First Post-Minkowskian approach to turbulent gravity

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) ∼k−n generates metric fluctuations scaling like xn−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 metric fluctuations, scaling like x, while 3d turbulence (n = 5/3) yields a weakly singular metric x−1/3and 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.