Comment on the exterior solutions and their geometry in scalar-tensor theories of gravity

TL;DR

This paper classifies and analyzes exterior solutions in scalar-tensor gravity theories, revealing new solutions, their physical significance, and unique null geodesic behaviors through geometric analysis.

Contribution

It provides a classification of spherical exterior solutions, derives new axisymmetric solutions, and explores null geodesic properties in scalar-tensor gravity.

Findings

Classification of spherical solutions into black hole and worm hole types

Derivation of new static, axisymmetric solutions related to Voorhees's solutions

Identification of anomalous null geodesic behaviors

Abstract

We study series of the stationary solutions with asymptotic flatness properties in the Einstein-Maxwell-free scalar system because they are locally equivalent with the exterior solutions in some class of the scalar-tensor theories of gravity. First, we classify spherical exterior solutions into two types of the solutions, an apparently black hole type solution and an apparently worm hole type solution. The solutions contain three parameters, and we clarify their physical significance. Second, we reduce the field equations for the axisymmetric exterior solutions. We find that the reduced equations are partially the same as the Ernst equations. As simple examples, we derive new series of the static, axisymmetric exterior solutions, which correspond to Voorhees's solutions. We then show a non-trivial relation between the spherical exterior solutions and our new solutions. Finally, since null geodesics have conformally invariant properties, we study the local geometry of the exterior solutions by using the optical scalar equations and find some anomalous behaviors of the null geodesics.