[Paper Review] A class of static spherically symmetric solutions in $f(Q)$-gravity

Marco Calzà, Lorenzo Sebastiani|arXiv (Cornell University)|Aug 27, 2022

Cosmology and Gravitation Theories4 citations

TL;DR

This paper presents a class of exact static spherically symmetric vacuum solutions in $f(Q)$-gravity by imposing an Ansatz that ensures the non-metricity scalar $Q$ is constant, leading to field equations being trivially satisfied. The key contribution is the identification of regular black hole and traversable wormhole solutions in vacuum without requiring matter violating energy conditions, leveraging the condition $f(Q_0) = f'(Q_0) = 0$.

ABSTRACT

We analyze a class of topological static spherically symmetric vacuum solutions in $f(Q)$-gravity. We considered an Ansatz ensuring that those solutions trivially satisfy the field equations of the theory when the non-metricity scalar is constant. In the specific, we provide and discuss local solutions in the form of black holes and traversable wormholes.

Motivation & Objective

• To identify exact vacuum solutions in $f(Q)$-gravity under the condition of constant non-metricity scalar $Q_0$.
• To explore whether such solutions can describe black holes and traversable wormholes without external matter sources.
• To demonstrate that the field equations are trivially satisfied when $f(Q_0) = f'(Q_0) = 0$, enabling a wide class of new solutions.
• To analyze the geometric and physical properties of these solutions, particularly their regularity and topology.
• To show that these vacuum solutions avoid the need for exotic matter by relying on the structure of $f(Q)$-gravity with constant $Q$.

Proposed method

• Imposes an Ansatz on the metric such that the non-metricity scalar $Q$ is constant, simplifying the field equations.
• Uses the condition $f(Q_0) = f'(Q_0) = 0$ to ensure that the field equations are satisfied identically in vacuum.
• Derives the general form of the metric for static spherically symmetric spacetimes with constant $Q_0$, including topological terms.
• Applies coordinate transformations to analyze regularity and asymptotic behavior, particularly for $z < 2$ and $z = 2$ cases.
• Analyzes the throat condition for traversable wormholes by requiring $g'(r_0) > 0$ at the throat radius $r_0$.
• Considers both flat ($k=0$) and curved ($k= /pm 1$) topological cases to classify solutions based on curvature and $Q_0$ sign.

Experimental results

Research questions

• RQ1 Can static spherically symmetric vacuum solutions in $f(Q)$-gravity be constructed when the non-metricity scalar $Q$ is constant?
• RQ2 What are the conditions under which such solutions describe regular black holes or traversable wormholes?
• RQ3 Can these solutions avoid the need for matter violating the null energy condition?
• RQ4 How does the functional form $f(Q)$ affect the existence and structure of these vacuum solutions?
• RQ5 What is the role of the topological parameter $k$ in classifying the resulting spacetime geometries?

Key findings

• Solutions with constant non-metricity scalar $Q_0$ satisfy the field equations trivially when $f(Q_0) = f'(Q_0) = 0$, enabling a wide class of exact solutions.
• For $z < 2$ and $c_1 > 0$, the metric becomes regular at $r=0$ after coordinate transformation, indicating the absence of central singularities.
• Traversable wormhole solutions are found in the flat topological case ($k=0$) when $Q_0 < 0$, $z < 2$, and $c_0 < 0$, or when $z > 2$ and $c_0 > 0$, with the throat at $r_0 = ig(rac{Q_0}{(2-z)c_0}ig)^{rac{1}{z-2}}$.
• The solution for $z=2$ yields an exponential metric form $ds^2 = -rac{e^{2 ilde{c}_1 r}}{r_0^2} dt^2 + dr^2 + ext{e}^{2 ilde{c}_1 r} ig(rac{d ho^2}{1-k ho^2} + ho^2 d heta^2ig)$, which is regular for $c_1 > 0$.
• The Schwarzschild solution is recovered as a special case when $z=0$ and $c_0 = -rac{Q_0}{2}$ in the flat case.
• The analysis shows that vacuum solutions in $f(Q)$-gravity can describe compact objects like black holes and wormholes without requiring exotic matter, due to the geometric nature of $f(Q)$-gravity with constant $Q$.