  Quadratic theories of gravity with second order equations of motion provide
an interesting model for testing deviations from general relativity in the
strong gravity regime. However, they can suffer from a loss of hyperbolicity,
even for initial data that is in the weak coupling regime and free from any
obvious pathology. This effect has been studied in a variety of cases including
isolated black holes and binaries. Here we explore the loss of hyperbolicity in
spin-induced scalarization of isolated Kerr black holes in a
scalar-Gauss-Bonnet theory of gravity, employing the modified CCZ4 formulation
that has recently been developed. We find that, as in previous studies,
hyperbolicity is lost when the scalar field and its gradients become large, and
identify the breakdown in our evolutions with the physical modes of the purely
gravitational sector. We vary the gauge parameters and find the results to be
independent of their value. This, along with our use of a different gauge
formulation to previous works, supports the premise that the loss of
hyperbolicity is dominated by the physical modes. Since scalar-Gauss-Bonnet
theories can be viewed as effective field theories (EFTs), we also examine the
strength of the coupling during the evolution. We find that at the moment when
hyperbolicity is lost the system is already well within the regime where the
EFT is no longer valid. This reinforces the idea that the theories should only
be applied within their regime of validity, and not treated as complete
theories in their own right.
