Disformal Maps: Classification and Singular Dynamics
Being agnostic about the field content of a gravitational system, we consider a general disformal transformation of the metric, $g_{μν}=C h_{μν}+D t_{μν}$, on a four-dimensional Lorentzian manifold. Using the Cayley-Hamilton theorem, we derive an explicit formula for the inverse disformed metric. Implementing the Hawking-Ellis classification, we categorize disformal transformations into four types: Type I, II, III, and IV, based on possible Jordan block structures. We further identify the Segre subclasses from the eigenvalues. We derive explicit relations between the Cayley-Hamilton degree of $t_{μν}$, its Hawking-Ellis type, and its Segre subclass, restricting the possible types from the degree alone. In some cases, the type can be determined without even performing a full Jordan decomposition. For singular transformations, when new dynamical degrees of freedom emerge, we obtain the form of their corresponding mimetic energy-momentum tensor $T^\star_{μν}$. Within the polynomial sector, the Hawking-Ellis types of $t_{μν}$ and $T^\star_{μν}$ always coincide for Type I. For Types II and III it can differ, while Type IV is preserved generically but can reduce to Type I when the complex pair is mapped to a repeated real eigenvalue. This makes it possible to infer physical properties of $T^\star_{μν}$ directly from the Hawking-Ellis type of $t_{μν}$. We apply our setup to scalar and gauge fields. We find a new pure-disformal scalar-gauge family yielding irrotational perfect fluids, including barotropic fluids with an arbitrary constant equation-of-state parameter. Two scalar fields realize all four types and the reductions from Types II and IV to Type I. This framework applies to the kinematical and dynamical study of invertible and non-invertible disformal transformations with different field content.