Disformal map and Petrov classification in modified gravity

Disformal transformation provides a map relating different scalar-tensor and vector-tensor theories and gives access to a powerful solution-generating method in modified gravity. In view of the vast family of new solutions one can achieve, it is crucial to design suitable tools to guide their construction. In this work, we address this question by revisiting the Petrov classification of disformally constructed solutions in modified gravity theories. We provide close formulas which relate the principal nulls directions as well as the Weyl scalars before and after the disformal transformation. These formulas allow one to capture if and how the Petrov type of a given seed geometry changes under a disformal transformation. Finally, we apply our general setup to three relevant disformally constructed solutions for which the seeds are respectively homogeneous and isotropic, static spherically symmetric and stationary axisymmetric. For the first two cases, we show that the Petrov type O and Petrov type D remain unchanged after a disformal transformation while we show that disformed Kerr black hole is no longer of type D but of general Petrov type I. The results presented in this work should serve as a new toolkit when constructing and comparing new disformal solutions in modified gravity.

Automated summary

This study explores disformal transformations in modified gravity by revisiting the Petrov classification, providing formulas for capturing changes in Petrov type under disformal transformation. The application of this method to various solutions reveals that Petrov type O and Petrov type D remain unchanged, while the disformed Kerr black hole shifts from type D to general Petrov type I. These results offer a new toolkit for constructing and comparing disformal solutions in modified gravity.