Existence of static spacetimes in higher-dimensional scalar–torsion theories with non-minimal derivative coupling

This paper investigates a class of static spacetimes within higher-dimensional (D ≥ 4) scalar–torsion theories featuring non-minimal derivative coupling and an active scalar potential. The spacetime structure is conformally related to a product space comprising a two-surface and a (D − 2)-dimensional submanifold. By analyzing the equations of motion, we demonstrate that the (D − 2)-dimensional submanifold must admit constant triplet structures, one of which corresponds to the torsion scalar. This condition allows the equations of motion to be reduced to a single highly nonlinear ordinary differential equation, referred to as the master equation. Our analysis reveals that the solutions to this model generically exhibit at least one naked singularity at the origin, ruling out the possibility of black hole or wormhole configurations. In the asymptotic region, the spacetimes converge to geometries with constant scalar curvature, which, in general, do not satisfy Einstein’s field equations. To further explore the behavior of the solutions, we employ a perturbative approach to linearize the master equation and construct first-order corrections. Additionally, we establish the local and global existence properties of the master equation and rigorously prove the non-existence of regular global solutions for D ≥ 4. These findings provide valuable insights into the structure and limitations of the scalar–torsion theory under consideration.