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, $g_{\mu\nu}=Ch_{\mu\nu}+Dt_{\mu\nu}$, 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. By examining the eigenvalues, we further classify each type into its corresponding Segre subclasses. We find explicit links between the Cayley-Hamilton degree of the disformal tensor $t_{\mu\nu}$, its Hawking-Ellis type, and its Segre subclass, which can restrict the possible Hawking-Ellis types once only the Cayley-Hamilton degree is known. 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 general form of their corresponding mimetic energy-momentum tensor $T^\star_{\mu\nu}$. We show that the Hawking-Ellis types of $t_{\mu\nu}$ and $T^\star_{\mu\nu}$ 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 $T^\star_{\mu\nu}$ directly from the Hawking-Ellis type of $t_{\mu\nu}$. We apply our setup to two specific cases: $t_{\mu\nu}=\partial_\mu\phi\partial_\nu\phi$ and $t_{\mu\nu}=F^{\alpha}{}_{\mu}F_{\alpha\nu}$, where $\phi$ is a scalar field and $F_{\mu\nu}$ is the field-strength tensor of a gauge field. This general framework can be used to systematically study the kinematical and dynamical properties of various invertible and non-invertible disformal transformations with different field content.