# On uniqueness of static spacetimes with non-trivial conformal scalar field

## TL;DR

This paper proves the uniqueness of a specific static spacetime with a conformal scalar field, showing it must be the known Bekenstein solution outside the photon surface, and that multiple photon surfaces with identical scalar fields do not exist.

## Contribution

It establishes the uniqueness of the Bocharova-Bronnikov-Melnikov-Bekenstein solution for static spacetimes with a conformal scalar field and rules out multiple photon surfaces with the same scalar field.

## Key findings

- The spacetime is uniquely the Bekenstein solution outside the photon surface.
- Multi-photon surfaces with identical scalar field values do not exist.
- The result clarifies the structure of static spacetimes with conformal scalar fields.

## Abstract

We discuss the uniqueness of the static spacetimes with non-trivial conformal scalar field. Then, we can show that the spacetime is unique to be the Bocharova-Bronnikov-Melnikov-Bekenstein solution outside the surface composed of the unstable circular orbit of photon(photon surface). In addition, we see that multi-photon surfaces having the same scalar field values do not exist.