  We study the Jordan frame formulation of generalizations of scalar-tensor
theories conceived by replacing the scalar with other fields such as vectors.
The generic theory in this family contains higher order time derivative terms
in the Jordan frame action which is indicative of ill-posedness. However, we
show that equations of motion can always be reduced to a second-order-in-time
form as long as the original Einstein frame formulation is well posed. The
inverse transformation from the Jordan frame back to the Einstein frame is not
possible for all field values in all theories, but we obtain a fully invertible
transformation for vector-tensor theories by a redefinition of the vector
field. Our main motivation is a better understanding of spontaneous
scalarization and its generalizations, however our conclusions are applicable
to a wide class of theories. Jordan frame has been traditionally used for
certain calculations in scalar-tensor theories of gravitation, and our results
will help researchers generalize these results, enabling comparison to
observational data.
