Generalized Fisher-Janis-Newman-Winicour (FJNW) and Yilmaz-Rosen solutions in the higher-dimensional scalar-tensor theory with nonminimal coupling

Abstract

This paper investigates higher-dimensional scalar-tensor theories of gravity with nonminimal coupling, focusing on the reconstruction of exact spherically symmetric solutions. We systematically apply the formalism developed in previous works, which demonstrates that any static spherically symmetric metric can be represented as an exact solution of a scalar-tensor theory with specific coupling functions $f(ϕ)$ and potential $U(ϕ)$. Our analysis centers on two important classes of solutions in arbitrary spacetime dimensions: the Fisher-Janis-Newman-Winicour (FJNW) metric and its limiting case, the generalized Yilmaz-Rosen metric. We derive the key relations for the coupling function $f(ϕ)$ and the scalar field potential $U(ϕ)$ for both solution families using the master equation formalism in the Jordan frame. For the FJNW metric, we find that in the special case $s = 2$, $D = 6$, the coupling function $f(ϕ)$ is positive in the domain $u^3 > M$, corresponding to gravitational attraction and a canonical scalar field with vanishing potential $U(ϕ) = 0$. In contrast, for the generalized Yilmaz-Rosen metric in arbitrary dimensions $D > 4$, the reconstructed coupling function is always negative, $f(ϕ) < 0$, indicating a phantom scalar field with negative kinetic energy and repulsive gravity.