Rotating BTZ-like Black Holes in Modified Gravity

• Rotating BTZ-like black holes are 2+1D AdS geometries that preserve key causal and kinematic structures while allowing deformations from quantum, matter, and higher-curvature corrections.
• They provide a benchmark for studies on null geodesics, unstable photon orbits, and precise thermodynamic behavior, including entropy quantization and fluctuation corrections.
• Modified gravity models and Lorentz-symmetry breaking introduce observable shifts in horizon structure, quasinormal mode spectra, and lensing dynamics, enriching the BTZ research program.

Searching arXiv for recent and foundational papers on rotating BTZ-like black holes to ground the article in published work. A rotating BTZ-like black hole is a $2+1$-dimensional stationary black-hole geometry that preserves the characteristic BTZ rotating structure—an AdS-like lapse sector, a nontrivial angular shift, and horizon-bearing circular symmetry—while incorporating deformations coming from modified matter content, Lorentz-symmetry breaking, running couplings, higher-curvature corrections, quantum backreaction, or analogue-gravity constructions. In the strict BTZ case, the geometry depends on mass, angular momentum, and a negative cosmological constant, and it provides the canonical rotating black hole of three-dimensional AdS gravity. In the broader “BTZ-like” sense, the term refers to rotating geometries that reduce to BTZ in an appropriate limit or retain its causal and kinematical architecture while modifying the radial sector, asymptotics, thermodynamics, or perturbative dynamics (Kala et al., 2020, Ding et al., 2023).

1. Canonical rotating BTZ geometry

The standard rotating BTZ black hole is written in the form

$$ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,$$

with

$$N^2 = \left(-M + \frac{r^2}{l^2} + \frac{J}{4r^2}\right), \qquad N_\phi = \frac{-J}{2r^2}.$$

Here $M$ is the black-hole mass, $J$ is the angular momentum, and $l$ is the AdS radius, related in the paper to the cosmological constant by

$l=-(\Lambda)^{-1/2},$

to be interpreted as the usual AdS relation for $\Lambda<0$ (Kala et al., 2020).

The horizons are located at

$$r_{\pm} = l M^{1/2} \left[\frac{1}{2}\left(1 \pm \sqrt{1-\left(\frac{J}{M l}\right)^2}\right)\right]^{1/2},$$

with the horizon-existence condition

$|J| \le Ml,$

and the extremal limit

$$ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,$$0

(Kala et al., 2020). Equivalent horizon formulas appear in other rotating BTZ treatments, including

$$ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,$$1

with $$ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,$$2 (Devi et al., 2023).

A frequently used equivalent BTZ form is

$$ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,$$3

which makes the angular-velocity function

$$ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,$$4

manifest (Devi et al., 2023). The standard thermodynamic quantities in this representation are

$$ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,$$5

together with the first law

$$ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,$$6

(Devi et al., 2023).

This canonical geometry is the baseline against which BTZ-like rotating deformations are compared. A plausible implication is that “BTZ-like” should be reserved for rotating $$ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,$$7-dimensional geometries that preserve at least the stationary circular form, an angular shift encoding frame dragging, and a horizon structure controlled by BTZ-type mass and spin parameters.

2. Null geodesics, photon orbits, and lensing structure

In the rotating BTZ background, null geodesics can be derived from conserved energy $$ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,$$8 and angular momentum $$ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,$$9. The first integrals quoted in the lensing analysis are

$$N^2 = \left(-M + \frac{r^2}{l^2} + \frac{J}{4r^2}\right), \qquad N_\phi = \frac{-J}{2r^2}.$$0

$$N^2 = \left(-M + \frac{r^2}{l^2} + \frac{J}{4r^2}\right), \qquad N_\phi = \frac{-J}{2r^2}.$$1

and

$$N^2 = \left(-M + \frac{r^2}{l^2} + \frac{J}{4r^2}\right), \qquad N_\phi = \frac{-J}{2r^2}.$$2

The radial motion factorizes as

$$N^2 = \left(-M + \frac{r^2}{l^2} + \frac{J}{4r^2}\right), \qquad N_\phi = \frac{-J}{2r^2}.$$3

with

$$N^2 = \left(-M + \frac{r^2}{l^2} + \frac{J}{4r^2}\right), \qquad N_\phi = \frac{-J}{2r^2}.$$4

(Kala et al., 2020).

The literature distinguishes direct and retrograde photon motion by the sign of $$N^2 = \left(-M + \frac{r^2}{l^2} + \frac{J}{4r^2}\right), \qquad N_\phi = \frac{-J}{2r^2}.$$5:

• direct motion: $$N^2 = \left(-M + \frac{r^2}{l^2} + \frac{J}{4r^2}\right), \qquad N_\phi = \frac{-J}{2r^2}.$$6,
• retrograde motion: $$N^2 = \left(-M + \frac{r^2}{l^2} + \frac{J}{4r^2}\right), \qquad N_\phi = \frac{-J}{2r^2}.$$7

(Kala et al., 2020). The turning point of a scattering trajectory is

$$N^2 = \left(-M + \frac{r^2}{l^2} + \frac{J}{4r^2}\right), \qquad N_\phi = \frac{-J}{2r^2}.$$8

The orbit equation is then expressed in terms of $$N^2 = \left(-M + \frac{r^2}{l^2} + \frac{J}{4r^2}\right), \qquad N_\phi = \frac{-J}{2r^2}.$$9 and decomposed into a form involving $M$0, $M$1, $M$2, $M$3, $M$4, and $M$5 (Kala et al., 2020).

A central result of this analysis is that the effective potential has no minimum, hence there is no stable photon orbit near the rotating BTZ horizon; only unstable null orbits occur (Kala et al., 2020). In the extremal case, the effective potential analysis yields the critical impact parameter

$M$6

with

$M$7

and the photon orbit is again interpreted as unstable (Kala et al., 2020).

The exact deflection angle is given by

$M$8

which can be integrated to a closed expression involving inverse hyperbolic functions and then rewritten logarithmically (Kala et al., 2020). The principal lensing trend is that the bending angle increases as the closest approach decreases, for both direct and retrograde trajectories, and that it “slightly decreases as the value of cosmological constant increases in the negative region” (Kala et al., 2020).

These features are specific to the BTZ lensing problem in $M$9-dimensional AdS gravity, but they supply a reference pattern for rotating BTZ-like geometries: frame-dragging splits direct and retrograde branches, unstable critical null orbits replace a stable photon sphere, and the AdS scale enters explicitly in both orbit structure and light bending.

3. Thermodynamics, quantization, and fluctuation corrections

The rotating BTZ black hole supports a standard thermodynamic description. In one formulation,

$J$0

(Devi et al., 2023). In another convention,

$J$1

with thermodynamic potentials

$J$2

(Upadhyay et al., 2019).

An independent line of work quantizes the horizon area via the adiabatic invariant and Bohr–Sommerfeld quantization. For the rotating BTZ metric

$J$3

with

$J$4

the action variable reduces to the entropy,

$J$5

Bohr–Sommerfeld quantization then yields an equally spaced entropy spectrum and hence an equally spaced area spectrum, with spacing independent of the rotation parameter (Liu et al., 2012). The horizon circumference is

$J$6

and the entropy is

$J$7

(Liu et al., 2012). The paper states that the area spectrum of the rotating BTZ black hole is equally spaced and irrelevant to the rotating parameter (Liu et al., 2012).

Thermal and statistical fluctuations provide a different modification. One corrected entropy formula is

$J$8

which for rotating BTZ becomes

$J$9

(Upadhyay et al., 2019). Another treatment adopts

$l$0

with

$l$1

for the rotating BTZ black hole (Pourhassan et al., 2017). In that framework the specific heat becomes

$l$2

and the paper concludes that higher-order quantum corrections affect stability, especially for small black holes (Pourhassan et al., 2017).

The thermodynamic literature therefore divides into three distinct BTZ-related themes: exact classical thermodynamics, semiclassical area quantization, and fluctuation-induced corrections. This suggests that “BTZ-like” should not be read as a purely geometric label; in current usage it often also implies a thermodynamic comparison class.

4. Rotating BTZ-like deformations in modified gravity and matter-coupled theories

Several exact and effective rotating BTZ-like families are now known. One prominent example is the rotating Einstein-bumblebee solution

$l$3

with

$l$4

Here $l$5 is the Lorentz-violating parameter induced by a radial bumblebee vacuum expectation value, and the solution exists only for a linear functional potential of the bumblebee field (Ding et al., 2023). The horizon radii are

$l$6

the ergosphere radius is

$l$7

and the thermodynamic quantities require modified area and volume,

$l$8

(Ding et al., 2023). The entropy product

$l$9

is universal, and the dual CFT central charges satisfy

$l=-(\Lambda)^{-1/2},$0

(Ding et al., 2023).

A closely related Einstein-bumblebee family is written as

$l=-(\Lambda)^{-1/2},$1

with Lorentz-symmetry-breaking parameter $l=-(\Lambda)^{-1/2},$2 (Quan et al., 27 Jan 2025). In this version the horizons

$l=-(\Lambda)^{-1/2},$3

the ergoregion radius

$l=-(\Lambda)^{-1/2},$4

and the horizon angular velocity

$l=-(\Lambda)^{-1/2},$5

remain BTZ-like, while the radial sector and scalar dynamics are deformed (Quan et al., 27 Jan 2025).

Einstein–Maxwell–Dilaton theory yields another exact rotating BTZ-like family. Starting from

$l=-(\Lambda)^{-1/2},$6

the solution takes

$l=-(\Lambda)^{-1/2},$7

and

$l=-(\Lambda)^{-1/2},$8

(Karakasis et al., 2022). The $l=-(\Lambda)^{-1/2},$9 limit recovers the charged rotating BTZ geometry, whereas for $\Lambda<0$0 the asymptotics cease to be AdS and the entropy becomes

$\Lambda<0$1

which is always positive (Karakasis et al., 2022).

Scale-dependent gravity supplies a different class of BTZ-like deformation. Preserving the BTZ rotating ansatz,

$\Lambda<0$2

one obtains

$\Lambda<0$3

$\Lambda<0$4

with

$\Lambda<0$5

(Rincon et al., 2018). The running coupling produces a curvature singularity at $\Lambda<0$6, since

$\Lambda<0$7

(Rincon et al., 2018).

Higher-curvature deformations need not preserve the full BTZ causal pattern. In three-dimensional Gauss–Bonnet gravity, the rotating solution

$\Lambda<0$8

with

$\Lambda<0$9

is BTZ-like but not of constant curvature (Hennigar et al., 2020). It possesses an ergoregion and outer horizon but does not have an inner horizon (Hennigar et al., 2020). This is a direct counterexample to the common misconception that every rotating BTZ-like geometry necessarily retains the BTZ pair of horizons.

5. Quantum, semiclassical, and perturbative rotating BTZ-like black holes

Quantum or semiclassical corrections generate further rotating BTZ-like spacetimes. The rotating quantum BTZ black hole, or qBTZ black hole, is a geometry that captures the exact backreaction of strongly coupled quantum conformal fields (Frassino et al., 2024). In the brane-induced description,

$$r_{\pm} = l M^{1/2} \left[\frac{1}{2}\left(1 \pm \sqrt{1-\left(\frac{J}{M l}\right)^2}\right)\right]^{1/2},$$0

with horizon condition

$$r_{\pm} = l M^{1/2} \left[\frac{1}{2}\left(1 \pm \sqrt{1-\left(\frac{J}{M l}\right)^2}\right)\right]^{1/2},$$1

or equivalently

$$r_{\pm} = l M^{1/2} \left[\frac{1}{2}\left(1 \pm \sqrt{1-\left(\frac{J}{M l}\right)^2}\right)\right]^{1/2},$$2

(Frassino et al., 2024). The extremality condition is

$$r_{\pm} = l M^{1/2} \left[\frac{1}{2}\left(1 \pm \sqrt{1-\left(\frac{J}{M l}\right)^2}\right)\right]^{1/2},$$3

and the paper shows that an extremal rotating qBTZ black hole cannot be overspun by test-particle capture: for a particle with maximal allowed angular momentum, the first-order change satisfies

$$r_{\pm} = l M^{1/2} \left[\frac{1}{2}\left(1 \pm \sqrt{1-\left(\frac{J}{M l}\right)^2}\right)\right]^{1/2},$$4

(Frassino et al., 2024).

Metric-affine modified gravity produces a slowly rotating charged BTZ-like family in Palatini Chern–Simons gravity. Starting from the charged non-rotating BTZ background

$$r_{\pm} = l M^{1/2} \left[\frac{1}{2}\left(1 \pm \sqrt{1-\left(\frac{J}{M l}\right)^2}\right)\right]^{1/2},$$5

rotation is introduced perturbatively as

$$r_{\pm} = l M^{1/2} \left[\frac{1}{2}\left(1 \pm \sqrt{1-\left(\frac{J}{M l}\right)^2}\right)\right]^{1/2},$$6

together with an induced magnetic field (Bombacigno et al., 3 Mar 2026). The asymptotic consistency condition

$$r_{\pm} = l M^{1/2} \left[\frac{1}{2}\left(1 \pm \sqrt{1-\left(\frac{J}{M l}\right)^2}\right)\right]^{1/2},$$7

ensures that

$$r_{\pm} = l M^{1/2} \left[\frac{1}{2}\left(1 \pm \sqrt{1-\left(\frac{J}{M l}\right)^2}\right)\right]^{1/2},$$8

so the geometry decays to BTZ-like form at large radius, while the horizon angular momentum and magnetic field remain finite (Bombacigno et al., 3 Mar 2026).

A different analytic continuation leads to the double Wick rotated rotating BTZ black hole, treated as a quotient geometry. The Euclidean rotating BTZ metric

$$r_{\pm} = l M^{1/2} \left[\frac{1}{2}\left(1 \pm \sqrt{1-\left(\frac{J}{M l}\right)^2}\right)\right]^{1/2},$$9

is double Wick rotated into a geometry that becomes equivalent to an ordinary rotating BTZ black hole after exchanging parameters and periodicities (Dai et al., 17 Apr 2026). The paper states

$|J| \le Ml,$0

and uses this identification to reproduce geometric entropy and time-like entanglement entropy (Dai et al., 17 Apr 2026).

These examples indicate that the modern BTZ-like literature spans exact quantum backreaction, parity-violating first-order deformations, and analytic continuations of the BTZ quotient structure. A plausible implication is that “BTZ-like” now functions as a family resemblance term rather than a single metric class.

6. Perturbations, quasinormal modes, clouds, and analogue realizations

Rotating BTZ-like black holes are a major testing ground for perturbation theory. In Einstein-bumblebee gravity, the rotating BTZ-like metric

$|J| \le Ml,$1

has horizon radii

$|J| \le Ml,$2

that do not depend on the Lorentz-breaking parameter $|J| \le Ml,$3 (Chen et al., 2023). Scalar perturbations yield exact right- and left-moving frequencies,

$|J| \le Ml,$4

$|J| \le Ml,$5

and the Lorentz-breaking parameter affects only the imaginary parts, not the real parts (Chen et al., 2023).

A later study extends this Einstein-bumblebee analysis to scalar, fermionic, and vector perturbations. In the coordinate system

$|J| \le Ml,$6

the exact spectra take the CFT form

$|J| \le Ml,$7

with

$|J| \le Ml,$8

(Quan et al., 26 Mar 2026). The paper emphasizes that the real parts remain exactly BTZ-like, while the Lorentz-symmetry-breaking parameter modifies the damping rates and conformal weights (Quan et al., 26 Mar 2026).

Stationary scalar clouds provide another diagnostic. For the Einstein-bumblebee metric

$|J| \le Ml,$9

the scalar ansatz

$$ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,$$00

reduces to a hypergeometric problem after

$$ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,$$01

(Quan et al., 27 Jan 2025). Under Robin boundary conditions

$$ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,$$02

stationary clouds occur at the synchronization threshold

$$ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,$$03

(Quan et al., 27 Jan 2025). The paper finds only fundamental stationary scalar clouds, $$ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,$$04, and reports degenerate cloud-existence lines for different $$ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,$$05 values (Quan et al., 27 Jan 2025).

The Kalb–Ramond BTZ-like black hole changes the perturbation problem qualitatively. With

$$ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,$$06

the scalar radial equation reduces to a general Heun equation after

$$ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,$$07

(Xia et al., 2 Nov 2025). Robin boundary conditions are imposed via

$$ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,$$08

and the quasinormal frequencies are determined numerically by the Wronskian condition

$$ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,$$09

(Xia et al., 2 Nov 2025). The paper states that only the fundamental left-branch mode becomes unstable and that the Kalb–Ramond parameter shifts the threshold and the range of Robin coupling where superradiance occurs (Xia et al., 2 Nov 2025).

Beyond gravitational theories, rotating BTZ-like structure has even been realized in analogue gravity. In a photon-fluid model, the acoustic metric

$$ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,$$10

is matched, up to a conformal factor, to the rotating BTZ geometry by choosing

$$ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,$$11

(Chen et al., 2023). A realizable optical vortex ansatz produces a radial flow that crosses the sound speed twice, thereby reproducing both inner and outer horizons, a defining BTZ feature absent from earlier optical rotating analogues (Chen et al., 2023).

Taken together, these studies show that rotating BTZ-like black holes are now used as laboratories for exact QNMs, AdS boundary-condition effects, superradiance, stationary clouds, and even experimental analogue horizons. This suggests that the phrase names not merely a geometry but a research program centered on $$ds^2 = -(N^2- r^2 N_\phi^2)dt^2 + \frac{1}{N^2} dr^2 + r^2d\phi^2 + 2r^2N_\phi dt d\phi,$$12-dimensional rotating AdS black-hole kinematics under controlled deformations.