  We give a necessary and sufficient condition for the projectivisation of a
slope semistable vector bundle to admit constant scalar curvature K\"ahler
(cscK) metrics in adiabatic classes, when the base admits a constant scalar
curvature metric. More precisely, we introduce a stability condition on vector
bundles, which we call adiabatic slope stability, which is a weaker version of
K-stability and involves only test configurations arising from subsheaves of
the bundle. We prove that, for a simple vector bundle with locally free graded
object, adiabatic slope stability is equivalent to the existence of cscK
metrics on the projectivisation, which solves a problem that has been open
since work of Ross--Thomas. In particular, this shows that the existence of
cscK metrics is equivalent to K-stability in this setting. We provide a
numerical criterion for the Donaldson-Futaki invariant associated to said test
configurations in terms of Chern classes of the vector bundle. This criterion
is computable in practice and we present an explicit example satisfying our
assumptions which is coming from a vector bundle that does not admit a
Hermite-Einstein metric.
