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.

• The paper derives an exact charged, static 2+1-dimensional f(Q) black hole without imposing g_tt = 1/g_rr, producing solutions with up to three horizons and recovering BTZ limits.
• The analysis finds a finite non-metricity scalar but divergent, stronger-than-GR curvature at the center, while numerical scans indicate positive temperature and heat capacity in selected parameter ranges.
• The topological study identifies one stable equilibrium branch with winding number w = +1, and geodesic analysis shows that charged null and timelike trajectories can reach the central singularity in finite affine or proper time.

Overview

This paper constructs an exact charged, static, circularly symmetric black hole solution in $(2+1)$-dimensional spacetime within symmetric teleparallel $f(Q)$ gravity, where the gravitational dynamics are governed by the non-metricity scalar $Q$ rather than curvature or torsion. The work of Nashed and Eid is notable for deriving the solution without imposing the common constraint $g_{tt} = 1/g_{rr}$ on the metric ansatz, which allows a richer causal structure than previously reported $f(Q)$ black hole models. The analysis proceeds through four complementary investigations: the geometric structure of the solution and its invariants, its thermodynamic stability, its topological classification via Duan's $\phi$-mapping formalism applied to a generalized free energy, and the geodesic completeness of the resulting singularity.

Geometric framework

The authors work within the symmetric teleparallel equivalent of general relativity (STEGR), where the affine connection is curvature-free and torsion-free, so that all gravitational effects are encoded in non-metricity. In the coincident gauge the connection vanishes identically, and the action takes the form

$$S = -\frac{1}{2\kappa}\int_{\mathcal{M}} f(Q)\sqrt{-g}\,d^4x + \int \sqrt{-g}\,\mathcal{L}_{em}\,d^4x,$$

with $\mathcal{L}_{em}$ the Maxwell Lagrangian. The theory reduces to STEGR when $f(Q) = Q$, and—unlike $f(R)$ gravity—the field equations remain second order.

The metric ansatz is

$f(Q)$0

with independent functions $f(Q)$1 and $f(Q)$2. For this geometry the non-metricity scalar evaluates to $f(Q)$3. Treating $f(Q)$4 via the chain rule converts the field equations into a closed system of four nonlinear ODEs in the four unknowns $f(Q)$5, which admits an exact analytic solution parameterized by six integration constants: $f(Q)$6 (mass), $f(Q)$7 (electric charge), $f(Q)$8 (background curvature term), $f(Q)$9 (non-metricity correction entering as $Q$0), $Q$1 (cosmological constant), and $Q$2 (gauge freedom in the electromagnetic potential).

The solution interpolates between several known limits. Setting $Q$3 yields an uncharged configuration; setting $Q$4 with $Q$5 recovers the BTZ metric with $Q$6. Importantly, when $Q$7 but $Q$8 (so that $Q$9 is constant), the charged solution still differs from the charged BTZ geometry of GR precisely because $g_{tt} = 1/g_{rr}$0—a direct consequence of relaxing the standard metric constraint. The full solution is asymptotically AdS even though de Sitter-type behavior emerges without an explicit cosmological-constant term in the field equations, and it admits up to three horizons depending on parameters.

Curvature, non-metricity, and singularities

A central finding concerns the behavior of geometric invariants near $g_{tt} = 1/g_{rr}$1. The Kretschmann scalar diverges asymptotically as

$g_{tt} = 1/g_{rr}$2

and the Ricci scalar behaves as $g_{tt} = 1/g_{rr}$3 at large $g_{tt} = 1/g_{rr}$4, while diverging more strongly toward the origin than in GR-type solutions. By contrast, the non-metricity scalar $g_{tt} = 1/g_{rr}$5 remains finite as $g_{tt} = 1/g_{rr}$6. The authors interpret this asymmetry as a redistribution of "gravitational load" from the curvature sector into the non-metricity sector: in the coincident gauge, the specific contractions defining $g_{tt} = 1/g_{rr}$7 cancel the leading radial divergences of the metric derivatives, whereas Levi-Civita-built scalars retain the imprint of steep central gradients. This is presented as improved regularity of the non-metricity sector rather than a resolution of the singularity itself—the curvature singularity persists and is in fact stronger than in GR. A related claim is that the electromagnetic potential $g_{tt} = 1/g_{rr}$8 remains finite at the origin, unlike the logarithmic divergence of the charged BTZ solution, suggesting a nonsingular effective charge distribution.

Thermodynamics

Thermodynamic quantities are evaluated at the outer horizon $g_{tt} = 1/g_{rr}$9, defined as the largest root of $f(Q)$0. The Hawking temperature follows from the surface gravity,

$f(Q)$1

and is found to be strictly positive over the scanned parameter range ($f(Q)$2, $f(Q)$3, $f(Q)$4). The Wald/Noether-charge entropy carries the characteristic $f(Q)$5 multiplicative factor:

$f(Q)$6

where $f(Q)$7 involves the inverse hyperbolic tangent combination fixed by the integration constants, subject to the reality condition $f(Q)$8. The heat capacity computed from these expressions remains positive throughout, which the authors take as evidence of local thermodynamic stability—a property they contrast with the instability regimes typical of many GR-based charged black holes. The horizon structure responds to parameters in a systematic way: increasing charge pushes horizons outward, while larger $f(Q)$9 or $\phi$0 tends to merge or eliminate horizons, reducing their number.

Topological classification

The topological analysis employs the generalized off-shell free energy $\phi$1, with $\phi$2 playing the role of an inverse temperature associated with Euclidean time periodicity. Following Wei–Liu–Mann, a vector field $\phi$3 is constructed on the $\phi$4 plane, whose zeros coincide with on-shell equilibrium states satisfying $\phi$5 at $\phi$6. Duan's $\phi$7-mapping topological current then assigns each zero a winding number determined by the sign of the Jacobian.

For representative parameters ($\phi$8, $\phi$9, $$S = -\frac{1}{2\kappa}\int_{\mathcal{M}} f(Q)\sqrt{-g}\,d^4x + \int \sqrt{-g}\,\mathcal{L}_{em}\,d^4x,$$0, $$S = -\frac{1}{2\kappa}\int_{\mathcal{M}} f(Q)\sqrt{-g}\,d^4x + \int \sqrt{-g}\,\mathcal{L}_{em}\,d^4x,$$1), a single fixed point exists at $$S = -\frac{1}{2\kappa}\int_{\mathcal{M}} f(Q)\sqrt{-g}\,d^4x + \int \sqrt{-g}\,\mathcal{L}_{em}\,d^4x,$$2 with winding number $$S = -\frac{1}{2\kappa}\int_{\mathcal{M}} f(Q)\sqrt{-g}\,d^4x + \int \sqrt{-g}\,\mathcal{L}_{em}\,d^4x,$$3: the condition $$S = -\frac{1}{2\kappa}\int_{\mathcal{M}} f(Q)\sqrt{-g}\,d^4x + \int \sqrt{-g}\,\mathcal{L}_{em}\,d^4x,$$4 admits only one positive root, and $$S = -\frac{1}{2\kappa}\int_{\mathcal{M}} f(Q)\sqrt{-g}\,d^4x + \int \sqrt{-g}\,\mathcal{L}_{em}\,d^4x,$$5 there, fixing the scaling exponent at $$S = -\frac{1}{2\kappa}\int_{\mathcal{M}} f(Q)\sqrt{-g}\,d^4x + \int \sqrt{-g}\,\mathcal{L}_{em}\,d^4x,$$6. The conclusion is that the thermodynamic phase space contains a single topologically stable branch. The authors note the dependence of this result on the bounded interval constraint $$S = -\frac{1}{2\kappa}\int_{\mathcal{M}} f(Q)\sqrt{-g}\,d^4x + \int \sqrt{-g}\,\mathcal{L}_{em}\,d^4x,$$7, which imposes $$S = -\frac{1}{2\kappa}\int_{\mathcal{M}} f(Q)\sqrt{-g}\,d^4x + \int \sqrt{-g}\,\mathcal{L}_{em}\,d^4x,$$8, $$S = -\frac{1}{2\kappa}\int_{\mathcal{M}} f(Q)\sqrt{-g}\,d^4x + \int \sqrt{-g}\,\mathcal{L}_{em}\,d^4x,$$9, and $\mathcal{L}_{em}$0; roots falling outside this interval would require negative $\mathcal{L}_{em}$1.

Multi-horizon configurations

The parameter space supports transitions among one-, two-, and three-horizon geometries. Varying the mass at fixed other parameters, $\mathcal{L}_{em}$2 produces three horizons, $\mathcal{L}_{em}$3 yields two coincident horizons (an extremal-like state), and $\mathcal{L}_{em}$4 leaves a naked singularity. Analogously, varying $\mathcal{L}_{em}$5 at fixed mass gives three horizons for $\mathcal{L}_{em}$6, degenerate horizons at $\mathcal{L}_{em}$7, and a single horizon for $\mathcal{L}_{em}$8. The effective potential for null and timelike geodesics exhibits the expected centrifugal barrier for photons and a deeper well for massive particles, consistent with the asymptotically AdS character of the spacetime.

Geodesic completeness

Near the origin the metric functions behave as $\mathcal{L}_{em}$9 and $f(Q) = Q$0 with $f(Q) = Q$1. Substituting into the radial geodesic equation shows that for nonzero angular momentum $f(Q) = Q$2 both null and timelike geodesics reach $f(Q) = Q$3 in finite affine parameter, while radial timelike geodesics ($f(Q) = Q$4) satisfy $f(Q) = Q$5, giving $f(Q) = Q$6 and hence termination at the center in finite proper time. Thus the central singularity is geodesically reachable in the charged case analyzed here. The abstract's statement that reachability depends on the presence of electric charge is not fully resolved within the body of the paper, which focuses on the charged branch; the uncharged case is asserted rather than demonstrated in detail.

Limitations and open questions

Several caveats qualify the results. First, the claimed stronger-than-GR central singularity coexists with the assertion of a possible "regularization effect" from non-metricity corrections; the paper does not reconcile these statements quantitatively, and the finiteness of $f(Q) = Q$7 alone does not constitute singularity resolution since curvature invariants still diverge. Second, the thermodynamic stability conclusions rest on numerical scans over restricted parameter sets rather than an analytic proof of positivity of $f(Q) = Q$8 and $f(Q) = Q$9 across the full admissible domain. Third, the topological classification yielding $f(R)$0 holds under the bounded-interval constraints on $f(R)$1; whether additional branches or bifurcation points (where the Jacobian vanishes) appear elsewhere in parameter space is left unexamined. Fourth, the geodesic-reachability dichotomy between charged and uncharged cases is stated in the abstract but only the charged case is worked out explicitly. Finally, the solution is static; rotating analogues, higher-dimensional extensions, and coupling to matter fields remain open problems identified by the authors.

Conclusion

This paper delivers an exact charged BTZ-like black hole in $f(R)$2-dimensional $f(R)$3 gravity derived without imposing $f(R)$4, showing that non-metricity corrections generate multi-horizon structures, stronger central curvature singularities alongside a finite non-metricity scalar, positive temperature, entropy, and heat capacity indicative of local stability, and a single topologically stable thermodynamic branch with winding number $f(R)$5. The solution reduces correctly to the BTZ geometry in the appropriate limits, confirming internal consistency, and its asymptotically AdS character preserves compatibility with holographic constructions. The main unresolved issues concern the precise status of the central singularity, analytic coverage of the stability claims, and the completeness of the topological classification beyond the constrained parameter regime studied.