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 EðkÞ ~ k-n generates
metric fluctuations scaling as xn-2, where x is the four-dimensional spacelike distance from an arbitrary
origin in Minkowski spacetime, highlighting the onset of metric singularities whenever n < 2. 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.