Four-scalar model and spherically symmetric solution in $f({\cal T})$ theory

TL;DR

This paper develops a four-scalar field model in $f({ m T})$ gravity to construct arbitrary spherically symmetric solutions, overcoming previous limitations and avoiding ghost instabilities, thus broadening the scope of modified teleparallel gravity.

Contribution

The authors introduce a four-scalar field formulation that generalizes previous models, enabling the realization of arbitrary spherically symmetric spacetimes in $f({ m T})$ gravity and ensuring ghost-free solutions.

Findings

Successfully reconstructs spherically symmetric solutions with quadratic $f({ m T})$ extension.

Demonstrates the model's ability to avoid ghost degrees of freedom.

Provides explicit examples of scalar fields for specific metrics.

Abstract

In this work, we investigate the construction of spherically symmetric solutions within the framework of modified teleparallel gravity, focusing in particular on $f({\cal T})$ theory, where ${\cal T}$ represents the torsion scalar. Conventional mimetic gravity and its extension with two scalar fields are unable to reproduce general spherically symmetric geometries in $f({\cal T})$ gravity, since consistency requires either a constant torsion scalar or a linear form of $f({\cal T})$, which corresponds to the teleparallel equivalent of General Relativity (TEGR). To overcome this restriction, we introduce a four-scalar field formulation that generalizes the two-scalar model and enables the realization of arbitrary spherically symmetric spacetimes. We further demonstrate that the model avoids ghost degrees of freedom by imposing appropriate constraints through Lagrange multipliers. As an application of the formalism, we examine a specific spherically symmetric metric and determine the corresponding four scalar fields, analyzing their properties. In the presence of a quadratic extension of the action, ${\cal T} + \tfrac{\alpha}{2}{\cal T}^2$, our approach successfully reconstructs physically viable solutions free from ghost instabilities. These findings highlight the potential of scalar-field-based constructions for generating consistent spacetime geometries in modified gravity theories, thereby extending beyond the standard formulations.