Black Hole Topologies and Geodesic Structures in Symmetric Teleparallel f(Q) Gravity

Abstract

Black hole solutions are studied here within the symmetric teleparallel formulation of gravity, employing the $f(Q)$ model in which the gravitational dynamics are governed by the non-metricity scalar $Q$. We focus on static, circularly symmetric spacetimes in $(2+1)$-dimensions, analyzing both charged and uncharged cases. By adopting a power-law form for $f(Q)$, we derive exact black hole solutions and explore their thermodynamic and geometric properties. Curvature and non-metricity scalars reveal central singularities stronger than those in general relativity. we find that the horizon radii increase with the charge parameter while higher values of the non-metricity coefficient, $c_{4}$, or the cosmological constant $Λ$ tend to merge or eliminate horizons, reducing their total number and altering the near-origin structure of the spacetime. We perform a detailed topological analysis based on the Euler characteristic and examine the geodesic completeness of the spacetime. Our findings show that, depending on the presence of electric charge, the singularity may or may not be reachable by geodesics. The thermodynamic stability is confirmed via temperature, entropy, and heat capacity calculations. This study highlights the rich structure of $f(Q)$ gravity in lower-dimensional settings and offers new insights into the nature of singularities and black hole topologies in modified gravity theories.

Figures (4)

• Figure 1: Systematic graphical representations of the spatial component of Eq. (\ref{['metric']}) are displayed in Fig. \ref{['fig:R']}., \ref{['fig:fr']} the temporal component of the metric given by Eq. (\ref{['metric']}).
• Figure 2: Panel \ref{['fig:R1']} illustrates the behavior of the temporal component of the metric obtained from Eq. (\ref{['metric']}), while panel \ref{['fig:R']} displays the corresponding spatial component of the same metric. Panel \ref{['fig:temp1']} presents the Hawking temperature calculated from Eq. (\ref{['kGR']}). The entropy behavior derived from Eq. (\ref{['ent1']}) is shown in panel \ref{['fig:ent1']}, whereas panel \ref{['fig:heat']} depicts the heat capacity obtained from Eq. (\ref{['heat1']}). All curves are generated by scanning the physical outer horizon $r_2$, defined as the largest root of $k(r)=0$. In these plots, the parameters are fixed to $\Lambda=0.1$ and $m=1$, while the remaining parameters are chosen according to the set specified in Sec. IV, ensuring that the Hawking temperature satisfies $T(r_2)>0$. In particular, the parameters are taken as $\Lambda=0.1$, $m=1$, and $c_2=10^{3}$.
• Figure 3: Fig. \ref{['fig2']}\ref{['fig2:ONE']} shows $\frac{d\zeta^r}{dr}$ at solution of $T-\frac{1}{\tau}$. Fig. \ref{['fig2']}\ref{['fig2:two']} presents the curve corresponding to the zero points of $T-\frac{1}{\tau}=0$. Fig. \ref{['fig2']}\ref{['fig2:three']} illustrates the vector field $\frac{\zeta^r}{|\zeta|}$ with one single fixed point.
• Figure 4: The three horizons of \ref{['sol']}; Fig. \ref{['Fig:3']}\ref{['fig3:a']} for different values of $M$; Fig. \ref{['Fig:3']}\ref{['fig3:c']} the effective potential given by Eq. \ref{['eq:Veff']} when $\delta=0$ and $\delta=1$.