Deformed BTZ Black Holes

• Deformed BTZ black holes are modified (2+1)-dimensional solutions incorporating quantum gravity, noncommutative geometry, and higher-derivative corrections.
• They exhibit altered horizon structures, thermodynamic properties, and causal behaviors driven by methods like noncommutative smearing, massive gravity, and string-theoretic deformations.
• These models provide a unified framework for exploring effective metric formalism, holographic dualities, and microstructural insights in quantum-corrected spacetimes.

Deformed BTZ black holes constitute a broad class of solutions and modifications to the standard (2+1)-dimensional Ba~nados–Teitelboim–Zanelli (BTZ) black hole. These deformations are motivated by quantum gravity, noncommutative geometry, exotic field couplings, higher-derivative theories, string-theoretic considerations, and effective descriptions capturing quantum corrections. Deformations typically alter fundamental aspects of BTZ spacetimes, such as their horizon structure, thermodynamics, causal properties, quantum stability, and topological invariants. The following sections provide a comprehensive account of the principal deformation mechanisms, their technical implementations, and associated physical phenomena.

1. Noncommutative and Quantum-Statistics-Induced Deformations

A key avenue of deformation is inspired by noncommutative geometry. Here, the classical commutativity of spacetime coordinates $[x^\mu, x^\nu] = 0$ is relaxed to $[x^\mu, x^\nu] = i\theta^{\mu\nu}$, introducing a minimal length scale $\sqrt{\theta}$ and requiring that matter distributions be "smeared"—often via a Gaussian profile—rather than point-like. For a (2+1)-dimensional spacetime, this leads to the energy density

$$\rho(r) = \frac{M}{4\pi\theta}\exp\left(-\frac{r^2}{4\theta}\right)$$

and a modified metric

$$f(r) = -M + 2M \exp\left(-\frac{r^2}{4\theta}\right) - \Lambda r^2$$

which asymptotically reduces to the standard BTZ form as $r/\sqrt{\theta}\to\infty$ (Rahaman et al., 2013). The horizon structure is typically enriched: for sufficiently large mass $M > M_0$ (with $M_0 \sim 0.214\sqrt{\theta}$), two horizons exist; $M=M_0$ yields a degenerate (extremal) horizon; $M<M_0$ gives no horizon, indicating a regular geometry.

Thermodynamic properties are significantly altered: $$T_H = -\frac{r_h}{4\pi} \left[2\Lambda + \frac{M}{\theta}\exp\left(-\frac{r_h^2}{4\theta}\right)\right]$$ such that $T_H\to 0$ as $r_h\to r_0$; entropy remains proportional to the perimeter, $S=4\pi r_h$, but heat capacity vanishes at extremality and is otherwise positive in the physical domain.

Quantum-statistics-driven deformations, such as those based on $q$- or $(q,p)$-deformed entropies, modify the Einstein field equations directly by incorporating non-classical entropy relations. The minimum possible mass for an extremal (charged) quantum black hole can be drastically reduced relative to the classical result: $$m^{(q)} = \left(\frac{h^3 G^{3/4}}{g(z,q)}\right)^{2/7} m_\text{classical}$$ where the function $g(z,q)$ embodies the deformation (Dil et al., 2016). For BTZ black holes, analogous relations are expected, yielding a parameter-controllable interpolation between classical and quantum black hole regimes and substantially modifying the mass–charge–horizon relationship and associated thermodynamics.

2. Deformations from Modified Gravity, Massive Gravity, and Torsion

BTZ black holes are also susceptible to deformations from modifications of the gravitational action, including massive gravity, higher-derivative corrections, and theories incorporating torsion. Within massive gravity—utilizing the dRGT mechanism or Vainshtein screening—the metric function is corrected by linear-in-$r$ terms arising from the graviton potential: $$f(r) = -\Lambda r^2 - m - 2q^2 \ln(r/l) + \tilde{M}^2 c c_1 r$$ where $\tilde{M}$ is the graviton mass scale (Chougule et al., 2018). Such deformations shift Hawking temperature and entropy and are associated with violation of the reverse isoperimetric inequality (super-entropic black holes), as well as the stabilization of black holes against thermal quantum fluctuations.

In noncommutative spaces, specifically in $2+1$-dimensional gravity reformulated as a Chern–Simons gauge theory, noncommutativity can be algebraically implemented via $[r^2,\varphi]=2i\theta$ (Kawamoto et al., 2017). The Seiberg–Witten map introduces gauge-field deformations, which, after appropriate coordinate transformations, leave the metric class invariant but induce new torsion components. The resulting affine connection is non-symmetric, and only the full metric-affine (Einstein–Cartan) framework consistently describes the dynamics: torsion, encoded in $T^a_{\mu\nu}$, cannot be eliminated via coordinate transformations and can have physical implications (for example, in coupling to spinor or other fields).

3. String Theory Motivated and $T\bar{T}$-Type Deformations

String theory models yield deformations of BTZ black holes and AdS$_3$ backgrounds through exactly marginal worldsheet deformations, coherent condensates of F1–NS5 systems, and symmetric product orbifolds. Deformed BTZ black holes can emerge near stacks of NS5 branes with $p$ fundamental strings; the metric, $B$-field, and dilaton take the form

$$ds^2 = -\frac{\rho^2 - \rho_0^2}{1 + R^2/\rho^2} d\tau^2 + \frac{\rho^2}{1 + R^2/\rho^2} d\theta^2 + N^2 d\rho^2$$

with the $B$-field and dilaton appropriately chosen (Chakraborty et al., 8 Feb 2024). Single-trace $T\bar{T}$ holography emerges, with the energy spectrum of winding strings and the BTZ background itself matching precisely the predictions of $T\bar{T}$-deformed CFT$_2$ (seed central charge $c=6k$ for $k$ NS5 branes). The energy is distributed among a symmetric product of $p$ CFTs, such that each receives a $1/p$ share of the total black hole energy, and the deformation is manifest as a universal $T\bar{T}$ flow: $$E(\lambda) = \frac{1}{\lambda R} \left[ -1 + \sqrt{1 + 2\lambda R E(0) + (\lambda R P)^2 } \right]$$ This structure is robust for both BTZ-type and global AdS$_3$ backgrounds.

Conically deformed AdS$_3$ orbifolds and their Lorentzian extensions are realized via SL$(2,\mathbb{R})$ elliptic elements; their collision yields BTZ black holes and their deformed generalizations. On the CFT side, these correspond holographically to "defect operators" and their OPEs, with twisted sectors organizing the microstate structure (Martinec, 2023).

In string-effective approaches, $\alpha'$-corrected, regular BTZ-like solutions have been constructed using homogeneous, Weyl-related backgrounds of negative Gaussian curvature, with higher-order $\alpha'$ terms from the two-loop $\beta$-functions systematically included (Naderi et al., 2023). Such solutions interpolate between deformed hyperbolic (Lobachevsky-type) geometries and standard BTZ, controlled by deformation parameters and the string length.

4. Effective Metric Formalism and Physical Constraints

A model-independent framework for quantum or general deformations of BTZ black holes utilizes an "effective metric" approach (Hohenegger et al., 20 Dec 2024). The BTZ metric coefficients are replaced by functions of a physical, coordinate-invariant quantity: $$\begin{aligned} f(r) &= -M - \left(r^2 \Lambda - \frac{J^2}{4r^2}\right) \Phi(d(r)) \ g(r) &= -M - \left(r^2 \Lambda - \frac{J^2}{4r^2}\right) \Psi(d(r)) \ h(r) &= -\frac{J}{2 r^2} \Omega(d(r)) \end{aligned}$$ where $d(r) = \int_0^r \frac{dz}{\sqrt{|f(z)|}}$ is the proper distance from the origin. These functions, $\Phi$, $\Psi$, $\Omega$, are determined by the underlying model or quantum corrections but must obey the asymptotic requirement $\to 1$ as $d\to\infty$ to guarantee classical AdS recovery.

Physical constraints are imposed:

• At the horizon, regularity of Ricci and Kretschmann scalars require, e.g., the first derivatives of these functions vanish and specific algebraic relations among horizon values and higher derivatives.
• The corrected Hawking temperature follows as

$$T = \frac{1}{2\pi} \sqrt{ \frac{M^2}{2 r_H^2} \theta_H (\theta_H + \kappa) + \frac{M}{2} \frac{\Psi_H^{(2)}}{\Psi_H} }$$

with $\theta_H$ and $\kappa$ determined by horizon data of the deformation functions.

This formalism unifies diverse deformed solutions, including the quantum-corrected quBTZ black hole, within a universal coordinate-invariant parametrization.

5. Topological, Thermodynamical, and Causal Features

Deformations can induce both qualitative and quantitative shifts in topological and thermodynamic properties.

• Topologically, BTZ black holes in (2+1)D exhibit only two possible classes (based on the number and winding of vector field defects in the free energy landscape): Winding number 1 for nontwisting, uncharged BTZ; winding number zero for rotating or charged BTZ, regardless of the mechanism generating rotation or charge (Du et al., 2023). Deformations modulating the free energy can, in principle, change the distribution or number of these defects, suggesting that new classes might arise depending on the deformation.
• Thermodynamically, rainbow gravity deformations—encoding energy dependence based on quantum gravity motivated dispersion relations—cause the Hawking temperature and entropy to acquire factors dependent on the horizon radius, with evaporation halting at a finite remnant size. However, global thermodynamical potentials like the Gibbs free energy remain unaffected due to canceling corrections (Alsaleh, 2017).
• The presence of noncommutative, torsion, or higher-derivative corrections can alter the stability, phase structure, and phase transition behavior (with, e.g., violation of inequalities like the reverse isoperimetric relation; appearance of first and second order phase transitions depending on the deformation parameters).
• Causally, backdrop deformations can eliminate or preserve event horizons depending on their functional form: generic Kerr–Schild deformations destroy the nonextremal BTZ horizon, while extremal horizons can persist under certain class of deformations (Gurses et al., 2019).
• For strong cosmic censorship, deformations to the BTZ interior geometry can restore or blunt the violation of the conjecture otherwise found for near-extremal rotating BTZ solutions, by breaking the detailed coincidence of quasinormal frequencies between interior and exterior, thus altering the regularity structure of the inner Cauchy horizon (Dias et al., 2019).

6. Microstructure and Holography

Exotic and deformed BTZ black holes reveal rich microstructure from the perspective of thermodynamic geometry. While a static, uncharged BTZ exhibits ideal-gas-like non-interacting microstructures (zero Ruppeiner curvature), the introduction of angular momentum, electric charge, or "exotic" actions (involving gravitational Chern–Simons terms) admits both repulsive and attractive regimes in the microstructure interaction landscape (Ghosh et al., 2020). Such deformations may yield crossovers, but not true criticality, in the Ruppeiner geometry.

In the context of holography, deformed BTZ spacetimes constructed via bulk scalar back-reaction (dual to CFT primary operator insertions) yield "micro thermofield" geometries mapping to nonlinearly deformed thermofield initial states in the dual CFT (Bak et al., 2017). The gravitational bulk is inherently coarse-grained relative to the dual CFT: entropy grows monotonically post-quench even when the fine-grained entanglement entropy is constant, and full recovery of microscopic information requires going beyond semiclassical gravity.

Symmetric product CFT$_2$ and $T\bar{T}$ deformations encode the microstate statistical structure of deformed BTZ black holes: the black hole energy is evenly distributed among $p$ component CFTs, matching the string-theoretic and holographic constructions (Chakraborty et al., 8 Feb 2024, Martinec, 2023).

7. Deformed Hyperbolic and Higher-Dimensional Analogues

An instructive structural parallel occurs in the paper of deformed hyperbolic black holes in 4D AdS. There, the deformation parameter sculpts the horizon away from exact hyperbolic geometry, as encoded in the metric

$$ds^2 = \frac{\ell^2}{(x-y)^2} \left[ F(y) dt^2 - \frac{dy^2}{F(y)} + \frac{dx^2}{G(x)} + G(x) d\phi^2 \right]$$

with domain-structure analysis revealing a triangular region in $(x,y)$ coordinates, and horizon curvature non-constant across $x$ (Chen et al., 2015). While the BTZ solution is maximally symmetric and hyperbolic in three dimensions, its deformed four-dimensional analogues allow for non-maximal symmetry and more intricate domain structures—including "black globules"—relating to possible deformed AdS/CFT duals.

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Deformed BTZ black holes thus constitute a laboratory for exploring the interplay between modified gravity, nonlocality, quantum gravity, holography, and microstructure. Technical realization of the deformations varies—from noncommutative smearing and generalized field equations to string-theoretic constructions and effective coordinate-invariant parametrizations—but their physical implications are profound: altered horizon and thermodynamic structure, new remnant and phase transition behavior, violation or restoration of classical cosmic censorship, and novel holographic dualities. These studies elucidate the landscape of quantum-corrected, non-Einsteinian, and otherwise nontrivial (2+1)-dimensional black hole spacetimes.