A Field-Theory Approach for Modeling Dissipative Relativistic Fluids

Breakthrough progress in gravitational-wave astronomy prompts us to revisit “old questions” in relativistic fluid dynamics. In order to provide robust models of binary neutron star mergers (like the celebrated GW170817 event [1,2]) and mixed binary systems involving a neutron star and a black hole (like the recently announced GW230529 event [3]) we need to carry out large scale numerical simulations incorporating as much of the extreme physics as we can manage [4,5]. In addition to the “live” spacetime of Einstein’s gravity, our simulations need to include the complex matter physics that comes into play at densities beyond nuclear saturation. These aspects must be represented faithfully in order to allow reliable parameter extraction from observed signals. Somewhat colloquially, the stated aim is to “constrain the equation of state” of supranuclear density matter. However, this aim includes a number of issues associated with the systematics of simulations and the extracted model waveforms. This, in turn, raises problems which become pressing for the development of the next-generation of gravitational-wave instruments (the Einstein Telescope in Europe and Cosmic Explorer in the USA). These instruments will be sensitive at higher frequencies than the current LIGO-Virgo-Kagra interferometers and are expected to observe the post-merger phase, in addition to the late inspiral phase currently seen.

State-of-the-art simulations tell us that binary mergers involve high-density matter at temperatures close to those reached in terrestrial collider experiments (up to 100 MeV) [6]. At these extreme temperatures, the fluid will be far from thermodynamical equilibrium and the role of neutrinos is expected to be paramount [6]. Recent numerical relativity experiments [7,8,9] indicate that out-of-equilibrium physics (in the form of bulk vicosity and/or neutrino transport) will affect the gravitational-wave signal at a “detectable” level. In order to explore the relevant physics we evidently need to incorporate non-equilibrium aspects in our numerical simulations. In effect, we need to consider dissipative general relativistic fluid dynamics [10].

The implementation of dissipation in relativistic fluid dynamics is known to be tricky, both conceptually and practically. While there has been important recent progress on issues relating to stability and causality [11,12], we still do not have a universally agreed “framework” that would allow us to consider the complete range of physics that comes into play in neutron star mergers. Mergers combine a highly-energetic, turbulent flow of beyond nuclear-density matter; strong magnetic fields; and a dynamical spacetime generating copious amounts of gravitational waves. These events are unique because they operate over an impressive range of spatial scales. At the smallest scales, they provide data for the matter equation of state [13,14,15,16], while on large scales they may form long-lived merger remnants (possibly eventually forming black holes [17,18,19]). Rapid nuclear reactions during low-density matter outflows may lead to observable kilonova signatures [20]. Observed short gamma-ray bursts may be explained as the twisting of the stars’ magnetic field which would help collimate an emerging jet [21]. Multi-messenger observations of these events will—at some level—encode dissipative aspects (ranging from the bulk viscosity in the merger remnant [6,8] to resistivity affecting the evolution of the magnetic field [22,23,24]).

Arguably, the most “complete” framework for modelling the physics we need to consider is the variational approach reviewed in [10]. Notably, recent developments of the variational strategy include dissipative effects [25]. This effort, motivated by the requirements from gravitational-wave astronomy1, provides an action principle for multi-fluid systems for which no explicit reference to an equilibrium state is required and as a result the field equations are fully non-linear. This is in sharp contrast to all other models for dissipative relativistic fluid dynamics which build on a phenomenological derivative expansion (away from a supposed equilibrium state). The main idea of the variational model is that the dynamical degrees of freedom of fluids are captured by fluxes, and if the flux for a fluid has non-zero covariant divergence, or, equivalently, its associated dual three-form is not closed, then there will be dissipation. Conceptually, the idea is clear but we are still quite far from turning this understanding into a complete “workable” model.

The aim of the present discussion is to take steps to improve the situation by building an explicit action principle which connects with the familiar Navier-Stokes equations. The discussion will introduce a number of “simplifications”. Most notably, we will restrict ourselves to a single-fluid model. In some sense, this is against “better judgement” because we know that issues like heat/entropy flow require a multi-fluid treatment [10]. Moreover, the variational framework readily allows for multi-fluid aspects to be incorporated. However, if we want to make contact with numerical simulations (and we do!) then it must be noted that such efforts reduce the analysis to a single fluid whenever this is possible. Hence, it makes sense to see how far we can get if we restrict the variational discussion in this sense from the outset. The obvious caveat to this statement of intent is that we should perhaps not expect the effort to be completely successful. We are cutting corners and this ought to impact on the model we arrive at. Having said that, we expect to learn useful lessons from the exercise. The calculation we present is perhaps mainly interesting from a conceptual perspective, but the derivation also highlights aspects that need to be included in more realistic models. For example, we will show that a new fluid variable (the proper time derivative of the matter space “metric”) must be included in the original Lagrangian of [25] in order to recover the expected terms associated with bulk- and shear-viscosity. This new inclusion, in turn, affects the field equations, the entropy creation rate, and the energy-momentum-stress tensor. Additionally, we provide an explicit formulation of the matter space entropy three-form, going beyond the phenomenology explored in previous work. The results show that evolution equations along world lines naturally arise in the model, as one might expect from a relativistic formulation.

In Section 2, the generic action is written down and a variation with respect to the field variables (particle and entropy flux and the spacetime metric) is given. In Section 3, abstract, three-dimensional “matter” spaces are introduced so that the fluxes can be reformulated in such a way that the action principle becomes viable. Section 4 uses the same approach as [25] to build the required variations of the field variables; in particular, the Lagrangian displacement in Section 4.2. While the approach is the same, derivatives of the matter space metrics are assumed in the generic functional form of the action. This is because models like traditional Navier-Stokes are not possible without such derivatives in the Lagrangian. In Section 5, all the ingredients are stirred together and poured back into the initial variation of Section 2. The fluid field equation, entropy creation rate, and the energy-momentum-stress tensor are derived. In Section 6, a specific form for the Lagrangian is written down. In the Appendix, we provide details of the derivations of key elements of the formalism. While the results of the derivations are essential to delivering the final product, the calculations themselves are not necessary during a first reading of the paper.

In the variational approach, the equations of motion are derived from an action principle which has as its Lagrangian the so-called “master” function $\mathsf{\Lambda }$ (see [10] for an extensive review). For a finite temperature single-component system (as considered here), the master function is a function of all the independent scalars which can be built using the spacetime metric $g_{ab}$, the particle flux $n^a$, and the entropy flux $s^a$. However, here we restrict ourselves by only considering $n^2=-g_{ab}{n}^a{n}^b$ and $s^2=-g_{ab}{s}^a{s}^b$ (excluding the quantity $g_{ab}{n}^a{s}^b$, known to be associated with entropy entrainment [10]). The action is then given by The variation of $S_F$ with respect to $n^a$, $s^a$, and $g_{ab}$ is where we have used the fact that and defined

As we restrict our analysis to systems with a single fluid degree of freedom, the two constituents, particles and entropy, must be comoving. We denote the corresponding unit four-velocity as $u^a$, with normalization $u_a{u}^a=-1$ (in geometric units). The particle flux is now $n^a=n{u}^a$, and the entropy flux is $s^a=s{u}^a$, where the particle density is given by $n=-u_a{n}^a$ and the entropy density is $s=-u_a{s}^a$. We also note that the chemical potential is given by $\mu =-u^a{\mu }_a$ and the temperature follows from $T=-u^a{\mathsf{\Theta }}_a$.

The derivation of the equations of motion is complicated by the fact that our variation of the fluxes $\delta n^a$ and $\delta s^a$ must involve, indirectly, the variation of the worldlines given by $u^a$. Because $u_a{u}^a=-1$ everywhere, it has only three degrees of freedom. The impact of this can be seen already in $\delta S_F$ above. The equations of motion result when arbitrary variations of the field degrees of freedom do not change $S_F$ to linear order; i.e. $\delta S_F=0$. If we consider arbitrary variations $\delta n^a$ and $\delta s^a$ then the equations of motion are simply ${\mu }_a={\mathsf{\Theta }}_a=0$, which do not recover the simplest perfect fluid equations.

As shown in [26], building a viable action for two different “particle” constituents, such as matter and entropy, and one four-velocity, is straight-forward in the non-dissipative (perfect fluid) regime; even the generalization to a non-dissipative system of, say, M-constituents and N-fluids follows naturally (see [10] for details). Building on this, Andersson and Comer [25] demonstrated how to take the basic principles built into these actions and develop a fully non-linear set of field equations for dissipative fluids. But, as we will demonstrate in the next section, it is not straight-forward, a priori, to extend single-fluid actions to dissipative systems (as represented by, for example, the traditional Navier-Stokes equations).

Let us introduce the necessary ingredients of a viable action principle for a single fluid of matter and entropy which has dissipation. The first step is to introduce two abstract three-dimensional Riemannian (“matter space”) manifolds, whose individual points correspond to individual fluid worldlines in spacetime. The second step is to assume that the two manifolds are diffeomorphic to each other. A fair bit of infrastructure will have to be built before getting to the action principle and the resulting field equations; in particular, a lot of detail on the so-called matter space metrics must be included as these were shown in [25] to be essential elements required for dissipation. Some of the more tedious details of the infrastructure construction are presented in the Appendix.

We will now show that the proper-time derivatives ${\dot{g}}^{AB}$, ${\dot{\overline{g}}}^{AB}$, and ${\dot{\widehat{g}}}^{AB}$ are directly connected to ${\sigma }_{ab}$, $\mathsf{\Theta }$, and $h_{ab}$. The implication of this is that any recovery of, say, the Navier-Stokes equations via the action principle means that ${\dot{g}}^{AB}$, ${\dot{\overline{g}}}^{AB}$, and ${\dot{\widehat{g}}}^{AB}$ must be included as independent variables in the field variations.

The result follows because the master function $\mathsf{\Lambda }$ is commonly left unspecified in the action-based approach: usually only its existence and the fields/fluxes it depends on are postulated. If an explicit master function can be provided, then the dependence of this on the fields’ derivatives will automatically be taken care of by the variational principle. We also note that [27] works around this issue by considering the dissipative fluxes as functionals of, say, the “metric” $g_{AB}$. In the present context, however, we try to avoid that as this would inevitably make the discussion somewhat phenomenological.

The “trick” that incorporates dissipation in the variational formulation is to specify that the functional dependencies of $n_{ABC}$ and $s_{ABC}$ are It is clear that $\Delta n_{ABC}=0$, since it only depends on $X^A$. Consequently, the particle flux creation rate ${\mathsf{\Gamma }}_{\mathrm{n}}$ is shown to vanish; i.e. using the fact that ${\nabla }_{[a}{\mathsf{\Psi }}_{b]}^B=0$, etc., we have However, the extra dependencies for $s_{ABC}$, as we will show below, lead to a non-zero entropy creation ${\mathsf{\Gamma }}_{\mathrm{s}}={\nabla }_a{s}^a$.