Causality and existence of solutions of relativistic viscous fluid dynamics with gravity

A new approach is described to help improve the foundations of relativistic viscous fluid dynamics and its coupling to general relativity. Focusing on neutral conformal fluids constructed solely in terms of hydrodynamic variables, we derive the most general viscous energy-momentum tensor yielding equations of motion of second order in the derivatives, which is shown to provide a novel type of generalization of the relativistic Navier-Stokes equations for which causality holds. We show how this energy-momentum tensor may be derived from conformal kinetic theory. We rigorously prove local existence, uniqueness, and causality of solutions of this theory (in the full nonlinear regime) both in a Minkowski background and also when the fluid is dynamically coupled to Einstein’s equations. Linearized disturbances around equilibrium in Minkowski spacetime are stable in this causal theory. A numerical study reveals the presence of an out-of-equilibrium hydrodynamic attractor for a rapidly expanding fluid. Further properties are also studied, and a brief discussion of how this approach can be generalized to nonconformal fluids is presented.

Figure 1

Illustration of causality. In curved spacetime $J^{-}(x)$ looks like a distorted light cone opening to the past (blue region); in flat spacetime the cone would be straight (dotted line). Points inside $J^{-}(x)$ can be joined to a point $x$ in spacetime by a causal past directed curve (e.g., the red line). The Cauchy surface $\mathrm{\Sigma }$ supports the initial data, and the value of the field $\phi (x)$ depends only on the initial data on $J^{-}(x)\cap \mathrm{\Sigma }$.

Figure 2

$|f_n{|}^{1/n}$ as a function of $n$, computed using (31), for $\eta /s=0.08$ and $\chi =4\eta$.

Figure 3

Hydrodynamic attractor solution for the causal tensor (4) with $\eta /s=0.08$ and $\chi =4\eta$. The black dashed lines represent solutions of (30) with different initial conditions, and the solid red line corresponds to the attractor solution. The NS solution is given by the purple dotted-dashed curve, while the dotted blue line denotes the equilibrium limit.

Figure 4

Temperature profile in $d{S}_3\otimes \mathbb{R}$ as a function of de Sitter time $\rho$. The red line is the solution of (32), the blue dotted-dashed line denotes the NS solution (33), and the ideal fluid case is shown in black (dashed). In this plot, $\overline{\eta }=0.2$, $\overline{\chi }=4\eta$, and $T_0=1.2$.

Figure 5

Dependence of the temperature with the transverse radius $r=\sqrt{x^2+y^2}$, evaluated at different Milne times $\tau =1$, 2, 3 fm, for the viscous fluid defined by (4) undergoing Gubser flow. The fluid also expands in the $z$ direction (not shown). In this plot, $\overline{\eta }=0.2$, $\chi =4\eta$, and $T_0=1.2$.