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

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.