A note on spherically symmetric, static spacetimes in Kanno-Soda on-brane gravity

TL;DR

This paper examines spherically symmetric, static solutions in the Kanno-Soda scalar-tensor on-brane gravity, highlighting constraints on classical solutions like Schwarzschild and Reissner-Nordstrom due to radion field behavior.

Contribution

It analyzes the viability of standard GR solutions within the KS effective theory, revealing conditions on the radion scalar for physical consistency.

Findings

Schwarzschild solution requires a constant radion for viability.

Reissner-Nordstrom naked singularity can have finite radion but violates energy conditions.

RN black hole solution is unphysical due to divergent inter-brane distance at the horizon.

Abstract

Spherically symmetric, static on-brane geometries in the Kanno-Soda (KS) effective scalar-tensor theory of on-brane gravity are discussed. In order to avoid brane collisions and/or an infinite inter-brane distance, at finite values of the brane coordinates, it is necessary that the radion scalar be everywhere finite and non-zero. This requirement constrains the viability of the standard, well-known solutions in General Relativity (GR), in the context of the KS effective theory. The radion for the Schwarzschild solution does not satisfy the above requirement. For the Reissner--Nordstrom (RN) naked singularity and the extremal RN solution, one can obtain everywhere finite, non-zero radion profiles, though the required on-brane matter violates the Weak Energy Condition. In contrast, for the RN black hole, the radion profile yields a divergent inter-brane distance at the horizon, which makes the solution unphysical. Thus, both the Schwarzschild and the RN solutions can be meaningful in the KS effective theory, only in the trivial GR limit, i.e. with a constant, non-zero radion.