Hybrid Fluid Models fromMutual Effective Metric Couplings

# Hybrid Fluid Models from Mutual Effective Metric Couplings

Aleksi Kurkela, Theoretical Physics Department, CERN,
CH-1211 Geneva, SwitzerlandFaculty of Science and Technology, University of Stavanger,
4036 Stavanger, NorwayDepartment of Physics, Indian Institute of Technology Madras,
Chennai 600036, India
Ayan Mukhopadhyay, Theoretical Physics Department, CERN,
CH-1211 Geneva, SwitzerlandFaculty of Science and Technology, University of Stavanger,
4036 Stavanger, NorwayDepartment of Physics, Indian Institute of Technology Madras,
Chennai 600036, India
Florian Preis, Theoretical Physics Department, CERN,
CH-1211 Geneva, SwitzerlandFaculty of Science and Technology, University of Stavanger,
4036 Stavanger, NorwayDepartment of Physics, Indian Institute of Technology Madras,
Chennai 600036, India
Anton Rebhan Theoretical Physics Department, CERN,
CH-1211 Geneva, SwitzerlandFaculty of Science and Technology, University of Stavanger,
4036 Stavanger, NorwayDepartment of Physics, Indian Institute of Technology Madras,
Chennai 600036, India
and Alexander Soloviev Theoretical Physics Department, CERN,
CH-1211 Geneva, SwitzerlandFaculty of Science and Technology, University of Stavanger,
4036 Stavanger, NorwayDepartment of Physics, Indian Institute of Technology Madras,
Chennai 600036, India
###### Abstract

Motivated by a semi-holographic approach to the dynamics of quark-gluon plasma which combines holographic and perturbative descriptions of a strongly coupled infrared and a more weakly coupled ultraviolet sector, we construct a hybrid two-fluid model where interactions between its two sectors are encoded by their effective metric backgrounds, which are determined mutually by their energy-momentum tensors. We derive the most general consistent ultralocal interactions such that the full system has a total conserved energy-momentum tensor in flat Minkowski space and study its consequences in and near thermal equilibrium by working out its phase structure and its hydrodynamic modes.

Institut für Theoretische Physik, Technische Universität Wien,
Wiedner Hauptstr. 8-10, A-1040 Vienna, AustriaInstitut für Theoretische Physik, Technische Universität Wien,
Wiedner Hauptstr. 8-10, A-1040 Vienna, AustriaInstitut für Theoretische Physik, Technische Universität Wien,
Wiedner Hauptstr. 8-10, A-1040 Vienna, Austria

## 1 Introduction

The hydrodynamic analysis of heavy-ion collisions performed at RHIC and the LHC suggests that a droplet of strongly interacting matter is generated in the collisions. The value of the specific viscosity that best describes these data is very low Heinz:2013th (); Romatschke:2017ejr (), , suggesting that the plasma is strongly coupled and does not have a description in terms of weakly interacting quasi-particles. This has encouraged much work in describing the plasma formed in terms of strongly coupled models, such as super Yang-Mills theory as described by AdS/CFT duality CasalderreySolana:2011us () (and references therein).

While the low value of implies that the system is strongly coupled, the collisions exhibit also hallmarks of weak coupling dynamics. In particular, it is seen that the hard components of high- jets go largely unmodified and resemble those created in - collisions Connors:2017ptx (). This suggests that the medium formed in heavy-ion collisions cannot be strongly coupled at all scales and even if some of the modes are strongly coupled, others are weakly coupled. Even more strikingly, the interpretation of the observed long range rapidity correlations in - and high multiplicity - collisions through final state interactions, combined with no signature of jet quenching in these systems may be seen to suggest the presence of both strongly and weakly coupled modes.

The simultaneous presence of strongly and weakly coupled modes poses a theoretical challenge. In absence of any fully developed non-perturbative method to access real-time properties of QCD in the non-perturbative regime, we may attempt to model the non-perturbative modes using a theory that we can solve in the strong coupling limit, while discussing the perturbative sector in a weak coupling approximation. Corresponding attempts have been made in Casalderrey-Solana:2014bpa (); Iancu:2014ava (); Mukhopadhyay:2015smb (). Such approaches in general pose the non-trivial question then, how the two sectors, described with different models describing different degrees of freedoms, be coupled.

A consistent coupling requires that the quantities that mediate the coupling should be well-defined in each theory and also gauge-invariant. In the context of jet-quenching, such couplings have been suggested for example in Liu:2006ug (); Casalderrey-Solana:2014bpa ().

In a different attempt to formulate a generic coupling between the two subsectors for the study of collective dynamics and equilibration, a local coupling of all the marginal operators of the two subsectors was proposed Iancu:2014ava (); Mukhopadhyay:2015smb (), following previous examples of a semi-holographic framework where only part of the dynamics is described by gauge/gravity duality Faulkner:2010tq (); Nickel:2010pr (); Jensen:2011af (); Mukhopadhyay:2013dqa (). This includes in particular a coupling between the energy-momentum tensors of the two subsectors, which can be induced by deforming the boundary metric of a holographic sector.

Specifically, as a semi-holographic model of the early stages of heavy-ion collisions, the perturbative sector was assumed to be described by classical Yang-Mills equations as in the glasma effective theory Gelis:2010nm () that describes the color-glass condensate initial conditions Kovner:1995ts (); Kovner:1995ja () of the deconfined gluonic matter liberated in the heavy-ion collisions, and the nonperturbative infrared sector by AdS/CFT, corresponding to strongly coupled super-Yang-Mills theory. The toy model studied in Mukhopadhyay:2015smb () demonstrated that in this way a closed system with a conserved energy-momentum tensor in Minkowski space can be obtained.111However, by only considering a gravitational coupling in a strictly homogeneous and isotropic situation which precludes propagating degrees of freedom in the bulk, the far-from-equilibrium system did not show any thermalization. When also a coupling between the gravitational dilaton field and the Yang-Mills Lagrangian density is turned on, the infrared sector turns out Ecker:2018ucc () to be heated up, thereby showing at least an onset of thermalization.

In this work, we explore the implications of the “democratic” couplings proposed in Banerjee:2017ozx () (and extensions thereof), where the effective metric of each subsystem depends on the energy-momentum fluctuations of the complements. Instead of far-from-equilibrium systems studied previously we concentrate on systems that are in equilibrium and the equilibration of near-equilibrium systems. Furthermore, we again restrict the couplings to that between the respective energy-momentum tensors of the subsystems.

We first address the question of what is the equilibrium state of two coupled conformal systems. As the system is assumed to be in thermal equilibrium, and the coupling depends only on the energy-momentum tensors of the subsystems, the microscopic features of the subsystems do not enter the discussion and therefore the results are generic for conformal subsystems and depend only on the properties of the coupling between the subsystems. We observe that requiring causality and ultraviolet completeness restricts the range the model parameters describing the coupling can take. In addition we find that the composite system – that breaks conformal symmetry due to dimensionful parameters of the coupling – exhibits a rich phase structure with a phase transition that takes the system from a sum of two separate conformal subsystems at low temperatures to a new emergent conformal system at high temperatures. As a function of the model parameters, this transition is either a cross-over or a first-order transition, and the two are separated by second-order critical endpoint with specific heat critical exponent .

Next we study in detail the collective dynamics of near-equilibrium systems. We first assume that each subsystem can be separately described as a conformal fluid in terms of first-order hydrodynamics. This assumption is generally valid if the length scale of the deviation of global equilibrium is sufficiently long for a well behaved gradient expansion and if no long-lived non-hydrodynamic modes are excited. Within this approximation we follow how linearized energy-momentum perturbations of the composite system approach global equilibrium and find a rich structure of two-fluid dynamics. In the shear sector we find that the overall viscosity interpolates between those of the subsystems and decreases with the coupling between the subsystems. In the sound sector we obtain two modes where only one is propagating with the thermodynamic speed of sound at large coupling. However both have attenuation vanishing with the square of momentum, implying that spatially homogeneous density perturbations of the individual subsystems are not attenuated and therefore more dynamics is required for the thermal equilibrium to be established between the two sectors. Indeed, this is in line with the findings in the semi-holographic toy model of Ref. Mukhopadhyay:2015smb (), where also interactions beyond the ones between the energy-momentum tensors are needed for thermalization EMPRS.

Finally, we study to what extent non-hydrodynamic modes in one subsystem are attenuated because of coupling to the other dissipative subsystem.

The organisation of the paper is as follows. In Section 2, we describe the general setup, its motivation in the semi-holographic context, as well as the concrete mutual metric coupling and how a total energy-momentum tensor that is conserved with respect to the (Minkowski) background metric of the full system arises. In Section 3, we discuss the requirements of causality and UV-completeness and study the consequences of our couplings for the thermodynamics and phase structure of the full system. In Section 4, we study the hydrodynamic limit of the full system, and in Section 5 we further study the case when the weakly coupled system can be described by kinetic theory and the strongly coupled sector as a conformal fluid with appropriate transport coefficients.

## 2 General setup

### 2.1 Semi-holography and democratic coupling

We consider a dynamical system in a fixed background metric (to be set to the Minkowski metric eventually) which consists of two subsystems and .

In the previous approach to semi-holography Mukhopadhyay:2015smb (), the full effective action of system was constructed as Mukhopadhyay:2015smb ():

 S=Spert[Aaμ,⋯]+Shol[g(b)μν=g(B)μν+γtpertμν,⋯], (1)

where is the effective perturbative action for , and is represented by , which is the holographic on-shell gravitational action in presence of sources. These sources, such as a non-trivial boundary metric , are functionals of the gauge-invariant operators of the perturbative sector, and is a hard-soft coupling. The full conserved energy-momentum tensor calculated by varying the action with respect to cannot be written in terms of the effective operators of each sector and therefore the low energy dynamics of the full system cannot be readily derived from the coarse-grained descriptions of the individual subsystems. Furthermore, the way the two sectors are coupled is somewhat asymmetric. On the one hand, the coupling amounts to deforming the metric of by the energy-momentum tensor of . On the other hand, the energy-momentum tensor of enters via the equations of motion of . Nevertheless the main improvement of Iancu:2014ava () made in Mukhopadhyay:2015smb () was that the full energy-momentum tensor is conserved, provided that the effective operators of satisfy a separate Ward identity and the equation of motion for the fields in are in effect.

Motivated by the semi-holographic approach in the democratic formulation Banerjee:2017ozx (), the two subsystems are assumed to have covariant dynamics with respect to individual effective metrics and , respectively.222In the following, quantities relating to the subsystems and will be distinguished either by indices 1 and 2 or by a tilde for those pertaining to (a tilde is used in particular when indices might be confusing). Interactions between the two subsystems are introduced by promoting each effective metric to functions that are locally determined by the state of the complement system,

 gμν=gμν[~tαβ,…],~gμν=~gμν[tαβ,…]. (2)

The two subsystems are assumed to share the same topological space so that we can use the same coordinates for both of them (and thus the total system)333Coordinate transformations would thus affect the background metric of the complete system and the effective metrics of the subsystems simultaneously.. Furthermore, the subsystems appear as closed systems with respect to their individual effective metrics, but they can exchange energy and momentum defined with respect to the actual physical background metric . Thus the effective metric tensors encode the interactions between the two subsystems.

The diffeomorphism invariance of the respective theories describing the two subsystems imply the Ward identities

 ∇μtμν=0,~∇μ~tμν=0, (3)

where and refer to the covariant derivatives with respect to the different effective metrics with the Levi-Civita connections

 Γμνρ=12gμσ(∂νgσρ+∂ρgσν−∂σgνρ)=Γμ(B)νρ+12gμσ(∇(B)νgσρ+∇(B)ρgσν−∇(B)σgνρ), ~Γμνρ=12~gμσ(∂ν~gσρ+∂ρ~gσν−∂σ~gνρ)=Γμ(B)νρ+12~gμσ(~∇(B)ν~gσρ+~∇(B)ρ~gσν−~∇(B)σ~gνρ). (4)

Above, is the covariant derivative with respect to and is the corresponding Levi-Civita connection, and the second equalities in (2.1) indicate that from the point of view of the actual physical background metric the identities (3) actually imply that work is done on the respective subsystems by external forces. In what follows, we restrict the forms (2) of the effective metrics and (in a generally covariant manner) such that there exists a for the full system that is locally conserved with respect to the physical background metric , i.e., we can enforce the Ward identity for the total system:

 ∇(B)μTμν=0. (5)

It turns out that the full energy-momentum tensor is a functional only of the effective operators and of the two sectors. Hence one can readily construct effective descriptions of the full dynamics from the effective description of each sector.444Additionally such couplings can generate expectation values of high-dimensional irrelevant operators without the need of introducing a non-trivial irrelevant deformation of the respective theory Banerjee:2017ozx (). This feature is needed for the cancellation of the Borel poles of the perturbative expansion. The main advantage of our method in the context of phenomenology is that it works even when we cannot invoke action principles for the effective descriptions of one or both subsystems. The full dynamics is obtained by solving the subsystems in a mutually self-consistent way as has been illustrated in case of the vacuum state in a toy example Banerjee:2017ozx ().

In the present paper, we utilize this to construct the low energy phenomenology by considering appropriate effective description of each subsector. First we assume that both sectors are described by fluids. Then we describe the perturbative sector by an effective kinetic theory and the non-perturbative sector by a strongly coupled fluid. We will be able to find consistent solutions for the full thermal equilibrium and also study its linear perturbations.

As a general remark, the principle of democratic coupling can be extended to other couplings such as that between scalar operators and Banerjee:2017ozx (). Let the theory describing the non-perturbative sector be also a (strongly coupled holographic) Yang-Mills theory with the coupling whereas is the coupling of the perturbative sector. These mutual deformations by scalar operators lead to the modified Ward identities (we turn off other couplings including the effective metric couplings for purpose of illustration)

 ∂μtμν=O∂νgYM,∂μ~tμν=~O∂ν~gYM. (6)

Then we may postulate a democratic coupling of the form:

 gYM=g0YM+α~O,~gYM=~g0YM+αO, (7)

where and are constants. It is clear then that the above Ward identities along with (7) imply the existence of the conserved energy-momentum tensor given by

 Tμν=tμν+~tμν−αO~Oδμν (8)

satisfying .

In Banerjee:2017ozx (), the most general scalar couplings of the form have been explored and a toy construction has been done to illustrate how these “hard-soft” couplings (such as ) along with the parameters of the holographic classical gravity determining can be derived as functions of the perturbative couplings in via simple consistency rules. In the following subsection, we extend and correct the democratic effective metric type couplings set up in Banerjee:2017ozx ().

### 2.2 Consistent mutual effective metric couplings

We start the construction of the coupling rules between the two subsystems by demanding that the total system a conserved energy-momentum tensor can be written for the total system in the flat background metric (from now on we choose unless explicitly mentioned otherwise)

 ∂μTμν=0, (9)

while simultaneously satisfying the Ward identities of the two subsystems in their respective curved metrics

 ∇μtμν=0,~∇μ~tμν=0, (10)

where and refer to the covariant derivatives with respect to the different metrics of the subsystems, with the corresponding Christoffel symbols (2.1).

For the rest of the paper, unless explicitly indicated otherwise, by we will mean and by we will mean , etc., with all lowering (and raising) of indices done by the effective metric (and its inverse) of the respective theory. The Ward identity of subsystem implies that

 ∇μtμμν=0, (11)

i.e.,

 ∂μtμμν+Γμμρtρμν−Γμνρtρμμ=0, (12)

or

 ∂μ(tμμν√−g)−12tμσ√−g∂νgμσ=0, (13)

where we have used

 Γμμν=∂ν(ln√−g),Γμνρtρμμ=12tμρ∂νgμρ, (14)

and multiplied both sides of (12) with to obtain (13). Similarly, the Ward identity for subsystem implies that

 ∂μ(~tμμν√−~g)−12~tμσ√−~g∂ν~gμσ=0. (15)

We require that both and are symmetric tensors. Using these Ward identities, it is straightforward to verify that the following local relations for the effective metrics

 gμν = ημν+γημα~tαβηβν√−~g+γ′ημνηαβ~tαβ√−~g, ~gμν = ημν+γημαtαβηβν√−g+γ′ημνηαβtαβ√−g, (16)

where and are coupling constants (with mass dimension ), allow us to construct a symmetric conserved tensor for the full system in flat space.

From (13) and (15) it follows that

 Kμμν=tμμν√−g+~tμμν√−~g+ΔKδμμν, (17)

with

 ΔK=−12[γ(tρα√−g)ηαβ(~tβσ√−~g)ησρ+γ′(tαβ√−g)ηαβ(~tσρ√−~g)ησρ] (18)

satisfies

 ∂μKμμν=0. (19)

Similarly it is easy to see that

 Lμνμ=tμνμ√−g+~tμνμ√−~g+ΔKδμνμ (20)

satisfies

 ∂νLμνμ=0. (21)

A symmetric and conserved total energy-momentum tensor with (also ) can therefore be defined by

 Tμμν=12(Kμμν+Lνμν). (22)

We can easily generalize the above construction for a curved background metric instead of the Minkowski metric using the second identities in (2.1) which imply

 Γμνμ−Γ(B)μ(B)νμ=∂ν(ln√−g)−∂ν(ln√−g(B))=∂ν(ln√−g√−g(B)) =√−g(B)√−g∂ν(√−g√−g(B))=√−g(B)√−g∇(B)ν(√−g√−g(B)), (23)

where we have used that is a scalar under general coordinate transformations.

With the help of these relations, one can readily see that the consistent coupling rules have the following general covariant forms

 gμν = g(B)μν+(γg(B)μα~tαβg(B)βν+γ′g(B)μν~tαβg(B)αβ)√−~g√−g(B), ~gμν = g(B)μν+(γg(B)μαtαβg(B)βν+γ′g(B)μνtαβg(B)αβ)√−g√−g(B). (24)

Then with

 ΔK = −γ2(tρα√−g√−g(B))g(B)αβ(~tβσ√−~g√−g(B))g(B)σρ (25) −γ′2(tαβ√−g√−g(B))g(B)αβ(~tσρ√−~g√−g(B))g(B)σρ,

the full conserved energy-momentum tensor is again given by (22), and it satisfies in the actual background where all degrees of freedom live. (Note ).

More general consistent couplings can be constructed if we permit higher powers of the energy-momentum tensors and together with new coupling constants carrying correspondingly higher inverse mass dimension. This is done in Appendix A (correcting and generalizing Ref. Banerjee:2017ozx () in this respect); in the following we will restrict ourselves to the above two coupling terms with coupling constants and .555Appendix A points out that in the most general set-up, where the total energy-momentum tensor satisfies thermodynamic consistency proven in Appendix B, each possible interaction term in the total energy-momentum tensor can be obtained via an appropriate coupling rule as a result of an interesting combinatoric identity. This is significant because a generic interaction term is not ruled out by any symmetry, and therefore it should indeed be reproduced by our way of introducing interactions via effective metrics.

While the dimensionful coupling constants introduced here appear to be arbitrary at this stage, we shall see that certain restrictions appear when further physical requirements are imposed. In particular, the complete dynamics should be such that causality remains intact. This means that the effective lightcone speed in the subsystems should not exceed the actual speed of light defined by . At least in the following applications to equilibrium and near-equilibrium situations, we can confirm that with just the two terms of in the consistent coupling rules corresponding to and causality can be ensured – at arbitrary energy scales – by choosing and . Interestingly enough, a positive value of the tensorial coupling constant was also found to be required in the semi-holographic study of Ref. Mukhopadhyay:2015smb () in order that interactions lead to a positive interaction measure, , which is a feature of (lattice) Yang-Mills theories at finite temperature Boyd:1996bx (); Borsanyi:2010cj ().

## 3 Thermodynamics

### 3.1 General equilibrium solution

We now assume that the full system , living in a flat Minkowski background metric and composed of two sectors and that interact through mutually determining their effective metrics, has reached a homogeneous and isotropic equilibrium state with temperature with total energy-momentum tensor

 Tμν=(E+P)UμUν+Pημν,Uμ=(1,0,0,0). (26)

Assuming furthermore that the subsystems and have also thermalized due to their internal dynamics taking place in their respective effective metrics, we expect that the latter will have a static, homogeneous and isotropic form for which we introduce the ansätze

 gμν=diag(−a2,b2,b2,b2),~gμν=diag(−~a2,~b2,~b2,~b2), (27)

with constants to be determined self-consistently.666If one of the systems is to be described by gauge/gravity duality, the simple metric ansatz above is of course not pertaining to the bulk, but to the boundary of the gravity dual.

The energy-momentum tensors of the subsystems are then of the form

 tμν=(ϵ1(T1)+P1(T1))uμuν+P1(T1)gμν,withuμ=(1/a,0,0,0), ~tμν=(ϵ2(T2)+P2(T2))~uμ~uν+P2(T2)~gμν,with~uμ=(1/~a,0,0,0), (28)

i.e.,

 tμν = diag(ϵ1(T1)a2,P1(T1)b2,P1(T1)b2,P1(T1)b2), ~tμν = diag(ϵ2(T2)~a2,P2(T2)~b2,P2(T2)~b2,P2(T2)~b2), (29)

with individual temperatures and .

The simplest coupling rules (2.2) now read

 1−a2=(γϵ2(T2)~a2−γ′(−ϵ2(T2)~a2+3P2(T2)~b2))~a~b3, b2−1=(γP2(T2)~b2+γ′(−ϵ2(T2)~a2+3P2(T2)~b2))~a~b3, 1−~a2=(γϵ1(T1)a2−γ′(−ϵ1(T1)a2+3P1(T1)b2))ab3, ~b2−1=(γP1(T1)b2+γ′(−ϵ1(T1)a2+3P1(T1)y2))ab3, (30)

and these determine , , and as functions of , and the coupling constants and . The full energy-density and pressure following from (22) are

 E = ϵ1(T1)ab3+ϵ2(T2)~a~b3 +γ2(ϵ1(T1)ϵ2(T2)a2~a2+3P1(T1)P2(T2)b2~b2)~a~b3ab3 +γ′2(−ϵ1(T1)a2+3P1(T1)b2)(−ϵ2(T2)~a2+3P2(T2)~b2)~a~b3ab3, P = P1(T1)ab3+P2(T2)~a~b3 (31) −γ2(ϵ1(T1)ϵ2(T2)a2~a2+3P1(T1)P2(T2)b2~b2)~a~b3ab3 −γ′2(−ϵ1(T1)a2+3P1(T1)b2)(−ϵ2(T2)~a2+3P2(T2)~b2)~a~b3ab3.

In a thermal equilibrium for the full system as well as for its individual subsystems, the physical temperature of living in Minkowski space is given by the inverse of , where is imaginary time and its period. The temperature of the subsystem , which effectively lives in a metric with constant , is then given by

 T−11=∫β0√−g00dτ=aβ=aT−1; (32)

by the same token we have . Hence,

 T=T1a=T2~a. (33)

Thus alone parametrizes the space of equilibrium solutions.

Using the thermodynamic identities

 ϵ1,2+P1,2=T1,2s1,2,E+P=TS, (34)

the result (3.1) implies

 TS=T1s1(T1)ab3+T2s2(T2)~a~b3=T[s1(T1)b3+s2(T2)~b3], (35)

showing that the total entropy density is the sum of the two entropy densities. Therefore, we identify the total entropy current as

 Sμ=√−gsμ1+√−~gsμ2 (36)

for , , and and .

This indeed makes perfect sense in a general non-equilibrium situation. When each sector has an entropy current satisfying

 ∇μsμ1≥0,~∇μsμ2≥0, (37)

this implies

 ∂μ(√−gsμ1)≥0,∂μ(√−~gsμ2)≥0, (38)

such that

 ∂μ(√−gsμ1+√−~gsμ2)=∂μSμ≥0. (39)

In thermal equilibrium, we also need to have

 dE=TdS (40)

or, equivalently, , for thermodynamic consistency. In Appendix B we prove this relation and the consistency of (33), (35) and (40) for the coupling discussed here as well as for the coupling rules that generalize (2.2). The mutual compatibility of the thermodynamic identities (34) and (40) of the full system with the global equilibrium condition (33) (along with the additivity of the total entropies that can be expected from the fact that each subsystem is closed in an effective point of view) provides a strong low-energy consistency check of our approach.

### 3.2 Causal structure of equilibrium solution

Since the causal structure of the dynamics taking place in the subsystems is dictated by the respective effective metrics only, causality in the full system, which is living in Minkowski space, is not guaranteed. For example, massless excitations from the point of view of subsystem with metric propagate with velocity with respect to the actual physical spacetime that the full system is occupying. (Recall that the two subsystems and the full system share the same topological space; the effective metrics of the subsystems just encode the effects of interactions between the two components of the full system.)

At least for the above solution for the equilibrium configuration obtained in the case of the simplest coupling rules (2.2) we can ensure the absence of superluminal propagation by requiring that the tensorial coupling constant together with and . To see this, take the sum of the first and second as well as of the third and fourth equation in (3.1). This leads to

 b2−a2 = γ(ϵ2(T2)~a2+P2(T2)~b2)~a~b3≥0, ~b2−~a2 = γ(ϵ1(T1)a2+P1(T1)b2)ab3≥0, (41)

independent of , implying that the effective lightcones defined by the metrics and are contained within the lightcone defined by the background Minkowski metric.

### 3.3 Conformal subsystems

In the following we shall consider the case of two conformal subsystems. For example one may think of a gas of nearly free massless particles for coupled to a strongly interacting quantum liquid for . The energy-momentum tensors and are assumed to be traceless with respect to the effective metrics and , thus the equations of state of the two subsystems are then simply

 ϵ1(T1)=3P1(T1)=3n1T41, ϵ2(T2)=3P2(T2)=3n2T42, (42)

with constant prefactors and .

The entropy is a simple expression in terms of the effective lightcone velocities associated with the effective metrics and , respectively,

 S=4T3(n1v3+n2~v3), (43)

where

 v:=ab,~v:=~a~b. (44)

Similarly we obtain for (3.1)

 1−v2b2 = 3γT4n21−r(1−~v2)~v5~b2, b2−1 = γT4n2~v2+3r(1−~v2)~v5~b2, 1−~v2~b2 = 3γT4n11−r(1−v2)v5b2, ~b2−1 = γT4n1v2+3r(1−v2)v5b2, (45)

where

 r:=−γ′γ (46)

is a dimensionless coupling constant that we shall use from now on in exchange for . Eliminating and yields the two equations

 n1γT4 = v5(1−~v2)(3+~v2)[3+v2~v2−3r(1−v2)(1−~v2)]2, (47) n2γT4 = ~v5(1−v2)(3+v2)[3+v2~v2−3r(1−v2)(1−~v2)]2. (48)

Since causality implies , we see that solutions exist for arbitrarily high only when the denominator on the right-hand side of (47) is able to reach a zero and is positive. This is the case when , which thus turns out to be a necessary (as well as sufficient) condition for ultraviolet completeness for the simplest coupling rules (2.2); otherwise this model would exist only up to some finite value of .

As shown in Appendix C, the high-temperature behavior of the total system is governed by the fact that the metric factors asymptote to linear functions of the physical temperature . Since the effective temperatures of the subsystems are given by and , they stop growing together with and and instead saturate at finite values proportional to . For figure 1 displays this behavior for equal and unequal subsystems, i.e., and , respectively.

Although the subsystems are conformal, when the two sectors interact, the full system in general is no longer conformally invariant. With the simplest coupling rules (2.2) and the resulting solution (3.1) we find

 E−3P=6γv2~v2n1n2T8a2~a2[(3+v2~v2)−3r(1−v2)(1−~v2)]. (49)

Note that the term in square brackets in (49) is the square root of the denominator in (48); it is positive in the uncoupled case where , and it cannot change sign for any finite value of . Therefore, the conditions for causality and condition for ultraviolet completeness, , imply that the interaction measure is positive (as is the case with lattice QCD results), and that the full system approaches conformality at large temperature .

While in general we have to resort to numerical evaluations, one can also derive perturbative expansions for all quantities (see Appendix C). For small couplings or for small temperature,777When writing down perturbative results, we shall assume that and are of the same order, i.e., that is of order 1. , the resulting , , , and are all close to unity, and thus , i.e., the full system approaches conformality at small temperature as expected.

This behavior can also be seen in the speed of sound (squared) of the full system, defined thermodynamically by

 c2s=dPdE=(dlnSdlnT)−1, (50)

which expanded up to third order in reads

 cs(T)=1√3−8√3γT4n1n2n1+n2−32√3γ2T8n1n2(n21+n22)(n1+n2)2+O((γT4)3). (51)

With conformal subsystems the dependence on appears only at third order; in quantities which only depend on and , as is the case for the entropy, also the third-order term is still independent of .

#### 3.3.1 Equal subsystems

For the special case where , the numerical solution of (47) is displayed in Fig. 2 for various values of .888Note that having equal equations of states does not imply that the subsystems are identical. Later on, we shall consider hydrodynamical results with subsystems that have but different transport coefficients.

It turns out that for more than one solution exists. This corresponds to a phase transition that will be discussed in section 3.3.3. Concentrating first on the case , the behavior of the pressure and the interaction measure (divided by ) is shown in the left panel of Fig. 3 for a typical case (). Intriguingly, shows an increase somewhat reminiscent of the deconfinement crossover transition in QCD.

The speed of sound (squared) (50) shown in the right panel of Fig. 3 exhibits a pronounced dip, indicating a crossover as opposed to a phase transition as is increased from the conformal situation at to large values, where it asymptotes again to conformal value .

Since , the entropy increases from its interaction free value at , where , in parallel to the drop in displayed in Fig. 2

In the case of two identical conformal subsystems, the relation between the effective lightcone velocity and is given by the roots of a polynomial equation of 9th degree (explicitly given in (187)), which in general can only be solved numerically. The asymptotic value of is however determined by a simple quadratic equation which yields

 v2∞:=limγT4→∞v2=3r−√3√4r−13r−1. (52)

Evidently, the entire physical range is covered as varies between unity and infinity.

Since the speed of sound approaches the conformal value at large , for sufficiently small values of (namely ), can be larger than the effective lightcone speed of the subsystems. This is no contradiction to causality, since besides dynamics within the subsystems, there is also collective dynamics between them. (In Section 4.2 this will be studied further.)

#### 3.3.2 Unequal subsystems

For unequal systems one can show (using formulae (47) and (48)) that there exist solutions for and in the limit for any value of and . They are given by the (sextic) equations

 3[r(1−v2∞)−1]5/2(4r−1)v5∞[r(1−v2∞)+v2∞/3]1/2=n2n1=(4r−1)~v5∞[r(1−~v2∞)+~v2∞/3]1/23[r(1−~v2∞)−1]5/2, (53)

which have a unique solution in the domain when