It is well known that the fluid equations (Euler equation, Navier-Stokes, …), being non-linear, may have highly turbulent solutions. Of course, these solutions are non-analytical. The laminar flow solutions (Couette flow for example) may be unstable to perturbations, depending on viscosity.
Also, low-viscosity fluids (water for example) are more turbulent than high-viscosity fluids (oil, for example).
I was wondering if something similar may happen with gravity and spacetime itself. The Einstein equations are highly non-linear: do turbulent solutions exist?
Or is gravity like some highly viscous fluid, i.e. without any turbulence?
What might a turbulent metric look like? Of course it would not be an analytic solution.
I imagine that spacetime turbulences may be relevant on a very large scale only (cosmological scales, or even at the Multiverse level). And maybe at the Planck scale too (quantum foam). But how could we define geometric turbulence?
The only reference I've found on this subject, which shows that the idea isn't crazy, is this :
https://www.perimeterinstitute.ca/news/turbulent-black-holes
EDIT : I have posted an answer below, which I think is very interesting. I don't know if this hypothesis was already studied before.
Best Answer
Gravity can, of course, become turbulent if it is coupled to a turbulent fluid. The interesting question is thus, as John Rennie points out, whether a vacuum solution can be "turbulent".
As far as I'm aware this is not known. If turbulence does occur in vacuum gravity, it is remarkably hard to stir up. Even in very extreme situations like colliding black hole binaries, which are now simulated pretty routinely, no turbulence has been observed.
EDIT: One approach one might take to study this is the "Post-Newtonian expansion", in which GR is formulated as an expansion in powers of some characteristic speed $\frac{v}{c}$. This has been conducted to extremely high order and the accuracy of the results at least for binary black holes rivals that of full nonlinear simulation. To all existing orders, the PN expansion is known to be exactly integrable. So if GR exhibits turbulent behaviour, it does so only in very extreme situations.
There are some theoretical reasons one might expect turbulence, which are hinted at in the press release you link to. Because of AdS/CFT one expects at least certain vacuum GR spacetimes to be equivalently modelled by a certain quantum field theory with a special symmetry. But that field theory, in some limit, should itself be approximately described by the Navier-Stokes equations. Therefore, again perhaps only in some weird and not-entirely-understood limit, the vacuum EFE's ought to described by the Navier-Stokes equations.
The point of the study you linked to was to investigate what sorts of behaviour in the gravitational theory one might get when the corresponding hydrodynamic theory is turbulent. The conclusion seems to be that certain turbulence-like behaviours appear in the gravitational theory. It seems to me a bit of an overstatement to say this group has discovered full-blown gravitational turbulence.
It's also, by the way, not yet known whether more pedestrian sorts of chaos can occur in the GR two-body problem. The Kerr spacetime is exactly integrable and geodesics are not chaotic. However, an actual particle will not move in the Kerr spacetime, but in a deformed spacetime including also its own gravitational field. An open question is if and when this perturbation can lead to chaotic motion.
EDIT2: There are also some theoretical reasons one might not expect turbulence. Basically what I'm imagining by turbulence is something like highly-nonlinear gravitational waves self-interacting strongly enough to excite vortex stretching, etc. But attempts to simulate such self-interactions (e.g. http://relativity.livingreviews.org/Articles/lrr-2007-5/) typically find that more or less generically, such strong gravitational fields either lead to rapid dispersal to infinity or the formation of a black hole. In closed spacetimes even small perturbations seem to eventually form a black hole more or less generically, although this is still unsettled. However, these studies are almost always done in high symmetry, so the question is far from resolved.