Dyonic AdS black holes from magnetohydrodynamics

Marco M. Caldarelli, Óscar J. C. Dias and Dietmar Klemm

Departament de Física Fonamental, Universitat de Barcelona,

Marti i Franquès 1, E-08028 Barcelona

[.3em] Instituut voor Theoretische Fysica, Katholieke Universiteit Leuven,

Celestijnenlaan 200D B-3001 Leuven, Belgium

[.3em] Dept. de Física e Centro de Física do Porto, Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre 687, 4169 - 007 Porto, Portugal

[.3em] Dipartimento di Fisica dell’Università di Milano,

Via Celoria 16, I-20133 Milano and

INFN, Sezione di Milano, Via Celoria 16, I-20133 Milano.

[.3em]

marco.caldarelli@fys.kuleuven.be, oscar.dias@fc.up.pt, dietmar.klemm@mi.infn.it

ABSTRACT

We use the AdS/CFT correspondence to argue that large dyonic black holes in anti-de Sitter spacetime are dual to stationary solutions of the equations of relativistic magnetohydrodynamics on the conformal boundary of AdS. The dyonic Kerr-Newman-AdS solution corresponds to a charged diamagnetic fluid not subject to any net Lorentz force, due to orthogonal magnetic and electric fields compensating each other. The conserved charges, stress tensor and R-current of the fluid are shown to be in exact agreement with the corresponding quantities of the black hole. Furthermore, we obtain stationary solutions of the Navier-Stokes equations in four dimensions, which yield predictions for (yet to be constructed) charged rotating black strings in AdS carrying nonvanishing momentum along the string. Finally, we consider Scherk-Schwarz reduced AdS gravity on a circle. In this theory, large black holes and black strings are dual to lumps of deconfined plasma of the associated CFT. We analyze the effects that a magnetic field introduces in the Rayleigh-Plateau instability of a plasma tube, which is holographically dual to the Gregory-Laflamme instability of a magnetically charged black string.

###### Contents

## 1 Introduction

The AdS/CFT correspondence (cf. [1] for a review)
has provided us with a powerful tool to get insight into the strong coupling
dynamics of certain field theories by studying classical gravity.
In the long wavelength limit, it is reasonable to expect that these
interacting field theories admit an effective description in terms of
hydrodynamics. One can thus conversely use the equations of fluid mechanics
(which, in simple contexts, are much easier to solve than the
full set of Einstein’s equations) in order to make predictions
for the gravity side. This was done in [2], where the
fluid configurations dual to (yet to be discovered) black rings in
(Scherk-Schwarz compactified) AdS were constructed^{1}^{1}1For a discussion
of plasma lumps dual to black saturns in AdS cf. [3]..

More recently, it was shown in [4] (cf. also [5] for the four-dimensional case) that the equations of hydrodynamics (i. e., the Navier-Stokes equations) can also be derived directly from Einstein’s equations with negative cosmological constant, without making any use of string theory, from which these ideas originally emerged.

The correspondence between gravity in asymptotically AdS spaces and fluid mechanics has interesting consequences: For instance, it is well known that under certain conditions fluids are affected by the so-called Rayleigh-Plateau instability (responsible e. g. for the pinch-off of thin water jets from kitchen taps), and one might ask what the gravity dual of (the relativistic analogue of) this instability is. This point was studied recently in [6], where it was argued that the gravity dual is the Gregory-Laflamme instability [7].

In this paper, we shall consider a more general setting in which the charged fluid moves in external magnetic fields. The fluid is thus described by the equations of magnetohydrodynamics (MHD). Such a generalization is of interest for several reasons: First, there are two possible fall-off conditions for a bulk U(1) gauge field: A normalizable mode that corresponds to a VEV of the dual operator (an R-current), and a non-normalizable one corresponding to the application of an external gauge field, that can be thought of as an electromagnetic field. (2+1)-dimensional field theories deformed in this way, that are dual to magnetically charged AdS black holes, have become fashionable recently in the context of a possible holographic realization of condensed matter phenomena like superconductivity [8, 9], the Hall effect [10] and the Nernst effect [11]. In the long wavelength regime a quantum field theory in presence of external gauge fields is expected to be described by MHD. Second, solving the MHD equations (which, under certain symmetry assumptions, might be much easier than solving the full set of Einstein-Maxwell equations) can be helpful for constructing new magnetically charged black hole solutions in AdS. Furthermore, the phase structure of such AdS black holes can be studied in a simplified setting using magnetohydrodynamics.

We should stress that the magnetic field entering the MHD equations is nondynamical, so that there are no Maxwell equations on the boundary. Note however that, in the same way in which one can promote the CFT metric to a dynamical field [12], it should be possible to add dynamics also to the magnetic field, although we will not try to do this here.

The remainder of this paper is organized as follows: In section 2, we review general aspects of MHD, like the equation of state and the stress tensor of the fluid under the influence of external magnetic fields. Moreover, we show how the MHD equations emerge from Einstein-Maxwell-AdS gravity. In the following section, we generalize the results of [13] to the case of nonvanishing magnetic fields. We first consider static dyonic black holes in AdS. In an appropriate limit (when the horizon radius is much larger than the AdS curvature radius ), these black holes are effectively described by magnetohydrodynamics. Conformal invariance and extensivity imply that the grandcanonical partition function has to take the form

(1.1) |

where denotes a chemical potential conjugate to the U(1) electric charge of the fluid, is the applied magnetic field, and and represent the volume and the overall temperature of the fluid respectively. We then show that, using the function that can be read off from the partition function of static black holes as an input into the MHD equations, we can exactly reproduce the conserved charges, boundary stress tensor and R-currents associated to the rotating dyonic Kerr-Newman-AdS solutions. Their thermodynamics is very simple: It is summarized by the partition function

(1.2) |

with , and is the angular velocity of the black hole.

After that, we solve the Navier-Stokes equations on , or . This yields predictions for (yet to be constructed) charged rotating black strings in AdS with momentum along the string.

In section 4 we consider perturbations around plasma tube solutions of the MHD equations on , that are dual to magnetic black strings on Scherk-Schwarz compactified AdS. These plasma tubes suffer from the long wavelength Rayleigh-Plateau instability, which is shown to be weakened by the presence of a magnetic background. This is exactly what one expects from the known studies of the Gregory-Laflamme instability for magnetically charged black strings in asymptotically flat space, to which the considered SS-compactified AdS black strings are similar.

Throughout this paper we use calligraphic letters to indicate local thermodynamic quantities, whereas
refer to the whole fluid configuration. and
are local and global chemical potentials respectively.

## 2 Magnetohydrodynamics

### 2.1 Equation of state

We would like to know how the equation of state of a conformal fluid in dimensions changes in presence of a magnetic field . To this end, consider the grandcanonical potential

(2.1) |

It follows from conformal invariance and extensivity that

(2.2) |

where we defined and . Note that and have mass dimension one and two respectively, so that and are dimensionless. Equation (2.2) defines the function . From (2.2) it is also clear that

(2.3) |

If we derive this with respect to , set and use

(2.4) |

where is the magnetization density, we obtain

(2.5) |

Using the definition (2.1) of , this yields the equation of state

(2.6) |

We will see below that large dyonic black holes in AdS indeed have a grandcanonical potential of the form (2.2), with .

### 2.2 Stress tensor of the fluid

We want to determine now the general form of the stress tensor for our fluid, and we will do this in a derivative expansion in the fluid velocity , following [15]. The zero order term corresponds to a perfect fluid (under the influence of external electromagnetic fields , labelled by the index ), with stress tensor, charge currents and entropy current given respectively by (cf. e. g. [11])

(2.7) |

where we have introduced the projection tensor

(2.8) |

on the directions orthogonal to . is the rest frame energy density, and are the rest frame charge- and entropy densities, while and denote the chemical potentials and magnetic fields respectively. The polarization tensor is defined by

(2.9) |

Note that the microscopic currents (polarization currents) are given by

(2.10) |

whereas represents the total current density, so that the combination appearing in (2.7) is the macroscopic or transport current. Notice also that the extra contribution to the stress tensor in (2.7) comes from the coupling of the polarization to the electromagnetic field.

The first subleading order in the derivative expansion of the MHD equations describes dissipative phenomena like viscosity and diffusion. Lorentz invariance and the requirement that the entropy is non-decreasing determine the form of the stress tensor and the currents. Let us introduce diffusion currents ,

(2.11) |

with the constraint , meaning that the diffusive process is purely spatial according to an observer comoving with the fluid element. Next we introduce the heat flux and the viscous stress tensor, decomposed in a symmetric traceless part and a trace , such that

(2.12) |

Again these quantities are purely spatial, and verify the constraints

(2.13) |

Note that the new fields , , and are first order in . Finally the entropy flux is a linear combination of all the available vectors that are at most first order in . Therefore

(2.14) |

where the scalars and are functions of the thermodynamic parameters.

To determine the most general form allowed for the newly introduced quantities we impose the second law of thermodynamics. To have a non-decreasing total entropy the four-divergence of the entropy flux must be positive,

(2.15) |

Projecting the MHD equations on we obtain

(2.16) |

which can be cast,
with some straightforward manipulations, in the
form^{2}^{2}2To get this, one has to use
, with the
susceptibilities , as well as the Bianchi
identities for .

(2.17) | |||||

with the expansion .

We have the Euler relation

(2.18) |

which implies the Gibbs-Duhem relation

(2.19) |

with the magnetization densities . This yields

(2.20) |

The entropy density is a function of , and , so that its gradient reads

(2.21) |

Then

(2.22) | |||||

The conservation of the charge currents yields the
relation^{3}^{3}3Note that the polarization current is separately conserved,
.

(2.23) |

that, together with (2.17), can be used to put the divergence of the entropy flux in the form

(2.24) |

Note that, in order to get (2.24), we assumed that the grandcanonical potential depends on only through the invariant (no summation over ), and we defined to be given (up to a prefactor) by . This leads to

(2.25) |

We ensure that the right hand side of (2.24) is positive or vanishing by requiring that each term be positive or vanishing. This can be easily achieved by choosing

(2.26) |

Then, the diffusion currents are given by

(2.27) |

where we have introduced the projector to ensure that the diffusion currents be spacelike, and the diffusion matrix is positive definite. Moreover, we set

(2.28) |

where is the bulk viscosity, and the heat flux is given by

(2.29) |

with the thermal conductivity of the fluid. Finally,

(2.30) |

with the shear viscosity of the fluid and the shear tensor,

(2.31) |

With these definitions, we find

(2.32) |

which is positive by construction. Notice finally that the stress tensor in (2.7) is traceless on account of the equation of state (2.6).

### 2.3 MHD equations from gravity

In this section, we consider Einstein-Maxwell-AdS gravity in dimensions and show that at leading order the boundary theory dual to a charged -dimensional AdS black hole reduces to magnetohydrodynamics in dimensions in the long wavelength sector. We follow the usual procedure to foliate asymptotically the spacetime with timelike hypersurfaces and regularize the action by adding appropriate boundary counterterms. Then, in the limit, the Brown-York stress tensor has a finite limit, the renormalized holographic stress tensor of the dual CFT. The projection of the Einstein-Maxwell equations on then shows that this stress tensor together with the boundary gauge fields satisfy the equations of magnetohydrodynamics on the boundary. We can interpret this as the leading contribution in the derivative expansion to the boundary MHD equations describing the long wavelength sector of gravity.

The Einstein-Maxwell action with negative cosmological constant reads

(2.33) |

where is the Chern-Simons term, present in the case. are bulk indices, whereas refer to the boundary. In this subsection (and only here), we refer to the bulk gauge field as to distinguish it from the boundary gauge field. In all other sections of the paper we do not need such a distinction. denotes the boundary hypersurface with outward pointing unit normal , and induced metric

(2.34) |

is the trace of the extrinsic curvature defined by (with the bulk covariant derivative)

(2.35) |

Finally, are the usual boundary counterterms needed to obtain a finite action in the limit . We shall not need the precise form of , but only the fact that their variation with respect to the metric is divergence-free,

(2.36) |

Here we have defined as the induced covariant derivative on . Also, notice that additional terms can be added to to handle the logarithmic divergences appearing for odd , corresponding to the Weyl anomaly of the dual CFT. We can use the Fefferman-Graham expansion to write the metric of any asymptotically AdS spacetime near spatial infinity in the form

(2.37) |

where

(2.38) |

The coefficients and depend only on the boundary coordinates and the coefficient , related to the holographic Weyl anomaly, is present for odd only. The normal vector to the constant hypersurfaces is and the induced metric is . Then, the conformal boundary metric of AdS is obtained by taking the limit

(2.39) |

Let us decompose the gauge field in the orthogonal component and its projection on ,

(2.40) |

where

(2.41) |

The analysis of the asymptotic behavior of vector gauge fields [16, 17] shows that has a finite limit, while the current goes to zero like . Therefore, the renormalized background gauge field in the dual field theory, and the -symmetry current read respectively

(2.42) |

Now, by varying with respect to the bulk metric and gauge field, we obtain the
equations of motion^{4}^{4}4In one has to take into account that there is a Chern-Simons term and the Maxwell equations read .

(2.43) |

where the stress tensor of the gauge field is given by

(2.44) |

Then, from the projection

(2.45) |

of the Einstein equations, we obtain, using Gauss-Codazzi

(2.46) |

We recognize in the term in parenthesis the Brown-York boundary stress tensor, which diverges as we take the limit, but will give a finite limit – the holographic stress tensor – once we take into account the counterterm contribution [18],

(2.47) |

Since the counterterm contribution is divergence-free by (2.36), we can add its divergence to the left hand side of (2.46) and, after multiplying the equation by , obtain a finite limit that reads

(2.48) |

where is the covariant derivative induced on the conformal boundary with metric . On the other hand, using the definition (2.41) it follows that , that becomes, in the limit, the conservation law for the -current,

(2.49) |

Note that equations (2.48) and (2.49) are simply the Ward identities associated to the bulk diffeomorphism and gauge invariance respectively [19].

These conservation equations for the boundary stress-energy tensor and -current become the equations for magnetohydrodynamics in the background field when a large black hole is present in the bulk. Indeed, as noted in [20], stationary Kerr-AdS black holes have a holographic stress tensor that assumes the perfect fluid form. We will show in the next section that this is still the case for dyonic Kerr-Newman-AdS black holes, and we believe it to be true in any dimension. This dual stress tensor, of the perfect fluid form, yields, when combined with equations (2.48) and (2.49), the equations of MHD. Then, if one perturbs these stationary solutions with long wavelength disturbances, in the spirit of [4], the horizon can still be decomposed into patches that tubewise extend to the boundary which are approximated by boosted pieces of black branes. Therefore, to leading order in the derivative expansion, large magnetically charged black holes are dual to a magnetohydrodynamic theory. We shall check this explicitely in the next section for dyonic AdS black holes. Higher orders in the perturbation theory produce higher-derivative dissipative terms in the stress tensor. A detailed analysis of this long wavelength sector of gravity, with a complete proof of the duality with MHD and the computation of the dual stress tensor up to third order in the derivative expansion has been performed by Hansen and Kraus in the AdS case and appeared during the last stages of preparation of this manuscript [14].

## 3 AdS black holes and black strings from MHD

### 3.1 Static dyonic black holes in AdS

The equations of motion following from the Einstein-Maxwell action with negative cosmological constant ,

(3.1) |

admit the static dyonic black hole solutions

(3.2) |

where

(3.3) |

and

(3.4) |

The horizon is thus (), () or (). , and denote the mass parameter, electric and magnetic charge respectively. The one-form gauge potential reads

(3.5) |

The strength of the magnetic field in the dual CFT can be obtained (up to rescaling by powers of ) by taking in the expression for the bulk field strength. This leads to . The electric charge density of the state in the field theory is given by , where is the R-current that can be computed as follows. On-shell we have for the variation of the action with respect to the gauge potential

(3.6) |

where denotes the induced metric on the boundary, and is the outward pointing unit normal to the boundary. One has thus in the limit of large

(3.7) |

To get the CFT R-current, one has to rescale this by , so that

(3.8) |

where we used the AdS/CFT dictionary

(3.9) |

Note that the result (3.8) was obtained for
in [10]. In order for the potential to be regular at the
horizon , must vanish there. This requires that we add the
pure gauge term to . This term is non-normalizable
and has the dual interpretation of adding a chemical potential for the electric
charge, , to the field theory [10].
The R-charge is , with the spatial volume^{5}^{5}5In order to
get a finite volume in the cases and , one has to compactify the
horizon to a torus or a higher genus Riemann surface respectively.

(3.10) |

This yields

(3.11) |

The entropy and energy of the black hole read

(3.12) |

(3.13) |

In what follows, we shall be interested in a magnetohydrodynamical
description of the above black holes. Like in [13],
we may estimate the mean free path for the fluid as
, where is the shear viscosity and
is the energy density of the fluid. For fluids described by a
gravitational dual, one has , where is the entropy
density [21]. Consequently, ,
and an MHD description will be valid if this value is much smaller than
the radius of the or the curvature radius^{6}^{6}6In
the (compactified) case, must be much smaller than
the length of any torus cycle., . Using the expressions
for and , this implies

(3.14) |

A sufficient condition for (3.14) to hold is , i. e. , for large black holes. In this case we can neglect the -contribution to the energy, and the thermodynamic fundamental relation becomes

(3.15) |

One easily checks that

(3.16) |

where

(3.17) |

is the Hawking temperature of the black hole. From (3.15), we get the grandcanonical potential

(3.18) |

with the function given by

(3.19) |

where is determined by the fourth order equation

(3.20) |

that follows from (3.17). Note that has indeed the form (2.2), as it must be.

We now want to obtain the dyonic AdS black holes from MHD on , or . The metric on the conformal boundary is given by

(3.21) |

so that the only nonzero Christoffel symbols are

(3.22) |

For stationary, translationally invariant and axisymmetric configurations one has (we also assume ), and the MHD equations become thus

(3.23) | |||

(3.24) | |||

(3.25) |

The macroscopic electric charge current and the entropy current are given respectively by

(3.26) |

where is the 3-velocity of the fluid, denotes the electric charge density and is the rest frame entropy density. Both currents are conserved,

(3.27) |

As there are no dissipative terms in the charge- and entropy currents, we have for the entropy

(3.28) |

and for the electric charge

(3.29) |

The Killing vectors of interest are (energy ) and (angular momentum on the , or ). The conserved charge related to a Killing vector is proportional to , and hence

(3.30) | |||||

(3.31) |

The shear tensor , heat flux and diffusion current
must vanish on any stationary solution of fluid dynamics. The
requirement means that the fluid motion should be just a
rigid rotation. By an SO transformation^{7}^{7}7Strictly speaking, an
applied electromagnetic field breaks SO invariance, so that our choice
of implies consistency conditions on (cf. (3.35)
below).
we can can choose this rotation
such that the 3-velocity of the fluid is for some constant . From and
one obtains then .

The equilibrium fluid flow is symmetric under translations of and , so that all thermodynamic quantities depend only on . We now evaluate the expansion, acceleration, shear tensor, heat flux (2.29) and diffusion current (2.27), with the result

(3.32) |

(3.33) |

(3.34) |

where and denote the thermal conductivity and the diffusion coefficient respectively, is the local temperature and the local chemical potential. The requirement that and vanish implies that

(3.35) |

with constant. If , the last equ. is solved by , where is constant. The conditions (3.35) determine all the thermodynamic quantities as a function of the coordinate . We now want to shew that this configuration solves the equations of magnetohydrodynamics. To this end, we first notice that the dissipative part of the energy-momentum tensor vanishes once (3.35) is imposed, so that all nonzero contributions to the stress tensor result from the perfect fluid part plus interaction with the external electromagnetic field, and read

(3.36) |

where denotes the local pressure and is given by

(3.37) |

Eqns. (3.23) and (3.25) imply , which are automatically satisfied if the susceptibility does not depend on and , which we assume in the following. The only nontrivial equation of motion (3.24) becomes

(3.38) |

Using the Gibbs-Duhem relation (2.19) as well as (2.25), equ. (3.38) can be cast into the form

(3.39) |

which is automatically solved using (3.35). Therefore, the rigidly rotating configurations are stationary solutions of the magnetohydrodynamic equations. Moreover, as the diffusion current and the heat flux vanish, the only nonzero contributions to the transport R-charge- and entropy currents come from the perfect fluid pieces (3.26), which are easily seen to be conserved as well.

From the grandcanonical potential (2.2) we find

(3.40) |

In the nonrotating case , we have , , , and thus (3.35) implies that the local temperature as well as and are constant. We can then easily compute the energy, entropy, R-charge and magnetization, with the result

(3.41) |

It is straightforward to verify^{8}^{8}8One can express
in terms of , , and check that it
vanishes. that the Hawking temperature of the black
hole is given by , and the chemical potential associated to the
R-charge is .
Using the relations

(3.42) |

following from (3.19), (3.20), one easily shows that , and coincide with the corresponding expressions (3.15), (3.12) and (3.11) for the static black hole.

In the next subsection we will show that, using the function determined from the static AdS black hole as an input into the MHD equations, one can exactly reproduce the thermodynamics, boundary stress tensor and R-current of the rotating dyonic Kerr-Newman-AdS solution.

### 3.2 Kerr-Newman-AdS

The Kerr-Newman-AdS solution to the equations of motion following from (3.1)
is given by^{9}^{9}9For the time being, we restrict to the case of spherical horizons. The
generalization to is straightforward.