Holographic Path to the Turbulent Side of Gravity

We study the dynamics of a $2+1$-dimensional relativistic viscous conformal fluid in Minkowski spacetime. Such fluid solutions arise as duals, under the “gravity/fluid correspondence,” to $3+1$-dimensional asymptotically anti–de Sitter (AAdS) black-brane solutions to the Einstein equation. We examine stability properties of shear flows, which correspond to hydrodynamic quasinormal modes of the black brane. We find that, for sufficiently high Reynolds number, the solution undergoes an inverse turbulent cascade to long-wavelength modes. We then map this fluid solution, via the gravity/fluid duality, into a bulk metric. This suggests a new and interesting feature of the behavior of perturbed AAdS black holes and black branes, which is not readily captured by a standard quasinormal mode analysis. Namely, for sufficiently large perturbed black objects (with long-lived quasinormal modes), nonlinear effects transfer energy from short- to long-wavelength modes via a turbulent cascade within the metric perturbation. As long-wavelength modes have slower decay, this transfer of energy lengthens the overall lifetime of the perturbation. We also discuss various implications of this behavior, including expectations for higher dimensions and the possibility of predicting turbulence in more general gravitational scenarios.

Popular Summary

Fluid flows are commonplace in our daily experience. A black hole subject to gravitational perturbations in the context of general relativity, however, requires intellectual imagination and is certainly devoid of actual fluids. The realization of “gravity/fluid correspondence,” that the dynamics of the latter actually finds an analogue in the former, therefore, is not only intellectually fascinating but also scientifically enriching, in terms of new predictions or new methods of analysis on one side of the correspondence as the result of transposing interesting known phenomena and methods on the other. In this theoretical paper, we explore the counterpart of fluid turbulence in the context of black holes and reveal that perturbed black holes in three spatial dimensions can display a turbulent cascade that transfers energy from short-wavelength modes to longer-wavelength ones and, in turn, a longer relaxation time.

The fluid we study is a relativistic, viscous conformal fluid in $2+1$ spacetime dimensions (where the space is flat), dual to a perturbed black hole in a $3+1$-dimensional asymptotically anti-de Sitter spacetime (i.e., the space has a constant negative scalar curvature). One of the hallmarks of fluid turbulence is the so-called energy cascade: a hierarchy of eddies over a wide range of length and energy scales, within which energy is transferred from larger-scale eddies to smaller-scale ones. Exploring the duality, we have found a particularly interesting and illuminating result: In the regime where turbulence is seen in the fluid dual, a turbulent cascade correspondingly occurs in the gravitational perturbations to the black hole—however, in an inverse way of energy transfer from short- to long-distance scales. This results in the formation of long-lived, large-scale “gravitational-wave tornadoes.” The onset of this turbulence regime also marks the range of applicability of the linear perturbation theory of black holes that produces the standard notion that the energy spectrum of a perturbed black hole is exponential as opposed to being “turbulent.”

One may speculate about the implications of this inverse turbulence cascade. One interesting possibility is that the slowly decaying longer-scale gravitational wave modes in near-extremal Kerr black holes may have observational consequences. Another is the potential of predicting turbulence in more general gravitational scenarios.

Figure 1

Vorticity field at various times for a turbulent run (${\rho }_0={10}^{10}$, $v_0=0.05$, $D=10$, $n=10$). The inverse cascade behavior is evident, leading to two counter-rotating, and slowly decaying, vortices at late times.

Figure 2

$L_2$ norms of (a) vorticity and (b) $u_i$, as a function of time for turbulent flow (same run as Fig. 1). In (a) there is an initial exponential decay, followed by a power law during the inverse cascade, followed by a slower exponential decay. In (b) $u_y$ grows exponentially until it is of similar amplitude to $u_x$.

Figure 3

The $u_y$ field at $t=300$ for the same run as Fig. 1.

Figure 4

$L_2$ norms of (a) vorticity and (b) $u_i$, as a function of time for a laminar flow. The norm of $u_y$ remains small throughout the decay (${\rho }_0={10}^7$, $v_0=0.02$, $D=10$, $n=10$).

Figure 5

Intermediate flow: (a) $u_i$, as a function of time for laminar flow and (b) growth rate of $\Vert u_y{\Vert }_2$ versus $\max (u_x)$. The zero of (b) corresponds to the critical Reynolds number (${\rho }_0={10}^7$, $v_0=0.015$, $D=4$, $n=10$).

Figure 6

Critical Reynolds number as a function of wave number $n$ (dark circles). The solid line is the linear fit (4.5).

Figure 7

Velocity profile of one of the vortices in Fig. 1f. The solid line is the fit to the Oseen vortex profile.

Figure 8

Contour plots of principal invariants of the Weyl tensor in the bulk (rescaled with suitable powers of $r$ as per Appendix ), computed from the zeroth-order metric (2.1), from the simulation snapshot in Fig. 1e. Notice that (a) is representative of the energy density $\rho$, while (b) is representative of the vorticity, as expected from Eq. (b6).

Figure 9

Contour plots for ${\Psi }_1$, $\Im ({\Psi }_2)$, and ${\Psi }_3$ (left to right) illustrating the vortex structure from the horizon to the boundary. Note that the “conical” structure is a result of the dependence upon the radial coordinate of the plotted quantities [see Eq. (b6)]. (In contrast to Fig. 8, we are not rescaling here by any powers of $r$.)

Figure 10

Radial profile of the enstrophy at the horizon, for a vortex solution, as calculated by Eq. (a24), the square of the Pontryagin density [12], and $[\Im ({\Psi }_2){]}^2$ [49] ( labeled $Z_1$, $Z_2$, and $Z_3$, respectively). Each curve has been rescaled by a trivial constant factor for easier comparison.