Scalar Field Static Spherically Symmetric Solutions in Teleparallel $F(T)$ Gravity

TL;DR

This work develops a covariant framework for static, spherically symmetric spacetimes in teleparallel $$F(T)$$ gravity with a scalar-field source, delivering a general, implementable formula for constructing $$F(T)$$ from a given scalar potential $$V(\phi)$$ and coframe ansatz. By adopting power-law, exponential, and logarithmic scalar-field forms, the authors derive explicit $$F(T)$$ solutions and corresponding $$V(T)$$ expressions, delineating conditions for quintessence and phantom regimes via the characteristic relation $$r(T)$$. The results yield rich classes of analytic solutions, including connections to Born–Infeld BHs and scalarized BH scenarios, and provide a practical toolkit for exploring DE-dominated astrophysical systems in the teleparallel setting. The framework unifies and extends previous linear-fluid results, enabling systematic generation of new exact solutions across broad parameter spaces with potential observational relevance.

Abstract

We investigate in this paper the static radial coordinate-dependent spherically symmetric spacetime in teleparallel $F(T)$ gravity for a scalar field source. We begin by setting the static field equations (FEs) to be solved and solve the conservation laws for scalar field potential solutions. We simplify the FEs and then find a general formula for computing the new teleparallel $F(T)$ solutions applicable for any scalar field potential $V(T)$ and coframe ansatz. We compute new non-trivial teleparallel $F(T)$ solutions by using a power-law coframe ansatz for each scalar potential case arising from the conservation laws. We apply this formula to find new exact teleparallel $F(T)$ solutions for several cases of coframe ansatz parameter. The new $F(T)$ solution classes will be relevant for {studying the models close to Born--Infeld and/or scalarized Black Hole (BH) solutions inside the} dark energy (DE) described by a fundamental scalar field such as quintessence, phantom energy or quintom system, to name only those types.

Figures (4)

• Figure 1: Plot of $F(T)$ versus $T$ for pure flat cosmological $a=b=0$ subcase ($p_0^2=1$, $b_0=2\delta$, $p=2$ (except for $p\gg 1$ case)).
• Figure 2: Plot of $F(T)$ versus $T$ for $b=0$ and $a\neq 0$ subcase ($p_0^2=1$, $b_0=2\delta$, $p=2$ (except for $p\gg 1$ case)).
• Figure 3: Plot of $F(T)$ versus $r(T)$ for $b=-1$ and $a\neq 0$ subcase ($p_0^2=1$, $b_0=2\delta$, $p=2$ (except for $p\gg 1$ case)). All other $b\neq 0$ will be similar to this last case.
• Figure 4: Plot of $V(T)$ versus $r(T)$ for $b=-1$ and $a\neq 0$ subcase ($p_0^2=1$, $b_0=2\delta$, $p=2$ (except for $p\gg 1$ case)).