Generalized theories of gravity and conformal continuations

TL;DR

This paper explores conformal continuations in scalar-tensor and $f(R)$ gravity theories, analyzing conditions for their existence and providing explicit examples, thereby enhancing understanding of singularity regularization across conformal frames.

Contribution

It identifies necessary and sufficient conditions for conformal continuations in scalar-tensor and $f(R)$ theories, including cases with regular spheres and Killing horizons, with explicit examples.

Findings

Conformal continuation can occur when a singularity in one frame corresponds to a regular surface in another.

Conditions for the existence of CC depend on the properties of the scalar field and the spacetime geometry.

Explicit examples demonstrate the realization of CC in specific scalar-tensor and $f(R)$ models.

Abstract

Many theories of gravity admit formulations in different, conformally related manifolds, known as the Jordan and Einstein conformal frames. Among them are various scalar-tensor theories of gravity and high-order theories with the Lagrangian $f(R)$ where $R$ is the scalar curvature and $f$ an arbitrary function. It may happen that a singularity in the Einstein frame corresponds to a regular surface S_trans in the Jordan frame, and the space-time is then continued beyond this surface. This phenomenon is called a conformal continuation (CC). We discuss the properties of vacuum static, spherically symmetric configurations of arbitrary dimension in scalar-tensor and $f(R)$ theories of gravity and indicate necessary and sufficient conditions for the existence of solutions admitting a CC. Two cases are distinguished, when S_trans is an ordinary regular sphere and when it is a Killing horizon. Two explicit examples of CCs are presented.