First Post-Minkowskian approach to turbulent gravity

TL;DR

This paper introduces a first-order Post-Minkowskian method to analyze how turbulent energy spectra influence spacetime metric fluctuations, revealing potential singularities and impacts on particle motion.

Contribution

It is the first to connect turbulence spectra with metric fluctuations in a relativistic framework using Post-Minkowskian approximation.

Findings

Turbulent energy spectra induce specific metric fluctuation scaling laws.

Power-law spectra with n<2 lead to metric singularities.

Metric fluctuations affect geodesic motion, offering a way to probe turbulence characteristics.

Abstract

We compute the metric fluctuations induced by a turbulent energy-matter tensor within the first order Post-Minkowskian approximation. It is found that the turbulent energy cascade can in principle interfere with the process of black hole formation, leading to a potentially strong coupling between these two highly nonlinear phenomena. It is further found that a power-law turbulent energy spectrum $E(k) \sim k^{-n}$ generates metric fluctuations scaling like $x^{n-2}$, where $x$ is a four-dimensional distance from an arbitrary origin in spacetime. This highlights the onset of metric singularities whenever $n <2$, meaning that $2d$ fluid turbulence ($n=3$) yields smooth %(differentiable) metric fluctuations, scaling like $x$, while $3d$ turbulence ($n=5/3$) yields a weakly singular metric $x^{-1/3}$and purely random fluctuations, $n=1$, generate a stronger $1/x$ singularity. Finally, the effect of metric fluctuations on the geodesic motion of test particles is also discussed as a potential technique to extract information on the spectral characteristics of fluctuating spacetime.