mode dispersion relations of the underlying theory, by appropriately summing the hydrodynamic series. Aside from direct access to dispersion relations, an advantage to this approach is that we are able to elucidate connections that exist between various modes through navigating the complex plane.
By way of a concrete example, we focus our attention on the dispersion relation of the shear-diffusion mode, whose hydrodynamic expansion can be written in the form of a series in spatial momentum ,
Note that as required for a hydrodynamic mode; at it is a zero-mode associated to translational symmetry and momentum conservation. The leading term in can be obtained from first-order relativistic hydrodynamics, as the dispersion relation that arises for transverse linear perturbations around equilibrium, finding,
where is the shear viscosity, the energy density and the pressure.111A review of this material and relativistic hydrodynamics in general can be found in . We may also include a finite charge density, , with chemical potential and the hydrodynamic variables become and a fluid velocity.
The coefficients of this series (1.1) are calculable once an equation of state and a set of transport coefficients (at first order this is just ) are given. This data may be fixed by matching to an underlying microscopic model if it has a hydrodynamic limit. In this paper we pick an underlying model and generate the expansion (1.1) in precisely this way. A particularly convenient theory to work with is one with a holographic dual – then it is relatively straightforward, at least numerically, to obtain higher order terms in the expansion (1.1) for hydrodynamics in dimensions by solving ODEs in a spacetime of dimension . In holography the first term has been computed exactly and the associated transport coefficient found to be, famously, [15, 16]
where is the entropy density. The Taylor expansion (1.1) proceeds in even powers of and one must go to third order in the hydrodynamic expansion to extract the next correction. For holography at zero charge density in this was carried out in , for the sound mode at third order see .
In this paper we pick the specific holographic model defined by Einstein-Maxwell theory in AdS. In this case the equilibrium state is given by a Reissner-Nordstrom black brane in the bulk. The linear modes of this theory, as poles of retarded correlators of the conserved currents, are quasinormal modes of this black brane. These modes exhibit a rich and interesting structure as a function of and , as was depicted beautifully in . Since we are first interested in the hydrodynamic Taylor series and its convergence properties, we are interested in the analytic structure in the complex plane. The exact222i.e. without performing the hydrodynamic expansion has a multi-sheeted structure, and we shall see that the longest-lived sheet (that which has the largest value of Im, the hydrodynamic sheet) contains branch points at
At least for the examples of that we study, these branch points are the closest non-analyticity to , and set a finite radius of convergence of the hydrodynamic expansion of the dispersion relation, (1.1). This point, and thus the physical origin of the scale , can be interpreted as the location of a collision between a hydrodynamic and non-hydrodynamic mode on the imaginary axis. Circumnavigating this point, we can move to a second, shorter-lived sheet, and that sheet is associated to a non-hydrodynamic mode.
The key result of this work is that we can first determine the location of these branch points from hydrodynamic series itself, and then we can accurately describe such additional sheets with a particular summation of the hydrodynamic expansion. In doing so, we tease out the dispersion relation of non-hydrodynamic modes from hydrodynamic data.333In the context of divergent series the non-hydrodynamic mode arises as from an ambiguity in performing the inverse Borel transformation, see e.g. [1, 2].
The remainder of the paper is structured as follows. In section 2 we detail the underlying model we will use to generate the hydrodynamic expansion of the shear-diffusion mode to high order in . In section 3 we explore the convergence of the hydrodynamic expansion, and in 4 we extend the expansion onto a second sheet by circumnavigating the branch points, extracting the frequency of a pair of non-hydrodynamic modes. We conclude with a discussion in section 5.
2 Hydrodynamic coefficients from holography
The spectrum of vector quasinormal modes of a Reissner-Nordstrom black brane in Einstein-Maxwell theory in AdS, contains a dispersion relation of the form (1.1) at small . This is because the model is holographically dual to a CFT at finite and , and this mode is precisely the shear diffusion mode in the hydrodynamic limit of that CFT. The goal of this section is to compute the small- Taylor expansion of the dispersion relation of this mode to high order, by solving an appropriate linear gravitational problem on the Reissner-Nordstrom background.
In Schwarzschild-like coordinates we have the following Reissner-Nordstrom black brane metric and gauge field, with holographic coordinate scaled such that the horizon is at ,
where the metric function
On this background we consider the following transverse metric and gauge field linear perturbations,
which satisfy a closed set of ODEs in the -coordinate of total differential order . It is possible to construct two linear combinations of the above variables, the master fields , which satisfy decoupled ODEs . Here we define,
With this change of variables we have the following equations of motion for the master fields,
The remaining equation is an equation that determines , but it is dependent on and so for the purposes of finding the values of we do not need to consider it.
Note that this master field equation reveals something about the analytic structure of solutions we will obtain, since it contains a square root involving . The two sectors are therefore connected through branch cuts, with a corresponding branch point occurring at (1.4). However, the purpose of the holographic model for this work is purely to compute the hydrodynamic dispersion relation (1.1) as a Taylor series in , and so we are strictly only interested in the hydrodynamic sheet in this section. This is the sheet whose frequency satisfies , and corresponds to the ‘’ choice above.
We impose boundary conditions for quasinormal modes, namely the absence of sources on the boundary of AdS at , and ingoing boundary conditions at the black hole horizon at . We therefore have a boundary value problem to solve, with eigenvalues which will depend on the momentum of interest . To compute the hydrodynamic as a Taylor series in , we expand both the perturbation and the eigenvalue , resulting in an infinite series of ODE boundary value problems, i.e.
Notice that the expansion for begins at in accordance with (1.1), selecting the hydrodynamic mode. The ingoing behaviour of near the horizon can be expressed as a series around ,
where we have fixed an arbitrary coefficient to unity. We subsequently expand this expression in to generate horizon boundary conditions for each of the . At level we can easily solve the equation, to obtain . Next order will determine and , and so on. We turn to numerics to evaluate the remaining and associated frequency coefficients for . To obtain one must solve ODEs. We do this using a shooting method, integrating out from the horizon with a guess for and iteratively improving this guess until we satisfy the boundary conditions at . The values we obtain are summarised in table 1.
3 Radius of convergence
Using the specific microscopic model detailed in the last section, we have determined the Taylor expansion of (up to some finite order , i.e. hydrodynamic order ),
Rendering the dimensionless using the scale , we present the coefficients obtained in table 1.
Next we can determine the radius of convergence by looking at the ratios of successive terms. Specifically we form the new dimensionless sequence,
The large- behaviour of this sequence is as illustrated in figure 1, so converges to confirming that as expected, the radius of convergence of the hydrodynamic expansion is governed by , i.e. the branch points at .
As a further diagnostic, we compute the diagonal Padé approximant of the series. i.e. defining a ratio of two polynomials,
calculate the coefficients from the coefficients by matching order-by-order in the Taylor series around . We plot the poles and the zeros of in figure 2 for the upper-half plane, showing an alternating sequence of poles and zeros along a radial ray. The closest pole to is given by , a good indicator of the branch point at (the same structure exists in the lower-half plane).
4 Through the branch cut
As we have established in the last section the Taylor expansion (3.1) has a finite radius of convergence set by a branch point in the complex- plane. We can determine the location of this point empirically from our hydrodynamic data in table 1, by looking at the radius of convergence, and also at the poles and zeros of the Padé approximant.
Armed with this knowledge of the nearest non-analytic obstacle to convergence of the hydrodynamic expansion, we adopt a new variable, , that can cover more than one sheet. In particular, the branch point is of square-root type and so we want to describe two sheets in the simplest possible way. Thus we are led to consider the following quadratic polynomial in z,
We fix the coefficients as follows. First we demand that at so we may consider instead a new Taylor series around , which implies . Next we require that the two solutions for have a branch point at a prescribed location, , which fixes . Finally we use the freedom to rescale to set , resulting in
For instance, on the first sheet is given by and on the second by . Thus we are matching the topology of the surface, and utilising some geometrical information: the location of the branch point itself.444For conformal maps in the context of the Borel plane, see for example . We set , the branch point discussed in section 3. With the transformation in place, we convert the expansion (3.1) to a new Taylor expansion in around . In order to see that there are now two sheets, one must convert back to the variable . However for this model we have found that we must go one step further, and sum the series as a Padé approximant in order to achieve good results. Specifically we construct the diagonal Padé approximant in the variable , . As can be seen in panel b) of figure 3 under comparison with the exact results, extends the hydrodynamic expansion onto a second sheet, with remarkable agreement.
Note however that there is a second branch point preventing us from accessing the real line on the second sheet, and is not sufficient. This is also indicated by a sequence of poles and zeros of in figure 4.
To pass this second branch point and get back to the real line, we simply repeat the above procedure transforming the variable to a new complex variable ,
so that we take into account the second branch point at . Note that we have now generated four sheets in total, though we will restrict our attention to the first three. As before, it is possible to estimate the value of purely from the hydrodynamic data. Specifically, we look for the pole of which occurs after we pass onto the second sheet, in figure 4. To the order considered we find the location of this pole , whilst from the exact solutions we compute the location as . We show the Padé approximant for the new variable , , resulting from both choices in the final two panels, c) and d), of figure 3. Whilst in panel c) we pay a penalty for not knowing the exact value of , remarkably it appears that the resulting deviations are restricted to the region near the branch point, decaying rapidly as we go towards . We have verified that this behaviour also persists after varying more significantly, and the value of the frequency on the real axis appears stable.
A comparison is made in table 2 of the first non-hydrodynamic mode frequency for various truncation orders, made using the estimated . In this table we see that by order 11 in the hydrodynamic expansion we have a reasonable estimate for the frequency, and by the time we reach order 79 the agreement is correct to four significant digits. Finally, in figure 5 we present plots of the two sheets of we have obtained in the complex plane from hydrodynamic data. This of course displays the dispersion relations of the non-hydrodynamic mode on the real- axis too, which we can effortlessly read off from the summed series.
By a suitable summation of the hydrodynamic Taylor expansion of the dispersion relation of the shear diffusion mode, we have extracted the frequencies of non-hydrodynamic modes. In particular at order 79 in the hydrodynamic expansion we extracted a pair of off-axis modes,
in agreement with a pair of off-axis short-lived modes coming from the underlying holographic theory, here up to four significant digits. These are the longest lived non-hydrodynamic modes in this channel. Our method explored the analytic structure of in complex , switching complex variables to cover multiple sheets where branch points were encountered. We designed the map of complex variables to be as simple as possible, built from a quadratic relation in order to produce the two sheets at a square-root branch point. From there we found that summing the series as a Padé approximant gave excellent results.
We were able to infer where the branch points occurred empirically, by the convergence properties of the Taylor expansion itself. However, the results we obtained on the real axis did not appear to be particularly sensitive to the precise location of the branch point – we demonstrated that the associated errors near this point, whilst very large, were damped as one approaches the real axis. This points to a simpler method to extract the modes, perhaps requiring only topological data.
We restricted our attention to the first few sheets, but it is an interesting question in general as to how much information about the spectrum of the underlying theory – in this case the quasinormal mode spectrum of a black hole – can be mined from the hydrodynamic data. It would of course also be desirable to have a systematic prescription to convert hydrodynamic data into a spectrum of non-hydrodynamic modes, perhaps one that does not rely on such a detailed understanding of the analytic properties, and the connectivity of the sheets in the complex plane. We postpone such questions to future work, except to say that at certain values of in this particular model, the level of branching can be very high, see figure 6. At this value of , along Re on the hydrodynamic sheet, one can seemingly access infinitely many other sheets via a series of branch cuts further down the Im axis.
Finally we conclude by observing that the structure of dispersion relations in the complex plane, which we exploited here, is not only of formal interest. Such modes can be physically relevant in situations where the resulting exponential growth in space is cut off by a physical source or boundary condition, as has been recently demonstrated in  for the case of steady flows over codimension-1 obstacles. In the relativistic context, instead of taking the Im slice of the dispersion relation as a surface in complex and complex , these modes may be viewed as the result of a different slicing that is defined by the velocity of the background fluid.
It is a pleasure to thank Massimiliano Ronzani, Julian Sonner, Peter Wittwer and Szabolcs Zakany for discussions. I am supported by the NCCR under grant number 51NF40-141869 ‘The Mathematics of Physics’ (SwissMAP).
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