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.
