Disformal transformation of stationary and axisymmetric solutions in modified gravity

TL;DR

This paper studies how disformal transformations affect stationary and axisymmetric solutions in various modified gravity theories, revealing changes in properties and conditions of the solutions, including black holes, and exploring their implications.

Contribution

It provides a systematic analysis of disformal transformations on stationary and axisymmetric solutions in scalar-tensor and vector-tensor theories, including black holes, and clarifies how these transformations alter solution properties.

Findings

Disformal transformations generally break circularity conditions in vector-tensor theories.

Disformed solutions in shift-symmetric scalar-tensor theories depend on scalar field coordinates.

Disformal transformations can modify the causal structure of spacetime solutions.

Abstract

The extended scalar-tensor and vector-tensor theories admit black hole solutions with the nontrivial profiles of the scalar and vector fields, respectively. The disformal transformation maps a solution in a class of the scalar-tensor or vector-tensor theories to that in another class, and hence it can be a useful tool to construct a new nontrivial solution from the known one. First, we investigate how the stationary and axisymmetric solutions in the vector-tensor theories without and with the $U(1)$ gauge symmetry are disformally transformed. We start from a stationary and axisymmetric solution satisfying the circularity conditions, and show that in both the cases the metric of the disformed solution in general does not satisfy the circularity conditions. Using the fact that a solution in a class of the vector-tensor theories with the vanishing field strength is mapped to that in a class of the shift-symmetric scalar-tensor theories, we derive the disformed stationary and axisymmetric solutions in a class of these theories, and show that the metric of the disformed solutions does not satisfy the circularity conditions if the scalar field depends on the time or azimuthal coordinate. We also confirm that in the scalar-tensor theories without the shift symmetry, the disformed stationary and axisymmetric solutions satisfy the circularity conditions. Second, we investigate the disformal transformations of the stationary and axisymmetric black hole solutions in the generalized Proca theory with the nonminimal coupling to the Einstein tensor, the shift-symmetric scalar-tensor theory with the nonminimal derivative coupling to the Einstein tensor, the Einstein-Maxwell theory, and the Einstein-conformally coupled scalar field theory. We show that the disformal transformations modify the causal properties of the spacetime.