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Stable black holes in lower dimensional $f(\mathbb{Q})$ non-metric gravity

G. G. L. Nashed, Salvatore Capozziello

TL;DR

This work derives exact charged and uncharged black hole solutions in (2+1)D within quadratic non-metricity gravity, characterized by $f(\mathbb{Q})=\mathbb{Q}+\frac{1}{2}\alpha\mathbb{Q}^{2}-2\Lambda$, and analyzes their geometry, thermodynamics, and geodesic stability. The authors show that quadratic corrections induce deviations from the BTZ geometry, including non-Einstein asymptotics and 1/$r$ hair in invariants, while yielding weaker singularities and allowing multi-horizon configurations on certain branches. Thermodynamic quantities such as the Hawking temperature, entropy, and heat capacity satisfy the first law, with entropy remaining positive for $1-12\alpha\Lambda\ge0$, and the first law extended to hair charges is confirmed. Geodesic analysis reveals modifications to effective potentials and photon orbits due to non-metricity, signaling observable differences from GR in strong-field regimes. Overall, the results demonstrate physically meaningful, stable black hole solutions in lower-dimensional $f(\mathbb{Q})$ gravity and illuminate their potential implications for holography and quantum gravity.

Abstract

We investigate exact charged and uncharged black hole solutions in a (2+1)-dimensional spacetime within the framework of quadratic form of $f(\mathbb{Q})$ symmetric teleparallel gravity, where $\mathbb{Q}$ is the non-metricity scalar. By adopting spherical symmetry and considering both vanishing and non-vanishing electromagnetic fields, we derive new classes of black hole solutions and analyze their geometric and physical properties. The study demonstrates that the inclusion of quadratic corrections in the gravitational Lagrangian significantly modifies the structure of solutions, producing deviations from the standard BTZ geometry. Invariants such as curvature and non-metricity scalars are calculated to classify the singularity structure and spacetime behavior. Thermodynamic quantities, including Hawking temperature, entropy, and heat capacity, are computed, showing consistency with the first law of black hole thermodynamics. Furthermore, we examine the geodesic motion of test particles and derive the effective potential to explore the stability of photon orbits. A notable outcome is the identification of weaker black hole singularities in comparison to General Relativity, attributed to the non-metricity corrections. The possibility of multi-horizon configurations is also explored. This study provides a comprehensive analysis of the gravitational, thermodynamic, and dynamical features of lower-dimensional black holes in $f(\mathbb{Q})$ gravity and highlights their distinct characteristics with respect to General Relativity.

Stable black holes in lower dimensional $f(\mathbb{Q})$ non-metric gravity

TL;DR

This work derives exact charged and uncharged black hole solutions in (2+1)D within quadratic non-metricity gravity, characterized by , and analyzes their geometry, thermodynamics, and geodesic stability. The authors show that quadratic corrections induce deviations from the BTZ geometry, including non-Einstein asymptotics and 1/ hair in invariants, while yielding weaker singularities and allowing multi-horizon configurations on certain branches. Thermodynamic quantities such as the Hawking temperature, entropy, and heat capacity satisfy the first law, with entropy remaining positive for , and the first law extended to hair charges is confirmed. Geodesic analysis reveals modifications to effective potentials and photon orbits due to non-metricity, signaling observable differences from GR in strong-field regimes. Overall, the results demonstrate physically meaningful, stable black hole solutions in lower-dimensional gravity and illuminate their potential implications for holography and quantum gravity.

Abstract

We investigate exact charged and uncharged black hole solutions in a (2+1)-dimensional spacetime within the framework of quadratic form of symmetric teleparallel gravity, where is the non-metricity scalar. By adopting spherical symmetry and considering both vanishing and non-vanishing electromagnetic fields, we derive new classes of black hole solutions and analyze their geometric and physical properties. The study demonstrates that the inclusion of quadratic corrections in the gravitational Lagrangian significantly modifies the structure of solutions, producing deviations from the standard BTZ geometry. Invariants such as curvature and non-metricity scalars are calculated to classify the singularity structure and spacetime behavior. Thermodynamic quantities, including Hawking temperature, entropy, and heat capacity, are computed, showing consistency with the first law of black hole thermodynamics. Furthermore, we examine the geodesic motion of test particles and derive the effective potential to explore the stability of photon orbits. A notable outcome is the identification of weaker black hole singularities in comparison to General Relativity, attributed to the non-metricity corrections. The possibility of multi-horizon configurations is also explored. This study provides a comprehensive analysis of the gravitational, thermodynamic, and dynamical features of lower-dimensional black holes in gravity and highlights their distinct characteristics with respect to General Relativity.

Paper Structure

This paper contains 16 sections, 60 equations, 4 figures.

Table of Contents

  1. Introduction
  2. A summary of non-metric gravity and its extension $f(\mathbb{Q})$
  3. Spherically symmetric black holes in quadratic $f(\mathbb{Q})$ gravity
  4. Uncharged black hole with $\alpha=0$
  5. Uncharged black hole with $\alpha\neq 0$
  6. Charged black hole with $\alpha=0$
  7. Charged black hole with $\alpha\neq0$
  8. Metric spacetime and invariants
  9. Metric and invariants of the case $\alpha\neq 0$ and $\varphi=0$
  10. Metric and invariants of case $\alpha\neq 0$ and $\varphi\neq0$
  11. Thermodynamics
  12. The case $\alpha\neq \frac{1}{12\Lambda}$
  13. The case $\alpha=\frac{1}{12\Lambda}$
  14. Comparing modified gravity and General Relativity by the geodesic equations
  15. Multi horizons
  16. ...and 1 more sections

Figures (4)

  • Figure 1: The plot \ref{['fig:R']} shows the metric given by Eqs. \ref{['met1']}. The plot \ref{['fig:fr1']} shows the behavior of the metric given by Eqs. \ref{['2t']}. The plot \ref{['fig:fr']} gives the behavior of the metric given by Eqs. \ref{['met2']}.
  • Figure 2: The plot \ref{['fig:temp1']} shows the behavior of Hawking temperature given by Eq. (\ref{['kGR']}); \ref{['fig:ent1']} shows the behavior of entropy given by (\ref{['ent1']}); \ref{['fig:heat']} shows the behavior of heat capacity given by Eq. (\ref{['heat1']}); \ref{['fig:temp']} shows the behavior of Hawking temperature given by Eq. (\ref{['temp']}); \ref{['fig:ent11']} shows the behavior of entropy given by (\ref{['ent11']}); \ref{['fig:heat11']} shows the behavior of heat capacity given by Eq. (\ref{['heat11']}). All the plots are derived using the following values of the constants, $\Lambda=-0.1$, $c_2=1$, $\alpha=-0.1$.
  • Figure 3: The plot \ref{['fig:VL']} shows the behavior of the potential given by Eq. \ref{['A8']} for different values of $L$ and $\epsilon=0$; \ref{['fig:VE']} shows the behavior of the potential given by Eq. \ref{['A8']} for different values of $\epsilon$ and $L\neq 0$; \ref{['fig:ddv']} shows the behavior of the second derivative of the potential given by Eq. \ref{['A8']}.
  • Figure 4: The plot \ref{['fig:metm']} shows the behavior of the ansatz given by Eq. \ref{['ggsol']}; \ref{['fig:metn']} shows the behavior of the ansatz given by Eq. \ref{['ggsol']}; \ref{['fig:tempm']} shows the behavior of the Hawking temperature of the solution given by Eq. \ref{['ggsol']}; \ref{['fig:entm']} shows the behavior of the entropy of the solution given by Eq. \ref{['ggsol']}; \ref{['fig:heatm']} shows the behavior of the heat capacity of the solution given by Eq. \ref{['ggsol']}. The values of the constants that used in these plots are $\Lambda=0.041$, $\alpha=-0.85$, $m=0.07$, $c_2=0.45$, and $c_3=0.6$.