  Spontaneous scalarization is a mechanism that allows a scalar field to go
undetected in weak gravity environments and yet develop a nontrivial
configuration in strongly gravitating systems. At the perturbative level it
manifests as a tachyonic instability around spacetimes that solve Einstein's
equations. The endpoint of this instability is a nontrivial scalar field
configuration that can significantly modify a compact object's structure and
can produce observational signatures of the scalar field's presence. Does such
a mechanism exists for vector fields? Here we revisit the model that
constitutes the most straightforward generalization of the original
scalarization model to a vector field and perform a perturbative analysis. We
show that a ghost appears as soon as the square of the naive effective mass
squared becomes negative anywhere. This result poses a serious obstacle in
generalizing spontaneous scalarization to vector fields.
