  Various groups recently demonstrated that the time evolution of simplest
self-interacting vector fields, those with self-interaction potentials, can
break down after a finite duration in what is called loss of hyperbolicity. We
establish that this is not an isolated issue, and other generalizations of the
Proca theory suffer from the same problem. Specifically, we show that vector
field theories with derivative self-interactions have a similar pathology. For
this, we derive the effective metric that governs the dynamics, and show that
it can change signature during time evolution. We also show that, generalized
Proca theories may suffer from tachyonic instabilities as well, which lead to
another form of unphysical behavior.
