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Turbulent spacetime from Einstein equation?

Hmmmn. Strictly speaking spacetime is an abstract mathematical "space" which models space at all time. It's the block universe. You can draw worldlines in it to represent motion through space over time, but there's nothing moving through it or in it. It's static. The wordlines aren't waving around like seaweed in the surf. However space can change over time. Gas clouds collapse to form stars and gravitational fields can become more pronounced. Since a gravitational field is "curved spacetime" we can reasonably say that spacetime changes. Let's call it space-time to distinguish it from the static block-universe spacetime. But can this change be a "turbulent" change? Hmmmn.

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 flows solutions (Couette flow for example) may be unstable to perturbations, depending on viscosity. Also, fluids which have a low viscosity (water for example) are more turbulent than fluids with high viscosity (oil, for example).

No problem there. Apart from the fact that space-time isn't a fluid.

I was wondering if something similar may happen with gravity and spacetime itself. Einstein equation being highly non-linear, do turbulent solutions exist ?

No. Because space-time isn't a fluid. Instead it's a gin-clear ghostly elastic solid! This is why you can see a shear stress term in Einstein's stress-energy-momentum tensor:

Public domain image by Maschen, based on an image by created by Bamse see Wikipedia

I kid ye not! Google on Einstein elastic. Then try to imagine a jelly on a plate. You can deform it, and you can curve it, and you can wibble it and you can wobble it. But you can't make it turbulent.

Or is gravity like some highly viscous fluid, i.e. without any turbulence ?

A gravitational field is where space is non uniform, this being modelled as curved space-time. Ah, here we go, this is what Einstein said:

"According to this theory the metrical qualities of the continuum of space-time differ in the environment of different points of space-time, and are partly conditioned by the matter existing outside of the territory under consideration. This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that 'empty space' in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials g$_{\mu

u}$), has, I think, finally disposed of the view that space is physically empty."

While being non-analytical, how a turbulent metric may look like? 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 ?

I think it's reasonable to propose a chaotic metric, but turbulent doesn't seem to fit in with General Relativity. Which as I'm sure you're aware, is one of the best-tested theories we've got. Meanwhile multiverses and quantum foam remain speculative. IMHO it's good to speculate and ask what if? It's good to think for yourself. But I'd say you're in danger of wandering away from hard science here into pseudoscience, where you won't find any evidence or answers.

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

The mention of the holographic conjecture does not auger well. And I'm afraid the fact that this comes from the Perimeter Institute doesn't mean it's correct. What we seem to have here is a rather speculative idea that looks as if it's at odds with general relativity. Ah, see the paper on the arXiv : Turbulent black holes. I've skimmed it, and I wonder if there could be some important issues. For example, the "coordinate" speed of light at the event horizon is zero. So if the black hole is spinning at half the speed of light, how fast is it spinning? What's half of zero? Anyway, it's late, and I have to go. You could try a new question asking for feedback on this paper. It's always better to refer to the actual paper rather than the reportage, because the latter can sometimes be misleading.