Nonlinear Dynamics of Tensor Modes in Conformal Real Relativistic Fluids
In the Second Order Theories (SOT) of real relativistic fluids, the non-ideal properties are described by a new set of dynamical tensor variables. In this work we explore the non-linear dynamics of those modes in a conformal fluid. Among all possible SOTs, we choose to work with the Divergence Type Theories (DTT) formalism, which ensures that the second law of thermodynamics is satisfied non-perturbatively. In considering a perturbative scheme within this formalism, at next to leading order a set of Maxwell-Cattaneo equations is obtained, as e.g. in Israel-Stewart theories. To study the dynamics of the tensor sector, we device a perturbative scheme, where tensor modes are linearly excited by an external stochastic force, without injecting energy. This can be understood as if only entropy were sourced into the system. The random stirring force we consider is Gaussian with a scale invariant spectrum. The tensor modes in turn excite vector modes, which backreact on the tensor sector, thus producing a consistent non-linear, second order description of the tensor dynamics. Using Martin-Siggia-Rose (MSR) formalism together with a renormalization group scheme, we obtain the two-point correlation function for the tensor modes at next to leading order. The corresponding spectrum deviates from scale invariance due to the induction of an anomalous exponent, proportional to the noise intensity.
Keywords:Quark-Gluon Plasma, Quantum Dissipative Systems, Renormalization Group, Thermal Field Theory
Fluid description of relativistic, high energy phenomena proved to be a powerful tool for a clearer understanding of them landau1 (); landau2 (). Examples are the thermalization romat17 () and isotropization Stri14 () of the quark-gluon plasma created in the Relativistic Heavy Ion Collider (RHIC) facilities; the behaviour of matter in the inner cores of Neutron Stars (NS) rishke10 (); FriedSterg13 (); sterg17 (); the state of the plasma around the cosmological phase transitions nikschlesigl18 (); etc. In general, the features of the phenomena observed in those systems cannot be explained using ideal relativistic fluids.
Unlike non-relativistic hydrodynamics, where there is a successful theory to describe non-ideal fluids, namely, the Navier-Stokes equation, there is no definite mathematical model to describe real relativistic fluids. The story of the development of such theory begins with the recognition of the parabolic character of Navier-Stokes and Fourier equations111Recall that the non-relativistic Fourier law allows for an instantaneous propagation of heat. israel88 (), which implies that they cannot be naively extended to relativistic regimes. In fact, the first attempts by Eckart and Landau Eck40 (); LL6 () to build a relativistic theory of dissipative fluids starting from the non-relativistic formulation, also encountered this pathology.
The paradox about the non-causal structure of Navier-Stokes and Fourier equations, known as First Order Theories (FOTs), was resolved phenomenologically in 1967 by I. Müller muller67 (). He showed that by including second order terms in heat flow and the stresses in the conventional expression for the entropy, it was possible to obtain a system of phenomenological equations which was consistent with the linearized form of Grad kinetic equations grad49 (), i.e., equations that describe transient effects that propagate with finite velocities. These equations, constitute the so-called Second Order Theories (SOTs), whose main difference with respect to FOTs is that the stresses are upgraded to dynamical variables that satisfy a set of Maxwell-Cattaneo equations Max67 (); Catt48 (); Catt58 (); JosPrez89 (). Latter on, Müller’s phenomenological theory was extended to the relativistic regime by W. Israel and others israel76 (); IsSte76 (); IsSte79a (); IsSte79b (); IsSte80 (); HisLind85 (); HisLind88a (); HisLind88b (); Ols90 (); OlsHis90 ().
An improved, more systematic description of relativistic thermodynamics was introduced in 1986 by Liu, Müller and Ruggieri LiMuRu86 (), who developed a field-like description of particle density, particle flux and energy-momentum components. The resulting field equations were the conservation of particle number, energy momentum and balance of fluxes, and were strongly constrained by the relativity principle, the requirement of hyperbolicity and the entropy principle. The only unknown functions of the formalism were the shear and bulk viscosities and the heat conductivity, and all propagation speeds were finite. Several years latter, Geroch and Lindblom extended the analysis of Liu et al. and wrote down a general theory were all the dynamical equations can be written as total-divergence equations GerLind90 (); GerLind91 (), see also Refs. cal98 (); ReNa97 (); PRCal09 (); PRCal10 (); cal15 (); LheReRu18 (). This theory, known as Divergence Type Theory (DTT) is causal in an open set of states around equilibrium states, can be cast in a simple mathematical form, and all the dynamics is determined by a single scalar generating functional of the dynamical variables. Moreover, besides the dynamical equations an extra vector four-current is introduced, the entropy four-current, whose divergence is non-negative and, by the sole virtue of the dynamical equations, is a function of the basic fields and not of any of their derivatives. This fact guarantees that the second law is automatically satisfied at all orders in a perturbative development. In contrast, as Israel-Stewart-like theories must be built order by order, the second law must be enforced in each step of the construction. In other words, DTTs are exact hydrodynamic theories that do not rely on velocity gradient expansions and therefore go beyond Israel-Stewart-like second-order theories. In Appendix A we elaborate this statement more formally.
The novelty of SOTs, either Israel-Stewart or DTT, is the introduction of tensor dynamical variables to account for non-ideal features of the flow which, at lowest order in a perturbation scheme, satisfy a set of Maxwell-Cattaneo equations. This means that besides the scalar (spin 0) and vector (spin 1) modes already found in Landau-Lifshitz or Eckart theories, it is possible to excite tensor (spin 2) perturbations. This fact enlarges the set of hydrodynamic effects that a real relativistic fluid can sustain. In this manuscript we begin to study the non-linear hydrodynamics of the tensor sector, within the framework of DTTs. We concentrate on the spectrum of tensor modes induced in the fluid by a stochastic scale invariant stirring force. If present in the Early Universe plasma, this stochastic flow could excite primordial gravitational waves, as shown recently in Ref. nahuel17 (), or seed primordial electromagnetic fluctuations calkan16 (). Another scenario where tensor modes could play a relevant role are high energy astrophysical compact objects as, e.g., Neutron Stars FriedSterg13 (). It is well known that tensor normal modes of those stars can source gravitational waves, however at present there is no compelling hydrodynamical model of those objects, or of their fluid internal layers.
The paper is organized as follows. In section 2 we give a brief description of second order DDT formalism for conformal fluids and write down the complete set of second order equations of the theory. In section 3 we introduce the scale-invariant stirring force spectrum and outline the field theory method that we shall use to handle the nonlinear response wyld61 (); mcomb90 (); mcomb14 (); cal09 (), concretely the Martin-Siggia-Rose (MSR) formalism msr-73 (); dedo-76 (); kam-11 (); eyink96 (); zancal02 (); mcomb14 () to calculate the two-point correlation function of the induced tensor perturbations. We write down the corresponding ‘one-particle irreducible effective action’ (1PIEA) rammer07 (); calhu08 () from which we shall calculate the mode correlations. In Section 4 the 1PIEA is evaluated perturbatively up to leading order, which amounts to considering only terms that are cubic in the tensor field. At this order, only an infrared divergence is present, for which a Renormalization Group (RG) zancal02 (); ma00 (); mcomb14 (); yaorz86 () resummation scheme is implemented. As a result an anomalous exponent, proportional to the noise intensity, is induced in the tensor modes spectrum, which thus effectively deviates from scale invariance. Finally, in Section 5 we outline our main conclusions. There are four Appendices where we put some miscellaneous calculations as well as some conceptual developments. In Appendix A we show that in DTTs the Second Law is satisfied non-perturbatively. In Appendix B we write down both the ideal hydrodynamics and the Landau-Lifhstiz non-ideal hydrodynamics as a DTT theory. We also build the minimal conformal DTT that we use in this manuscript. In Appendix C we introduce the decomposition into scalar, vector and tensor modes and write the last two components in the base of eigenfunctions of the curl operator. This has the advantage that the dynamical variables are scalars, a fact that facilitates the calculations. In Appendix D we find the solutions of the equations for the scalar and vector sectors induced by tensor perturbations at lowest non-linear order. Of all scalar modes, only temperature fluctuations are induced, while of vector modes only velocity perturbations are considered222The vector sector consists of the incompressible velocity modes and the vector modes of the dissipative tensor function of DTTs. In this first work on non-linear dynamics of DTTs we only consider the former because they are the lowest order non-linear contribution.. In Appendix E we express the two-point correlation function of the tensor modes in terms of the curl eigenfunctions and explicitate its properties. Finally in Appendix F we outline some aspects of the diagramatics of the Effective Action. We work in natural units () and signature .
2 The model
We shall work within a theory which is arguably the minimal extension of Landau-Lifshitz hydrodynamics which enforces the second law of thermodynamics non-perturbatively (see Appendices A and B). We consider real neutral conformal fluids, whose dynamics is given by the conservation laws of the energy-momentum tensor (EMT) and of a third order tensor that encodes the non-ideal properties of the flow. Besides the mentioned tensors, we also consider an entropy current whose conservation equation enforces the second law of thermodynamics. is symmetric and traceless, and is totally symmetric and traceless on any two indices. The set of hydrodynamic equations is
while the second law is (see Appendix A)
with a new tensor variable that describes the non-ideal behavior of the flow.
Both and are local functions of the true hydrodynamical degrees of freedom, which are the Landau-Lifshitz four-velocity (namely, the only time-like proper vector of ), the Landau-Lifshitz temperature , which is the only dimensionful variable, and the tensor which is zero in local thermal equilibrium (LTE). The four velocity is normalized as , and the tensor degrees of freedom satisfy the constraints . Discounting Lorentz invariance we therefore have true degrees of freedom.
According to the developments of Appendix B, we decompose the EMT into ideal and viscous parts as
is the projection tensor onto surfaces orthogonal to and is the Minkowski space time metric. is the Stefan-Boltzmann constant, which depends on the number and statistics of the fields in the theory and the Landau-Lifshitz temperature. For a single particle obeying Maxwell-Jüttner statistics, . The non-ideal part is given by
where and . The tensor is a dimensionless version of defined as
In equilibrium the constants and may be parameterized in terms of the Landau-Lifshitz shear viscosity and the fluid’s relaxation time as (see Appendix B)
We may estimate from the AdS/CFT bound PolSonStar01 (), , whereby . is an equilibrium temperature. Causality requires .
The tensor can also be decomposed as (see Appendix B)
The conservation equations for the energy and for the momentum are obtained as usual, by projecting the EMT conservation equation along , and onto the surfaces defined by . The energy conservation equation reads
and the momentum conservation equation is
Therefore we need supplementary equations to close the system. These are obtained as the transverse, traceless components of the conservation law for , namely
is a stochastic source which we use to excite the tensor modes in the theory, and will be described in more detail below. For now, only observe that this force sources entropy, and not energy, as there is no stirring in the equations that stem from the conservation of . This rather simple model, however, will permit to study the basic features of the dynamics of tensor modes. We leave the more realistic treatment, where also energy is injected, for a forthcoming work.
We normalize the force as
and adopt the Anderson-Witting prescription for the deterministic source AndWit ()
This yields the new equations
where is the shear tensor
The entropy current is with
and the entropy production
with is the stochastic source of entropy, associated to .
In the equations above, the degrees of freedom are , and . However, because of the constraints , , these are not all independent. To identify the independent degrees of freedom, we assume a fiducial equilibrium configuration with velocity and temperature . We also write for the projection onto three dimensional surfaces orthogonal to . We can write , with and . We further write the space-like components of as . Both and vanish in equilibrium. Observe that as is transverse with respect to , this means that . Then we may adopt the spatial components as the independent degrees of freedom, and write
We also parametrize . The temperature fluctuation , the three components of and the five independent components of together form the nine true degrees of freedom of the theory. After replacing the expressions of and defined above in eqs. (16) and (17) we obtain the corresponding equations for and . We quote them here for future use:
where . Of course, is a spatial scalar, may be decomposed into one scalar and two vector degrees of freedom, and in one scalar, two vector and two tensor components. To isolate these components with well defined tensorial character, we first introduce Fourier transforms according to the convention
and then decompose the Fourier transforms of , and into linear combinations of angular momentum eigenstates (see Appendix C). This allows us to express the dynamical content of the theory in terms of the following scalar functions: the temperature scalar fluctuation ; the scalar (compressible) parts and the vector (incompressible) parts of ; the scalar part , the vector part and the tensor part of of . In all the above expressions .
3 Nonlinear response of tensor modes
In this section we study the nonlinear response of tensor modes to the stochastic forcing in eqs. (21). has neither scalar nor vector components, i.e., . Its Fourier transform may be written in terms of two polarization amplitudes as
Each amplitude is an independent, equally distributed, rotation, parity and scale invariant Gaussian process with zero mean and correlation (see Appendix E)
where . We aim to see whether nonlinear effects break the sale invariance of the forcing. Notice also that we are forcing the fluid by injecting entropy rather than energy or momentum. Although simplistic, this toy model will allow us to begin to understand the basic features of the non-linear behavior of tensor modes. Moreover we assume weak forcing .
3.1 Generating functional
Being a forced, classical (i.e., not quantum) system, we calculate the correlation function of the tensor modes using the Martin-Siggia-Rose (MRS) prescription msr-73 (); dedo-76 (); kam-11 (). Formally, we have a theory of nine fields obeying equations of the form (cfr. eqs. (26)-(29) and (21))
where is the coefficients matrix of the linear terms and represents all the nonlinear terms in eqs. (26), (28) and (21). Since we are interested specifically in the tensor modes, we further discriminate , where , , are the amplitudes for the two tensor polarizations. Then we have the system
where we used the fact that the linear equations for fields of different tensorial character decouple. Observe that we have generalized the coupling to the deterministic source by including a dimensionless parameter in the linear part of eq. (21) with a dependence on and .
with the solutions to eqs. (35) for given tensor amplitudes , and are the solutions of (36) for a given noise realization, and given fields . is the Gaussian probability density of the forcing noise . The correlation functions of the fields are obtained from this generating functional as functional derivatives with respect to the sources .
We now change the into a delta function of the equations of motion
It can be shown that the functional determinant is constant and so shall be disregarded zinnjustin (). Following MSR procedure, we exponentiate this delta function by adding an auxiliary field and its corresponding source . We then have
Finally we integrate over and to get
To obtain a formal expression for we rewrite the equation of motion as
Iterating this equation, using as initial condition that , we obtain an expansion of the scalar and vector modes in powers of the tensor ones. Since by definition it is not possible to extract a scalar or a vector linearly from a tensor mode, the leading term in this development is at least quadratic in . With solution (42) we can write
where is the quadratic action
is the interaction action which has an infinite number of vertices, each vertex having one leg and legs, with . Here represents the tensor part of the nonlinear term in the hydrodynamic equation (21). In the section 4 we give the explicit form of to lowest non-trivial order.
3.2 The effective action
Our goal is to find the lowest order moments of the tensor modes, namely the mean fields
and the two-point correlations for the fluctuations and , which can be accommodated in a compact notation as
Rather than computing these derivatives directly, we introduce the 1-Particle Irreducible Effective Action(1PIEA) calhu08 ()
The function is the generating function of 1-particle irreducible correlation functions. In graphical language, these are functions that cannot be separated into two independent correlations by just cutting one internal line (or propagator). The mean fields are obtained as the solution to the equations of motion
with , and the two-point correlations are the inverse of the Hessian of the effective action
Some properties of the generating functional are relevant to the implementation of resummation techniques. It is possible to show that cal09 (). This implies that and the correlation functions of -fields alone all vanish on-shell. It also implies that, even off-shell, in order to obtain we need to set . All this means that we have
whereby all derivatives of with respect to vanish on-shell, where . Then the equations of motion reduce to
In order to evaluate we split it as , where is the classical action eq. (43) which depends only on the mean fields (, ) and the is the correction coming from the fluctuations. Explicitly, for its computation we need to replace and in (40) and drop the sources and linear terms in and . Then the action becomes
Stated in Quantum Field Theory (QFT) language, represents the sum of all one particle irreducible vacuum graphs for the theory with action .
4 Perturbative evaluation of the effective action
Since by symmetry the mean fields must vanish, our goal is to compute the correlations of the fluctuations, for which we only need to know the terms quadratic in the mean fields in the effective action (cfr. eqs. (54)-(56)). As we shall see, there is a quadratic term which is infrared divergent, and must be renormalized by a suitable counterterm in the classical action. To this effect, we shall regard the parameter introduced in eq. (36) as a bare parameter, yielding a finite effective action after suitable renormalization.
We must then seek for terms in which are cubic in ; these may only come from terms where one tensor combines with a quadratic term coming from either a scalar or vector degree of freedom. We analyze the scalar and vector degrees of freedom in Appendix D. We conclude that no such term can arise from the scalar sector, but there is a suitable term coming from the incompressible part of the velocity (see eq. (155)). Using this lowest order term we obtain
In QFT language, we may say that corresponds to the sum of three tadpole Feynman graphs, where the propagator in the internal line is the “classical” correlation function
Replacing this propagator in the loop integral of we see that the resulting expressions are infrared divergent. The divergent term may be regularized by introducing an infrared cutoff , whereby