On the fractal dimension of turbulent black holes

TL;DR

This paper measures the fractal dimension of turbulent black hole horizons using fluid-gravity duality, finding values consistent with theoretical bounds and critically examining previous estimates, with implications for understanding horizon geometry.

Contribution

It provides new measurements of the fractal dimension of turbulent black hole horizons and introduces a covariant method for defining and calculating this dimension.

Findings

Fractal dimensions D=2.584 and D=2.645 for different energy spectra.

Results are consistent with the upper bound D=3.

Critique of previous fractal dimension estimates and proposal for covariant calculation.

Abstract

We present measurements of the fractal dimension of a turbulent asymptotically anti-deSitter black brane reconstructed from simulated boundary fluid data at the perfect fluid order using the fluid-gravity duality. We argue that the boundary fluid energy spectrum scaling as $E(k)\sim k^{-2}$ is a more natural setting for the fluid-gravity duality than the Kraichnan-Kolmogorov scaling of $E(k) \sim k^{-5/3}$, but we obtain fractal dimensions $D$ for spatial sections of the horizon $H\cap\Sigma$ in both cases: $D=2.584(1)$ and $D=2.645(4)$, respectively. These results are consistent with the upper bound of $D=3$, thereby resolving the tension with the recent claim in Adams, Chesler, Liu (2014) that $D=3+1/3$. We offer a critical examination of the calculation which led to their result, and show that their proposed definition of fractal dimension performs poorly as a fractal dimension estimator on $1$-dimensional curves with known fractal dimension. Finally, we describe how to define and in principle calculate the fractal dimension of spatial sections of the horizon $H\cap \Sigma$ in a covariant manner, and we speculate on assigning a `bootstrapped' value of fractal dimension to the entire horizon $H$ when it is in a statistically quasi-steady turbulent state.