Measuring the properties of homogeneous turbulence in curved spacetimes

TL;DR

This work addresses turbulence in curved spacetimes, where standard flat-spacetime analyses can misrepresent statistical properties. It introduces a curvature-aware framework based on the proper second-order structure function $S_{2,\mathcal{P}}(\ell)$ and the proper PSD $\mathrm{PSD}_{\mathcal{P}}(\chi)$, computed with proper length $\ell$ and proper volume $\mathcal{V}$ in a 3+1 manifold using the lapse $\alpha$ and spatial metric $\gamma_{ij}$. Applying this to GRMHD simulations of a Kerr BH accretion disc (SANE regime) around $a_*=0.9375$, the authors analyze four turbulent zones—near-horizon (NH), disc, wind, and jet—and show that a Kolmogorov-like inertial range emerges when measurements are made with proper geometry, with Kerr-vs-flat differences reaching $40$–$80\%$ near the horizon. The findings emphasize the importance of curvature-aware turbulence statistics in strong gravity and chart a path toward fully covariant, higher-dimensional analyses across astrophysical systems.

Abstract

Turbulence in curved spacetimes in general, and in the vicinity of black holes (BHs) in particular, represents a poorly understood phenomenon that is often analysed employing techniques developed for flat spacetimes. We here propose a novel approach to study turbulence in strong gravitational fields that is based on the computation of structure functions on generic manifolds and is thus applicable to arbitrary curved spacetimes. In particular, we introduce, for the first time, a formalism to compute the characteristic properties of turbulence, such as the second-order structure function or the power spectral density, in terms of proper lengths and volumes and not in terms of coordinate lengths and volumes, as customarily done. By applying the new approach to the turbulent rest-mass density field from simulations of magnetised disc accretion onto a Kerr BH, we inspect in a rigorous way turbulence in regions close to the event horizon, but also in the disc, the wind, and in the jet. We demonstrate that the new approach can capture the typical behavior of an inertial-range cascade and that differences up to $40-80\%$ emerge in the vicinity of the event horizon with respect to the standard flat-spacetime approach. While these differences become smaller at larger distances, our study highlights that special care needs to be paid when analysing turbulence in strongly curved spacetimes.

Figures (11)

• Figure 1: Schematic cartoon contrasting the measurement of turbulent structures, e.g., vortices indicated in blue, in a flat spacetime (top panel) and in a curved one (bottom panel). Note the difference between coordinate and proper lengths between two points $A$ and $B$ in the turbulent field.
• Figure 2: Spatial distributions in a polar slice of some of the most relevant plasma quantities at two representative times, $t=8000 \, M$ (top row) and $t=10000 \, M$ (bottom row). From left to right, are reported the rest-mass density $\rho$ [panel $(a)$], the temperature $T$ [panel $(b)$], the magnetisation $\sigma$, and the plasma $\beta$ [panel $(d)$].
• Figure 3: Highlighting of the four different regions in which the statistical properties of rest-mass density turbulent field are studied. The left panel reports the boundaries of the "disc" region while the inset zooms-in onto the "near-horizon" (NH) region. The right panel is the same as on the left but shows the "wind" and "jet" regions, respectively. The data refers to the snapshot at $t = 8000 \, M$.
• Figure 4: Representative example of the procedure to build the second-order structure function starting from a given point $A$ and reports with red solid lines the set of curves leading to points $B$ at a proper distance $\ell=0.76\,M$ from $A$, while the turbulent rest-mass density field is shown as a transparent background. To create a contrast, we also show with blue solid lines the corresponding curves in a flat spacetime. The data refers to the snapshot at $t = 8000 \, M$.
• Figure 5: Left panel: proper second-order structure functions normalised to the maximum proper length measured, i.e., $S_{2, \mathcal{P}}(\ell)/S_{2, \mathcal{P}}(\ell_{\rm max})$, as computed for the NH region (full-red line), the disc (black line), the wind (blue line), and the jet region (green line). Also shown as a comparison the classical second-order structure function for the NH region (light-red line). Right panel: using the same notation as on the left, we report the proper auto-correlation functions normalised by the variance $C_{\mathcal{P}}(\ell) / C_{\mathcal{P}}(0)$. The data refers to the snapshot at $t = 8000 \, M$.