Disformal map and Petrov classification in modified gravity

TL;DR

This paper develops tools to analyze how disformal transformations affect the Petrov classification of solutions in modified gravity, enabling better understanding and construction of new solutions.

Contribution

It provides formulas relating Petrov types before and after disformal transformations, applied to various solutions including black holes, to guide solution classification.

Findings

Petrov type remains unchanged for certain solutions after disformal transformation.

Disformed Kerr black hole changes from type D to type I.

Formulas relate Weyl scalars and null directions pre- and post-transformation.

Abstract

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.